View source for Vector space

{{short description|Basic algebraic structure of linear algebra}}
{{Distinguish|Vector field}}
{{about|linear (vector) spaces|the structure in incidence geometry|Linear space (geometry)|the space technology company|Vector Space Systems}}
[[File:Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector {{math|'''v'''}} (blue) is added to another vector {{math|'''w'''}} (red, upper illustration). Below, '''w''' is stretched by a factor of 2, yielding the sum {{math|'''v''' + 2'''w'''}}.]]

A '''vector space''' (also called a '''linear space''') is a collection of objects called '''vectors''', which may be [[Vector addition|added]] together and [[Scalar multiplication|multiplied]] ("scaled") by numbers, called ''[[scalar (mathematics)|scalars]]''. Scalars are often taken to be [[real number]]s, but there are also vector spaces with scalar multiplication by [[complex number]]s, [[rational number]]s, or generally any [[field (mathematics)|field]]. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''[[axiom]]s'', listed below, in {{slink||Definition}}. For specifying that the scalars are real or complex numbers, the terms '''real vector space''' and '''complex vector space''' are often used.

[[Euclidean vector]]s are an example of a vector space.  They represent [[physics|physical]] quantities such as [[force]]s: any two forces (of the same type) can be added to yield a third, and the multiplication of a [[force vector]] by a real multiplier is another force vector. In the same vein, but in a more [[geometry|geometric]] sense, vectors representing displacements in the plane or in [[three-dimensional space]] also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.

Vector spaces are the subject of [[linear algebra]] and are well characterized by their [[dimension (vector space)|dimension]], which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in [[mathematical analysis]], as [[function space]]s, whose vectors are [[function (mathematics)|functions]]. These vector spaces are generally endowed with additional structure, which may be a [[topology]], allowing the consideration of issues of proximity and [[continuous function|continuity]]. Among these topologies, those that are defined by a [[norm (mathematics)|norm]] or [[inner product]] are more commonly used, as having a notion of [[metric (mathematics)|distance]] between two vectors. This is particularly the case of [[Banach space]]s and [[Hilbert space]]s, which are fundamental in mathematical analysis.

Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's [[analytic geometry]], [[matrix (mathematics)|matrices]], systems of [[linear equation]]s, and Euclidean vectors. The modern, more abstract treatment, first formulated by [[Giuseppe Peano]] in 1888, encompasses more general objects than [[Euclidean space]], but much of the theory can be seen as an extension of classical geometric ideas like [[line (geometry)|line]]s, [[plane (geometry)|plane]]s and their higher-dimensional analogs.

Today, vector spaces are applied throughout [[mathematics]], [[science]] and [[engineering]]. They are the appropriate linear-algebraic notion to deal with [[system of linear equations|systems of linear equations]]. They offer a framework for [[Fourier series|Fourier expansion]], which is employed in [[image compression]] routines, and they provide an environment that can be used for solution techniques for [[partial differential equation]]s. Furthermore, vector spaces furnish an abstract, [[coordinate-free]] way of dealing with geometrical and physical objects such as [[tensor]]s. This in turn allows the examination of local properties of [[manifold (mathematics)|manifolds]] by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and [[abstract algebra]].

{{Algebraic structures |Module}}

== Introduction and definition ==
The concept of vector space will first be explained by describing two particular examples:

===First example: arrows in the plane===
The first example of a vector space consists of [[arrow (symbol)|arrow]]s in a fixed [[plane (geometry)|plane]], starting at one fixed point. This is used in physics to describe [[force]]s or [[velocity|velocities]]. Given any two such arrows, {{math|'''v'''}} and {{math|'''w'''}}, the [[parallelogram]] spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the ''sum'' of the two arrows and is denoted {{math|'''v''' + '''w'''}}. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive [[real number]] {{math|''a''}}, the arrow that has the same direction as {{math|'''v'''}}, but is dilated or shrunk by multiplying its length by {{math|''a''}}, is called ''multiplication'' of {{math|'''v'''}} by {{math|''a''}}. It is denoted {{math|''a'''''v'''}}. When {{math|''a''}} is negative, {{math|''a'''''v'''}} is defined as the arrow pointing in the opposite direction, instead.

The following shows a few examples: if {{math|1=''a'' = 2}}, the resulting vector {{math|''a'''''w'''}} has the same direction as {{math|'''w'''}}, but is stretched to the double length of {{math|'''w'''}} (right image below). Equivalently, {{math|2'''w'''}} is the sum {{math|'''w''' + '''w'''}}. Moreover, {{math|1=(−1)'''v''' = −'''v'''}} has the opposite direction and the same length as {{math|'''v'''}} (blue vector pointing down in the right image).

{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|-
|width=50%|[[File:Vector addition3.svg|180px|Vector addition: the sum {{math|'''v''' + '''w'''}} (black) of the vectors {{math|'''v'''}} (blue) and {{math|'''w'''}} (red) is shown.]]
|width=50%|[[File:Scalar multiplication.svg|230px|Scalar multiplication: the multiples {{math|−'''v'''}} and {{math|2'''w'''}} are shown.]]
|}

===Second example: ordered pairs of numbers===
A second key example of a vector space is provided by pairs of real numbers {{math|''x''}} and {{math|''y''}}. (The order of the components {{math|''x''}} and {{math|''y''}} is significant, so such a pair is also called an [[ordered pair]].) Such a pair is written as {{math|(''x'', ''y'')}}. The sum of two such pairs and multiplication of a pair with a number is defined as follows:
:{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>)}} + {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>)}} {{math|1== (''x''<sub>1</sub> + ''x''<sub>2</sub>, ''y''<sub>1</sub> + ''y''<sub>2</sub>)}}
and 
:{{math|1=''a''&thinsp;(''x'', ''y'') = (''ax'', ''ay'')}}.

The first example above reduces to this one if the arrows are represented by the pair of [[Cartesian coordinates]] of their end points.

===Definition===
In this article, vectors are represented in boldface to distinguish them from scalars.<ref group=nb>It is also common, especially in physics, to denote vectors with an arrow on top: {{math|{{vec|''v''}}}}.</ref>

A vector space over a [[field (mathematics)|field]] {{mvar|F}} is a [[set (mathematics)|set]]&nbsp;{{mvar|V}} together with two operations that satisfy the eight axioms listed below. In the following, {{math| ''V'' × ''V''}} denotes the [[Cartesian product]] of {{math| ''V''}} with itself, and {{math| →}} denotes a [[Function (mathematics)#Map|mapping]] from one set to another.

* The first operation, called '''vector addition''' or simply '''addition''' {{math| + : ''V'' × ''V'' → ''V''}}, takes any two vectors&nbsp;{{math|'''v'''}} and {{math|'''w'''}} and assigns to them a third vector which is commonly written as {{math|'''v''' + '''w'''}}, and called the sum of these two vectors. (The resultant vector is also an element of the set {{mvar|V}}.)
* The second operation, called '''[[scalar multiplication]]''' {{math| · : ''F'' × ''V'' → ''V''}}， takes any scalar&nbsp;{{mvar|a}} and any vector&nbsp;{{math|'''v'''}} and gives another vector&nbsp;{{math|''a'''''v'''}}. (Similarly, the vector {{math|''a'''''v'''}} is an element of the set {{mvar|V}}. Scalar multiplication is not to be confused with the [[scalar product]], also called ''inner product'' or ''dot product'', which is an additional structure present on some specific, but not all vector spaces. Scalar multiplication is a multiplication of a vector ''by'' a scalar; the other is a multiplication of two vectors ''producing'' a scalar.)

Elements of {{mvar|V}} are commonly called ''vectors''. Elements of&nbsp;{{mvar|F}} are commonly called ''scalars''.

In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.

To qualify as a vector space, the set&nbsp;{{mvar|V}} and the operations of addition and multiplication must adhere to a number of requirements called [[axiom]]s.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 1, p. 27}}</ref> In the list below, let {{math|'''u'''}}, {{math|'''v'''}} and {{math|'''w'''}} be arbitrary vectors in {{mvar|V}}, and {{mvar|a}} and {{mvar|b}} scalars in {{mvar|F}}.

{| border="0" style="width:100%;" class="wikitable"
|-
! Axiom || Meaning
|-
| [[Associativity]] of addition || {{math|1='''u''' + ('''v''' + '''w''') = ('''u''' + '''v''') + '''w'''}}
|- style="background:#F8F4FF;"
| [[Commutativity]] of addition || {{math|1='''u''' + '''v''' = '''v''' + '''u'''}}
|-
| [[Identity element]] of addition || There exists an element {{math|'''0''' ∈ ''V''}}, called the ''[[zero vector]]'', such that {{math|1='''v''' + '''0''' = '''v'''}} for all {{math|'''v''' ∈ ''V''}}.
|- style="background:#F8F4FF;"
| [[Inverse element]]s of addition || For every {{math|'''v''' ∈ ''V''}}, there exists an element {{math|−'''v''' ∈ ''V''}}, called the ''[[additive inverse]]'' of {{math|'''v'''}}, such that {{math|1='''v''' + (−'''v''') = '''0'''}}.
|-
| [[Semigroup action|Compatibility]] of scalar multiplication with field multiplication || {{math|1=''a''(''b'''''v''') = (''ab'')'''v'''}} <ref group=nb>This axiom and the next refer to two different operations: scalar multiplication: {{math|''b'''''v'''}}; and field multiplication: {{math|''ab''}}. They do not assert the associativity of either operation. More formally, scalar multiplication is a [[monoid action]] of the multiplicative monoid of the field {{mvar|F}} on the vector space {{mvar|V}}.</ref>
|- style="background:#F8F4FF;"
| Identity element of scalar multiplication ||  {{math|1=1'''v''' = '''v'''}}, where {{math|1}} denotes the [[multiplicative identity]] in {{mvar|F}}.
|-
| [[Distributivity]] of scalar multiplication with respect to vector addition&emsp;&emsp;|| {{math|1=''a''('''u''' + '''v''') = ''a'''''u''' + ''a'''''v'''}}
|- style="background:#F8F4FF;"
| Distributivity of scalar multiplication with respect to field addition || {{math|1=(''a'' + ''b'')'''v''' = ''a'''''v''' + ''b'''''v'''}}
|}

These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands:
:{{math|1=(''x''<sub>'''v'''</sub>, ''y''<sub>'''v'''</sub>) + (''x''<sub>'''w'''</sub>, ''y''<sub>'''w'''</sub>) = (''x''<sub>'''w'''</sub>, ''y''<sub>'''w'''</sub>) + (''x''<sub>'''v'''</sub>, ''y''<sub>'''v'''</sub>)}}.

Likewise, in the geometric example of vectors as arrows, {{math|1='''v''' + '''w''' = '''w''' + '''v'''}} since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space.

Subtraction of two vectors and division by a (non-zero) scalar can be defined as
: <math>\begin{align}
  \mathbf{v} - \mathbf{w} &= \mathbf{v} + (-\mathbf{w}) \\
    \frac{\mathbf{v}}{a} &= \frac{1}{a}\mathbf{v}
\end{align}</math>.

When the scalar field {{mvar|F}} is the [[real number]]s {{math|'''R'''}}, the vector space is called a ''real vector space''. When the scalar field is the [[complex number]]s {{math|'''C'''}}, the vector space is called a ''complex vector space''. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed [[field (mathematics)|field]] {{mvar|F}}. The notion is then known as an {{mvar|F}}-''vector spaces'' or a ''vector space over {{mvar|F}}''. A field is, essentially, a set of numbers possessing [[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]] operations.<ref group=nb>Some authors (such as {{Harvard citations|last = Brown|year = 1991|nb=yes}}) restrict attention to the fields {{math|'''R'''}} or {{math|'''C'''}}, but most of the theory is unchanged for an arbitrary field.</ref> For example, [[rational number]]s form a field.

In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of [[neighborhood (topology)|nearness]], [[angle]]s or [[distance]]s. To deal with such matters, particular types of vector spaces are introduced; see [[#Vector spaces with additional structure|below]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. -->

===Alternative formulations and elementary consequences===
Vector addition and scalar multiplication are operations, satisfying the [[closure (mathematics)|closure]] property:  {{math|'''u''' + '''v'''}} and {{math|''a'''''v'''}} are in {{math|''V''}} for all {{math|''a''}} in {{math|''F''}}, and {{math|'''u'''}}, {{math|'''v'''}} in {{math|''V''}}. Some older sources mention these properties as separate axioms.<ref>{{Harvard citations|last = van der Waerden|year = 1993|nb = yes|loc=Ch. 19}}</ref>

In the parlance of [[abstract algebra]], the first four axioms are equivalent to requiring the set of vectors to be an [[abelian group]] under addition. The remaining axioms give this group an {{math|''F''}}-[[Module (mathematics)|module]] structure. In other words, there is a [[ring homomorphism]] {{math|''f''}} from the field {{math|''F''}} into the [[endomorphism ring]] of the group of vectors. Then scalar multiplication {{math|''a'''''v'''}} is defined as {{math|(''f''(''a''))('''v''')}}.<ref>{{Harvard citations|last=Bourbaki|year=1998|nb=yes|loc=§II.1.1}}. Bourbaki calls the group homomorphisms {{math|''f''(''a'')}} ''homotheties''.</ref>

There are a number of direct consequences of the vector space axioms. Some of them derive from [[elementary group theory]], applied to the additive group of vectors: for example, the zero vector {{math|'''0'''}} of {{math|''V''}} and the additive inverse {{math|−'''v'''}} of any vector {{math|'''v'''}} are unique. Further properties follow by employing also the distributive law for the scalar multiplication, for example {{math|''a'''''v'''}} equals {{math|'''0'''}} if and only if {{math|''a''}} equals {{math|0}} or {{math|'''v'''}} equals {{math|'''0'''}}.

==History==
{{See|History of algebra}}
Vector spaces stem from [[affine geometry]], via the introduction of [[coordinate]]s in the plane or three-dimensional space. Around 1636, French mathematicians [[René Descartes]] and [[Pierre de Fermat]] founded [[analytic geometry]] by identifying solutions to an equation of two variables with points on a plane [[curve]].<ref>{{Harvard citations|last = Bourbaki|year = 1969|nb = yes|loc = ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91}}.</ref> To achieve geometric solutions without using coordinates, [[Bernhard Bolzano|Bolzano]] introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors.<ref>{{Harvard citations|last = Bolzano|year = 1804|nb = yes}}.</ref> This work was made use of in the conception of [[Barycentric coordinates (mathematics)|barycentric coordinates]] by [[August Ferdinand Möbius|Möbius]] in 1827.<ref>{{Harvard citations|last = Möbius|year = 1827|nb = yes}}.</ref> The foundation of the definition of vectors was [[Giusto Bellavitis|Bellavitis]]' notion of the bipoint, an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of [[complex numbers]] by [[Jean-Robert Argand|Argand]] and [[William Rowan Hamilton|Hamilton]] and the inception of [[quaternion]]s by the latter.<ref>{{Harvard citations|last = Hamilton|year = 1853|nb = yes}}.</ref> They are elements in '''R'''<sup>2</sup> and '''R'''<sup>4</sup>; treating them using [[linear combination]]s goes back to [[Laguerre]] in 1867, who also defined [[system of linear equations|systems of linear equations]].

In 1857, [[Arthur Cayley|Cayley]] introduced the [[matrix notation]] which allows for a harmonization and simplification of [[linear map]]s. Around the same time, [[Grassmann]] studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.<ref>{{Harvard citations|last = Grassmann|year = 2000|nb = yes}}.</ref> In his work, the concepts of [[linear independence]] and [[dimension]], as well as [[scalar product]]s are present. Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called [[algebra]]s. Italian mathematician [[Guiseppe Peano|Peano]] was the first to give the modern definition of vector spaces and linear maps in 1888.<ref>{{Harvard citations|last = Peano|year = 1888|nb = yes |loc = ch. IX}}.</ref>

An important development of vector spaces is due to the construction of [[function spaces]] by [[Henri Lebesgue]]. This was later formalized by [[Stefan Banach|Banach]] and [[David Hilbert|Hilbert]], around 1920.<ref>{{Harvard citations|last = Banach|year = 1922|nb = yes}}.</ref> At that time, [[algebra]] and the new field of [[functional analysis]] began to interact, notably with key concepts such as [[Lp space|spaces of ''p''-integrable functions]] and [[Hilbert space]]s.<ref>{{Harvard citations|last = Dorier|year = 1995|nb = yes}}, {{Harvard citations|last = Moore|year = 1995|nb = yes}}.</ref> Also at this time, the first studies concerning infinite-dimensional vector spaces were done.

== Examples ==
{{main|Examples of vector spaces}}

===Coordinate spaces===
{{Main|Coordinate space}}
The simplest example of a vector space over a field {{math|''F''}} is the field itself, equipped with its standard addition and multiplication. More generally, all [[tuple|{{math|''n''}}-tuples]] (sequences of length {{math|''n''}}) 
:{{math|(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>)}}
of elements of {{math|''F''}} form a vector space that is  usually denoted {{math|''F''<sup>''n''</sup>}} and called a ''[[coordinate space]]''.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. I.1|nb = yes}}</ref> 
The case {{math|1=''n'' = 1}} is the above-mentioned simplest example, in which the field {{math|''F''}} is also regarded as a vector space over itself. The case {{math|1=''F'' = '''R'''}} and {{math|1=''n'' = 2}} was discussed in the introduction above.

===Complex numbers and other field extensions===
The set of [[complex numbers]] {{math|'''C'''}}, that is, numbers that can be written in the form {{math|1=''x'' + ''iy''}} for [[real numbers]] {{math|''x''}} and {{math|''y''}} where {{math|''i''}} is the [[imaginary unit]], form a vector space over the reals with the usual addition and multiplication: {{math|1=(''x'' + ''iy'') + (''a'' + ''ib'') = (''x'' + ''a'') + ''i''(''y'' + ''b'')}} and {{math|1=''c'' ⋅ (''x'' + ''iy'') = (''c'' ⋅ ''x'') + ''i''(''c'' ⋅ ''y'')}} for real numbers {{math|''x''}}, {{math|''y''}}, {{math|''a''}}, {{math|''b''}} and {{math|''c''}}.  The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic.

In fact, the example of complex numbers is essentially the same (that is, it is ''isomorphic'') to the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number {{math|''x'' + ''i'' ''y''}} as representing the ordered pair {{math|(''x'', ''y'')}} in the [[complex plane]] then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.

More generally, [[field extension]]s provide another class of examples of vector spaces, particularly in algebra and [[algebraic number theory]]: a field {{math|''F''}} containing a [[Field extension|smaller field]] {{math|''E''}} is an {{math|''E''}}-vector space, by the given multiplication and addition operations of {{math|''F''}}.<ref>{{Harvard citations|last = Lang|year =2002|loc = ch. V.1|nb = yes}}</ref> For example, the complex numbers are a vector space over {{math|'''R'''}}, and the field extension <math>\mathbf{Q}(i\sqrt{5})</math> is a vector space over {{math|'''Q'''}}.  <!--A particularly interesting type of field extension in [[number theory]] is {{math|'''Q'''(''α'')}}, the extension of the rational numbers {{math|'''Q'''}} by a fixed complex number {{math|''α''}}. {{math|'''Q'''(''α'')}} is the smallest field containing the rationals and a fixed complex number ''α''. Its dimension as a vector space over {{math|'''Q'''}} depends on the choice of {{math|''α''}}.-->

===Function spaces===
{{Main|Function space}}
[[File:Example for addition of functions.svg|thumb|Addition of functions: The sum of the sine and the exponential function is <math>\sin+\exp:\R\to\R</math> with <math>(\sin+\exp)(x)=\sin(x)+\exp(x)</math>]]

Functions from any fixed set {{math|Ω}} to a field {{math|''F''}} also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions {{math|''f''}} and {{math|''g''}} is the function {{math|(''f'' + ''g'')}} given by
:{{math|1=(''f'' + ''g'')(''w'') = ''f''(''w'') + ''g''(''w'')}},
and similarly for multiplication. Such function spaces occur in many geometric situations, when {{math|Ω}} is the [[real line]] or an [[interval (mathematics)|interval]], or other [[subset]]s of {{math|'''R'''}}. Many notions in topology and analysis, such as [[continuous function|continuity]], [[integral|integrability]] or [[differentiability]] are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.<ref>{{Harvard citations|last = Lang|year = 1993|loc = ch. XII.3., p. 335|nb = yes}}</ref>  Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of [[functional analysis]], see [[#labelFunctionalAnalysis|below]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> Algebraic constraints also yield vector spaces: the <span id=labelPolynomialRing>[[polynomial ring|vector space {{math|''F''[x]}}]]</span> is given by [[polynomial function]]s:
:{{math|1=''f''(''x'') = ''r''<sub>0</sub> + ''r''<sub>1</sub>''x'' + ... + ''r''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ''r''<sub>''n''</sub>''x''<sup>''n''</sup>}}, where the [[coefficient]]s {{math|''r''<sub>0</sub>, ..., ''r''<sub>''n''</sub>}} are in {{math|''F''}}.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. IX.1|nb = yes}}</ref>

===Linear equations===
{{Main|Linear equation|Linear differential equation|Systems of linear equations}}
Systems of [[homogeneous linear equation]]s are closely tied to vector spaces.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. VI.3.|nb = yes}}</ref> For example, the solutions of 
:{|
|-
| style="text-align:right;"|{{math|''a''}}
|{{math|+}}
|{{math|3''b''}}
|{{math|+}}
| style="text-align:right;"|{{math|''c''}}
|{{math|{{=}} 0}}
|-
|{{math|4''a''}}
|{{math|+}}
|{{math|2''b''}}
|{{math|+}}
|{{math|2''c''}}
|{{math|{{=}} 0}}
|}
are given by triples with arbitrary {{math|''a''}}, {{math|1=''b'' = ''a''/2}}, and {{math|1=''c'' = −5''a''/2}}. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. [[matrix (mathematics)|Matrices]] can be used to condense multiple linear equations as above into one vector equation, namely
:<span id=equation3>{{math|1=''A'''''x''' = '''0'''}}</span>,
where {{math|1=''A'' =}} <math>\begin{bmatrix}
1 & 3 & 1 \\
4 & 2 & 2\end{bmatrix}</math> is the matrix containing the coefficients of the given equations, {{math|'''x'''}} is the vector {{math|(''a'', ''b'', ''c'')}}, {{math|''A'''''x'''}} denotes the [[matrix product]], and {{math|1='''0''' = (0, 0)}} is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example,
:<span id=equation1>{{math|1=''f''′′(''x'') + 2''f''′(''x'') + ''f''(''x'') = 0}}</span>
yields {{math|1=''f''(''x'') = ''a&thinsp;e''<sup>−''x''</sup> + ''bx&thinsp;e''<sup>−''x''</sup>}}, where {{math|''a''}} and {{math|''b''}} are arbitrary constants, and {{math|''e''<sup>''x''</sup>}} is the [[natural exponential function]].

== Basis and dimension ==
{{Main|basis (linear algebra)|l1=Basis|dimension (linear algebra)|l2=Dimension}}
[[File:Vector components and base change.svg|A vector {{math|'''v'''}} in {{math|'''R'''<sup>2</sup>}} (blue) expressed in terms of different bases: using the [[standard basis]] of {{math|1='''R'''<sup>2</sup> '''v''' = ''x'''''e'''<sub>1</sub> + ''y'''''e'''<sub>2</sub>}} (black), and using a different, non-[[orthogonal vector|orthogonal]] basis: {{math|1='''v''' = '''f'''<sub>1</sub> + '''f'''<sub>2</sub>}} (red).|thumb|200px]]
<span id=label1>''Bases''</span> allow one to represent vectors by a [[sequence]] of scalars called ''coordinates'' or ''components''. A basis is a (finite or infinite) set {{math|1=''B'' = {'''b'''<sub>''i''</sub>}<sub>''i'' ∈ ''I''</sub>}} of vectors {{math|'''b'''<sub>''i''</sub>}}, for convenience often indexed by some [[index set]] {{math|''I''}}, that spans the whole space and is [[linearly independent]]. "Spanning the whole space" means that any vector {{math|'''v'''}} can be expressed as a finite sum (called a ''[[linear combination]]'') of the basis elements:
{{NumBlk|:|<math>\mathbf{v} = a_1 \mathbf{b}_{i_1} + a_2 \mathbf{b}_{i_2} + \cdots + a_n \mathbf{b}_{i_n},</math>|{{EquationNote|1}}}}
where the {{math|''a''<sub>''k''</sub>}} are scalars, called the coordinates (or the components) of the vector {{math|'''v'''}} with respect to the basis {{math|''B''}}, and {{math|'''b'''<sub>''i''<sub>''k''</sub></sub>}} {{math|1=(''k'' = 1, ..., ''n'')}} elements of {{math|''B''}}. Linear independence means that the coordinates {{math|''a''<sub>''k''</sub>}} are uniquely determined for any vector in the vector space.

For example, the [[coordinate vector]]s {{math|1='''e'''<sub>1</sub> = (1, 0, ..., 0)}}, {{math|1='''e'''<sub>2</sub> = (0, 1, 0, ..., 0)}}, to {{math|1='''e'''<sub>''n''</sub> = (0, 0, ..., 0, 1)}}, form a basis of {{math|''F''<sup>''n''</sup>}}, called the [[standard basis]], since any vector {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} can be uniquely expressed as a linear combination of these vectors:
:{{math|1=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) = ''x''<sub>1</sub>(1, 0, ..., 0) + ''x''<sub>2</sub>(0, 1, 0, ..., 0) + ... + ''x''<sub>''n''</sub>(0, ..., 0, 1) = ''x''<sub>1</sub>'''e'''<sub>1</sub> + ''x''<sub>2</sub>'''e'''<sub>2</sub> + ... + ''x''<sub>''n''</sub>'''e'''<sub>''n''</sub>}}.
The corresponding coordinates {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|...}}, {{math|''x''<sub>''n''</sub>}} are just the [[Cartesian coordinates]] of the vector.

Every vector space has a basis. This follows from [[Zorn's lemma]], an equivalent formulation of the [[Axiom of Choice]].<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=Theorem 1.9, p. 43}}</ref> Given the other axioms of [[Zermelo–Fraenkel set theory]], the existence of bases is equivalent to the axiom of choice.<ref>{{Harvard citations|last = Blass|year = 1984|nb=yes}}</ref> The [[ultrafilter lemma]], which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or [[cardinality]] (cf. ''[[Dimension theorem for vector spaces]]'').<ref>{{Harvard citations|last = Halpern|year = 1966|nb = yes|loc=pp. 670–673}}</ref> It is called the ''dimension'' of the vector space, denoted by dim ''V''. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.<ref>{{Harvard citations|last = Artin|year = 1991|nb = yes|loc=Theorem 3.3.13}}</ref>

The dimension of the coordinate space {{math|''F''<sup>''n''</sup>}} is {{math|''n''}}, by the basis exhibited above. The dimension of the polynomial ring ''F''[''x''] introduced [[#labelPolynomialRing|above]]{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> is [[countably infinite]], a basis is given by {{math|1}}, {{math|''x''}}, {{math|''x''<sup>2</sup>}}, {{math|...}} [[A fortiori]], the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.<ref group=nb>The [[indicator function]]s of intervals (of which there are infinitely many) are linearly independent, for example.</ref> Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous [[ordinary differential equation]] equals the degree of the equation.<ref>{{Harvard citations|last = Braun|year = 1993|nb = yes|loc=Th. 3.4.5, p. 291}}</ref> For example, the solution space for the [[#equation1|above equation]]{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> is generated by {{math|''e''<sup>−''x''</sup> and ''xe''<sup>−''x''</sup>}}. These two functions are linearly independent over {{math|'''R'''}}, so the dimension of this space is two, as is the degree of the equation.

A field extension over the rationals {{math|'''Q'''}} can be thought of as a vector space over {{math|'''Q'''}} (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of {{math|'''Q'''}}, and otherwise ignoring the field multiplication). The dimension (or [[degree of a field extension|degree]]) of the field extension {{math|'''Q'''(α)}} over {{math|'''Q'''}} depends on {{math|α}}. If {{math|α}} satisfies some polynomial equation
<math display=block> q_n \alpha^n + q_{n - 1} \alpha^{n - 1} + \ldots + q_0 = 0</math>
with rational coefficients {{math|''q''<sub>''n''</sub>, ..., ''q''<sub>0</sub>}} (in other words, if α is [[algebraic number|algebraic]]), the dimension is finite. More precisely, it equals the degree of the [[minimal polynomial (field theory)|minimal polynomial]] having α as a [[root of a function|root]].<ref>{{Harvard citations|last = Stewart|year = 1975|nb = yes|loc=Proposition 4.3, p. 52}}</ref> For example, the complex numbers '''C''' are a two-dimensional real vector space, generated by 1 and the [[imaginary unit]] ''i''. The latter satisfies ''i''<sup>2</sup> + 1 = 0, an equation of degree two. Thus, '''C''' is a two-dimensional '''R'''-vector space (and, as any field, one-dimensional as a vector space over itself, '''C'''). If α is not algebraic, the dimension of '''Q'''(α) over '''Q''' is infinite. For instance, for α = [[pi|π]] there is no such equation, in other words π is [[transcendental number|transcendental]].<ref>{{Harvard citations|last = Stewart|year = 1975|nb = yes|loc=Theorem 6.5, p. 74}}</ref>

==Linear maps and matrices==
{{Main|Linear map}}
The relation of two vector spaces can be expressed by ''linear map'' or ''linear transformation''. They are [[function (mathematics)|functions]] that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
:<math>f(\mathbf v + \mathbf w) = f(\mathbf v) + f(\mathbf w)</math> and {{math|''f''(''a'' · '''v''')}} <math>=</math> {{math|''a'' · ''f''('''v''')}} for all {{math|'''v'''}} and {{math|'''w'''}} in {{math|''V''}}, all {{math|''a''}} in {{math|''F''}}.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 2, p. 45}}</ref>

An ''[[isomorphism]]'' is a linear map {{math|''f'' : ''V'' → ''W''}} such that there exists an [[inverse map]] {{math|''g'' : ''W'' → ''V''}}, which is a map such that the two possible [[function composition|compositions]] {{math|''f'' ∘ ''g'' : ''W'' → ''W''}} and {{math|''g'' ∘ ''f'' : ''V'' → ''V''}} are [[Identity function|identity maps]]. Equivalently, {{math|''f''}} is both one-to-one ([[injective]]) and onto ([[surjective]]).<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. IV.4, Corollary, p. 106}}</ref>  If there exists an isomorphism between {{math|''V''}} and {{math|''W''}}, the two spaces are said to be ''isomorphic''; they are then essentially identical as vector spaces, since all identities holding in {{math|''V''}} are, via {{math|''f''}}, transported to similar ones in {{math|''W''}}, and vice versa via {{math|''g''}}.

[[File:Vector components.svg|180px|right|thumb|Describing an arrow vector {{math|'''v'''}} by its coordinates {{math|''x''}} and {{math|''y''}} yields an isomorphism of vector spaces.]]
For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow {{math|'''v'''}} departing at the [[origin (mathematics)|origin]] of some (fixed) [[coordinate system]] can be expressed as an ordered pair by considering the {{math|''x''}}- and {{math|''y''}}-component of the arrow, as shown in the image at the right. Conversely, given a pair {{math|(''x'', ''y'')}}, the arrow going by {{math|''x''}} to the right (or to the left, if {{math|''x''}} is negative), and {{math|''y''}} up (down, if {{math|''y''}} is negative) turns back the arrow {{math|'''v'''}}.

Linear maps {{math|''V'' → ''W''}} between two vector spaces form a vector space {{math|Hom<sub>''F''</sub>(''V'', ''W'')}}, also denoted {{math|L(''V'', ''W'')}}.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=Example IV.2.6}}</ref> The space of linear maps from {{math|''V''}} to {{math|''F''}} is called the ''[[dual vector space]]'', denoted {{math|''V''<sup>∗</sup>}}.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. VI.6}}</ref> Via the injective [[natural (category theory)|natural]] map {{math|''V'' → ''V''<sup>∗∗</sup>}}, any vector space can be embedded into its ''bidual''; the map is an isomorphism if and only if the space is finite-dimensional.<ref>{{Harvard citations|last = Halmos|year =1974|nb = yes|loc=p. 28, Ex. 9}}</ref>

Once a basis of {{math|''V''}} is chosen, linear maps {{math|''f'' : ''V'' → ''W''}} are completely determined by specifying the images of the basis vectors, because any element of {{math|''V''}} is expressed uniquely as a linear combination of them.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=Theorem IV.2.1, p. 95}}</ref> If {{math|1=dim ''V'' = dim ''W''}}, a [[bijection|1-to-1 correspondence]] between fixed bases of {{math|''V''}} and {{math|''W''}} gives rise to a linear map that maps any basis element of {{math|''V''}} to the corresponding basis element of {{math|''W''}}. It is an isomorphism, by its very definition.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=Th. 2.5 and 2.6, p. 49}}</ref> Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is ''completely classified'' ([[up to]] isomorphism) by its dimension, a single number. In particular, any ''n''-dimensional {{math|''F''}}-vector space {{math|''V''}} is isomorphic to {{math|''F''<sup>''n''</sup>}}. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism {{math|''φ'' : ''F''<sup>''n''</sup> → ''V''}} is equivalent to the choice of a basis of {{math|''V''}}, by mapping the standard basis of {{math|''F''<sup>''n''</sup>}} to {{math|''V''}}, via {{math|''φ''}}. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see [[#basis in inf dim context|below]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. -->

===Matrices===
{{Main|Matrix (mathematics)|l1=Matrix|Determinant}}
[[Image:Matrix.svg|right|thumb|200px|A typical matrix]]
''Matrices'' are a useful notion to encode linear maps.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. V.1}}</ref>  They are written as a rectangular array of scalars as in the image at the right. Any {{math|''m''}}-by-{{math|''n''}} matrix {{math|''A''}} gives rise to a linear map from {{math|''F''<sup>''n''</sup>}} to {{math|''F''<sup>''m''</sup>}}, by the following
:<math>\mathbf x = (x_1, x_2, \cdots, x_n) \mapsto \left(\sum_{j=1}^n a_{1j}x_j, \sum_{j=1}^n a_{2j}x_j, \cdots, \sum_{j=1}^n a_{mj}x_j \right)</math>, where <math>\sum</math> denotes [[summation]],
or, using the [[matrix multiplication]] of the matrix {{math|''A''}} with the coordinate vector {{math|'''x'''}}:
:<span id=equation2>{{math|'''x''' ↦ ''A'''''x'''}}</span>.
Moreover, after choosing bases of {{math|''V''}} and {{math|''W''}}, ''any'' linear map {{math|''f'' : ''V'' → ''W''}} is uniquely represented by a matrix via this assignment.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. V.3., Corollary, p. 106}}</ref>

[[Image:Determinant parallelepiped.svg|200px|right|thumb|The volume of this [[parallelepiped]] is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, and {{math|''r''<sub>3</sub>}}.]]
The [[determinant]] {{math|det (''A'')}} of a [[square matrix]] {{math|''A''}} is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=Theorem VII.9.8, p. 198}}</ref> The linear transformation of {{math|'''R'''<sup>''n''</sup>}} corresponding to a real ''n''-by-''n'' matrix is [[Orientation (mathematics)|orientation preserving]] if and only if its determinant is positive.

===Eigenvalues and eigenvectors===
{{Main|Eigenvalues and eigenvectors}}
[[Endomorphism]]s, linear maps {{math|''f'' : ''V'' → ''V''}}, are particularly important since in this case vectors {{math|'''v'''}} can be compared with their image under {{math|''f''}}, {{math|''f''('''v''')}}. Any nonzero vector {{math|'''v'''}} satisfying {{math|1=''λ'''''v''' = ''f''('''v''')}}, where {{math|''λ''}} is a scalar, is called an ''eigenvector'' of {{math|''f''}} with ''eigenvalue'' {{math|''λ''}}.<ref group=nb>The nomenclature derives from [[German language|German]] "[[wikt:eigen|eigen]]", which means own or proper.</ref><ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 8, p. 135–156}}</ref> Equivalently, {{math|'''v'''}} is an element of the [[Kernel (linear algebra)|kernel]] of the difference {{math|''f'' − ''λ'' · Id}} (where Id is the [[identity function|identity map]] {{math|''V'' → ''V'')}}. If {{math|''V''}} is finite-dimensional, this can be rephrased using determinants: {{math|''f''}}  having eigenvalue {{math|''λ''}} is equivalent to
:{{math|1=det(''f'' − ''λ'' · Id) = 0}}.
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in {{math|''λ''}}, called the [[characteristic polynomial]] of {{math|''f''}}.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. IX.4}}</ref>  If the field {{math|''F''}} is large enough to contain a zero of this polynomial (which automatically happens for {{math|''F''}} [[algebraically closed field|algebraically closed]], such as {{math|1=''F'' = '''C'''}}) any linear map has at least one eigenvector. The vector space {{math|''V''}} may or may not possess an [[eigenbasis]], a basis consisting of eigenvectors. This phenomenon is governed by the [[Jordan canonical form]] of the map.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 8, p. 140}}.</ref><ref group = nb> See also [[Jordan–Chevalley decomposition]].</ref> The set of all eigenvectors corresponding to a particular eigenvalue of {{math|''f''}} forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and {{math|''f''}}) in question. To achieve the [[spectral theorem]], the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see [[#labelSpectralTheorem|below]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. -->

==Basic constructions==
In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by [[universal property|universal properties]], which determine an object {{math|''X''}} by specifying the linear maps from {{math|''X''}} to any other vector space.

===Subspaces and quotient spaces===
{{Main|Linear subspace|Quotient vector space}}
[[File:Linear subspaces with shading.svg|thumb|250px|right|A line passing through the [[origin (mathematics)|origin]] (blue, thick) in [[Euclidean space|{{math|'''R'''<sup>3</sup>}}]] is a linear subspace. It is the intersection of two [[plane (mathematics)|planes]] (green and yellow).]]

A nonempty [[subset]] ''W'' of a vector space ''V'' that is closed under addition and scalar multiplication (and therefore contains the '''0'''-vector of ''V'') is called a ''linear subspace'' of ''V'', or simply a ''subspace'' of ''V'', when the ambient space is unambiguously a vector space.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 1, p. 29}}</ref><ref group=nb>This is typically the case when a vector space is also considered as an [[affine space]]. In this case, a linear subspace contains the [[zero vector]], while an affine subspace does not necessarily contain it.</ref> Subspaces of ''V'' are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set ''S'' of vectors is called its [[linear span|span]], and it is the smallest subspace of ''V'' containing the set ''S''. Expressed in terms of elements, the span is the subspace consisting of all the [[linear combination]]s of elements of ''S''.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 1, p. 35}}</ref>

{{anchor|vector line|vector plane|vector hyperplane}}A linear subspace of dimension 1 is a '''vector line'''.  A linear subspace of dimension 2 is a '''vector plane'''. A linear subspace that contains all elements but one of a basis of the ambient space is a '''vector hyperplane'''. In a vector space of finite dimension {{math|''n''}}, a vector hyperplane is thus a subspace of dimension {{math|''n'' – 1}}.

The counterpart to subspaces are ''quotient vector spaces''.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 3, p. 64}}</ref> Given any subspace {{math|''W'' ⊂ ''V''}}, the quotient space ''V''/''W'' ("''V'' [[modular arithmetic|modulo]] ''W''") is defined as follows: as a set, it consists of {{math|1='''v''' + ''W'' = {'''v''' + '''w''' : '''w''' ∈ ''W''},}} where '''v''' is an arbitrary vector in ''V''. The sum of two such elements {{math|'''v'''<sub>1</sub> + ''W''}} and {{math|'''v'''<sub>2</sub> + ''W''}} is {{math|('''v'''<sub>1</sub> + '''v'''<sub>2</sub>) + ''W'',}} and scalar multiplication is given by {{math|1=''a'' · ('''v''' + ''W'') = (''a'' · '''v''') + ''W''}}. The key point in this definition is that {{math|1='''v'''<sub>1</sub> + ''W'' = '''v'''<sub>2</sub> + ''W''}} [[if and only if]] the difference of '''v'''<sub>1</sub> and '''v'''<sub>2</sub> lies in ''W''.<ref group=nb>Some authors (such as {{Harvard citations|last = Roman|year = 2005|nb = yes}}) choose to start with this [[equivalence relation]] and derive the concrete shape of ''V''/''W'' from this.</ref> This way, the quotient space "forgets" information that is contained in the subspace ''W''.

The [[kernel (algebra)|kernel]] ker(''f'') of a linear map {{math|''f'' : ''V'' → ''W''}} consists of vectors '''v''' that are mapped to '''0''' in ''W''.<ref>{{Harvard citations|last = Lang|year = 1987|nb = yes|loc=ch. IV.3.}}</ref> Both kernel and [[image (mathematics)|image]] {{math|1=im(''f'') = {''f''('''v''') : '''v''' ∈ ''V''} }} are subspaces of ''V'' and ''W'', respectively.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 2, p. 48}}</ref> The existence of kernels and images is part of the statement that the [[category of vector spaces]] (over a fixed field ''F'') is an [[abelian category]], that is, a corpus of mathematical objects and structure-preserving maps between them (a [[category (mathematics)|category]]) that behaves much like the [[category of abelian groups]].<ref>{{Harvard citations|last = Mac Lane|year = 1998|nb = yes}}</ref> Because of this, many statements such as the [[first isomorphism theorem]] (also called [[rank–nullity theorem]] in matrix-related terms)
:''V'' / ker(''f'') ≡ im(''f'').
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for [[group (mathematics)|groups]].

An important example is the kernel of a linear map {{math|'''x''' ↦ ''A'''''x'''}} for some fixed matrix ''A'', as [[#equation2|above]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> The kernel of this map is the subspace of vectors '''x''' such that {{math|1=''A'''''x''' = 0}}, which is precisely the set of solutions to the system of homogeneous linear equations belonging to ''A''. This concept also extends to linear differential equations
:<math>a_0 f + a_1 \frac{d f}{d x} + a_2 \frac{d^2 f}{d x^2} + \cdots + a_n \frac{d^n f}{d x^n} = 0</math>, where the coefficients ''a''<sub>''i''</sub> are functions in ''x'', too.
In the corresponding map
:<math>f \mapsto D(f) = \sum_{i=0}^n a_i \frac{d^i f}{d x^i}</math>,
the [[derivative]]s of the function ''f'' appear linearly (as opposed to ''f''′′(''x'')<sup>2</sup>, for example). Since differentiation is a linear procedure (that is, {{math|1=(''f'' + ''g'')′ = ''f''′ + ''g''&thinsp;′}} and {{math|1=(''c''·''f'')′ = ''c''·''f''′}} for a constant {{math|''c''}}) this assignment is linear, called a [[linear differential operator]]. In particular, the solutions to the differential equation {{math|1=''D''(''f'') = 0}} form a vector space (over {{math|'''R'''}} or {{math|'''C'''}}).

===Direct product and direct sum===
{{Main|Direct product|Direct sum of modules}}
The ''direct product'' of vector spaces and the ''direct sum'' of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

The ''direct product''<!--explain direct--> <math>\textstyle{\prod_{i \in I} V_i}</math> of a family of vector spaces ''V''<sub>''i''</sub> consists of the set of all tuples ({{math|'''v'''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}}, which specify for each index ''i'' in some [[index set]] ''I'' an element '''v'''<sub>''i''</sub> of ''V''<sub>''i''</sub>.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=ch. 1, pp. 31–32}}</ref> Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' <math>\oplus_{i \in I} V_i</math> (also called [[coproduct]] and denoted <math>\textstyle{\coprod_{i \in I}V_i}</math>), where only tuples with finitely many nonzero vectors are allowed. If the index set ''I'' is finite, the two constructions agree, but in general they are different.

===Tensor product===
{{Main|Tensor product of vector spaces}}
The ''tensor product'' {{math|''V'' ⊗<sub>''F''</sub> ''W''}}, or simply {{math|''V'' ⊗ ''W''}}, of two vector spaces ''V'' and ''W'' is one of the central notions of [[multilinear algebra]] which deals with extending notions such as linear maps to several variables. A map {{math|''g'' : [[Cartesian product|''V'' × ''W'']] → ''X''}} is called [[bilinear map|bilinear]] if ''g'' is linear in both variables '''v''' and '''w'''.  That is to say, for fixed '''w''' the map {{math|'''v''' ↦ ''g''('''v''', '''w''')}} is linear in the sense above and likewise for fixed '''v'''.

The tensor product is a particular vector space that is a ''universal'' recipient of bilinear maps ''g'', as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called [[tensor]]s
:'''v'''<sub>1</sub> ⊗ '''w'''<sub>1</sub> + '''v'''<sub>2</sub> ⊗ '''w'''<sub>2</sub> + ... + '''v'''<sub>''n''</sub> ⊗ '''w'''<sub>''n''</sub>,
subject to the rules 
: ''a'' &middot; ('''v''' ⊗ '''w''') = (''a'' &middot; '''v''') ⊗ '''w''' = '''v''' ⊗ (''a'' &middot; '''w'''), where ''a'' is a scalar,
:('''v'''<sub>1</sub> + '''v'''<sub>2</sub>) ⊗ '''w''' = '''v'''<sub>1</sub> ⊗ '''w''' + '''v'''<sub>2</sub> ⊗ '''w''', and
:'''v''' ⊗ ('''w'''<sub>1</sub> + '''w'''<sub>2</sub>) = '''v''' ⊗ '''w'''<sub>1</sub> + '''v''' ⊗ '''w'''<sub>2</sub>.<ref>{{Harvard citations|last = Lang|year = 2002|loc = ch. XVI.1|nb = yes}}</ref>
[[Image:Universal tensor prod.svg|right|thumb|200px|[[Commutative diagram]] depicting the universal property of the tensor product.]]
These rules ensure that the map ''f'' from the {{math|''V'' × ''W''}} to {{math|''V'' ⊗ ''W''}} that maps a [[tuple]] {{math|('''v''', '''w''')}} to {{math|'''v''' ⊗ '''w'''}} is bilinear. The universality states that given ''any'' vector space ''X'' and ''any'' bilinear map {{math|''g'' : ''V'' × ''W'' → ''X''}}, there exists a unique map ''u'', shown in the diagram with a dotted arrow, whose [[function composition|composition]] with ''f'' equals ''g'': {{math|1=''u''('''v''' ⊗ '''w''') = ''g''('''v''', '''w''')}}.<ref>{{Harvard citations|last = Roman|year = 2005|nb = yes|loc=Th. 14.3}}. See also [[Yoneda lemma]].</ref> This is called the [[universal property]] of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.

==Vector spaces with additional structure==
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces ''per se'' do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions [[Limit of a sequence|converges]] to another function. Likewise, linear algebra is not adapted to deal with [[infinite series]], since the addition operation allows only finitely many terms to be added. <span id=labelFunctionalAnalysis>Therefore, the needs of [[functional analysis]] require considering additional structures.</span>

A vector space may be given a [[partial order]] ≤, under which some vectors can be compared.<ref>{{Harvard citations|last1 = Schaefer|last2=Wolff|year = 1999|loc = pp. 204–205|nb = yes}}</ref> For example, ''n''-dimensional real space '''R'''<sup>''n''</sup> can be ordered by comparing its vectors componentwise. [[Ordered vector space]]s, for example [[Riesz space]]s, are fundamental to [[Lebesgue integration]], which relies on the ability to express a function as a difference of two positive functions
:''f'' = ''f''<sup>+</sup> − ''f''<sup>−</sup>,
where ''f''<sup>+</sup> denotes the positive part of ''f'' and ''f''<sup>−</sup> the negative part.<ref>{{Harvard citations|last = Bourbaki|year = 2004|nb = yes|loc=ch. 2, p. 48}}</ref>

===Normed vector spaces and inner product spaces===
{{Main|Normed vector space|Inner product space}}
"Measuring" vectors is done by specifying a [[norm (mathematics)|norm]], a datum which measures lengths of vectors, or by an [[inner product]], which measures angles between vectors. Norms and inner products are denoted <math>| \mathbf v|</math> and <math>\lang \mathbf v , \mathbf w \rang</math>, respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm <math>|\mathbf v| := \sqrt {\langle \mathbf v , \mathbf v \rangle}</math>. Vector spaces endowed with such data are known as ''normed vector spaces'' and ''inner product spaces'', respectively.<ref>{{Harvard citations|last =Roman|year = 2005|nb = yes|loc=ch. 9}}</ref>

Coordinate space ''F''<sup>''n''</sup> can be equipped with the standard [[dot product]]:
:<math>\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n.</math>
In '''R'''<sup>2</sup>, this reflects the common notion of the angle between two vectors '''x''' and '''y''', by the [[law of cosines]]:
:<math>\mathbf x \cdot \mathbf y = \cos\left(\angle (\mathbf x, \mathbf y)\right) \cdot |\mathbf x| \cdot |\mathbf y|.</math>
Because of this, two vectors satisfying <math>\lang \mathbf x , \mathbf y \rang = 0</math> are called [[orthogonal]]. An important variant of the standard dot product is used in [[Minkowski space]]: '''R'''<sup>4</sup> endowed with the Lorentz product
:<math>\lang \mathbf x | \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4.</math><ref>{{Harvard citations|last =Naber|year = 2003|nb = yes|loc=ch. 1.2}}</ref>
In contrast to the standard dot product, it is not [[positive definite bilinear form|positive definite]]: <math>\lang \mathbf x | \mathbf x \rang</math> also takes negative values, for example for <math>\mathbf x = (0, 0, 0, 1)</math>. Singling out the fourth coordinate—[[timelike|corresponding to time]], as opposed to three space-dimensions—makes it useful for the mathematical treatment of [[special relativity]].

===Topological vector spaces===
{{Main|Topological vector space}}
Convergence questions are treated by considering vector spaces ''V'' carrying a compatible [[topological space|topology]], a structure that allows one to talk about elements being [[neighborhood (topology)|close to each other]].<ref>{{Harvard citations|last = Treves|year = 1967|nb = yes}}</ref><ref>{{Harvard citations|last = Bourbaki|year = 1987|nb = yes}}</ref> Compatible here means that addition and scalar multiplication have to be [[continuous map]]s.  Roughly, if '''x''' and '''y''' in ''V'', and ''a'' in ''F'' vary by a bounded amount, then so do {{math|'''x''' + '''y'''}} and {{math|''a'''''x'''}}.<ref group=nb>This requirement implies that the topology gives rise to a [[uniform structure]], {{Harvard citations|last = Bourbaki|year = 1989|loc = ch. II|nb = yes}}</ref> To make sense of specifying the amount a scalar changes, the field ''F'' also has to carry a topology in this context; a common choice are the reals or the complex numbers.

In such ''topological vector spaces'' one can consider [[series (mathematics)|series]] of vectors.  The [[infinite sum]]
:<math>\sum_{i=0}^{\infty} f_i</math>
denotes the [[limit of a sequence|limit]] of the corresponding finite partial sums of the sequence (''f''<sub>''i''</sub>)<sub>''i''∈'''N'''</sub> of elements of ''V''. For example, the ''f''<sub>''i''</sub> could be (real or complex) functions belonging to some [[function space]] ''V'', in which case the series is a [[function series]]. The [[modes of convergence|mode of convergence]] of the series depends on the topology imposed on the function space. In such cases, [[pointwise convergence]] and [[uniform convergence]] are two prominent examples.

[[Image:Vector norms2.svg|thumb|right|250px|[[Unit ball|Unit "spheres"]] in '''R'''<sup>2</sup> consist of plane vectors of norm 1.  Depicted are the unit spheres in different [[Lp norm|''p''-norm]]s, for ''p'' = 1, 2, and ∞. The bigger diamond depicts points of 1-norm equal to 2.]]
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any [[Cauchy sequence]] has a limit; such a vector space is called [[Completeness (topology)|complete]]. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the [[topology of uniform convergence]] is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the [[Weierstrass approximation theorem]].<ref>{{harvnb|Kreyszig|1989|loc=§4.11-5}}</ref>  In contrast, the space of ''all'' continuous functions on [0,1] with the same topology is complete.<ref>{{harvnb|Kreyszig|1989|loc=§1.5-5}}</ref> A norm gives rise to a topology by defining that a sequence of vectors '''v'''<sub>''n''</sub> converges to '''v''' if and only if
:<math>\text{lim}_{n \rightarrow \infty} |\mathbf v_n - \mathbf v| = 0.</math>
Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of [[functional analysis]]—focusses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.<ref>{{Harvard citations|last =Choquet|year = 1966|nb = yes|loc=Proposition III.7.2}}</ref> The image at the right shows the equivalence of the 1-norm and ∞-norm on '''R'''<sup>2</sup>: as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called [[functional (mathematics)|functional]]s) {{math|''V'' → ''W''}}, maps between topological vector spaces are required to be continuous.<ref>{{Harvard citations|last = Treves|year = 1967|nb = yes|loc=p. 34–36}}</ref> In particular, the <span id=label2>(topological) dual space {{math|''V''<sup>∗</sup>}}</span> consists of continuous functionals {{math|''V'' → '''R'''}} (or to {{math|'''C'''}}). The fundamental [[Hahn–Banach theorem]] is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.<ref>{{Harvard citations|last = Lang|year =1983|nb = yes|loc=Cor. 4.1.2, p. 69}}</ref>

====Banach spaces====
{{Main|Banach space}}
''Banach spaces'', introduced by [[Stefan Banach]], are complete normed vector spaces.<ref>{{Harvard citations|last = Treves|year = 1967|nb = yes|loc=ch. 11}}</ref> 

A first example is [[Lp space|the vector space <math>\ell^{p}</math>]] consisting of infinite vectors with real entries 
<math>\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n},\ldots\right)</math>
whose [[p-norm|<math>p</math>-norm]] 
<math>(1\leq{p}\leq\infty)</math>
given by 
:<math>\left\Vert\mathbf{x}\right\Vert_{p} := \left(\sum_{i} \left\vert x_{i}\right\vert^{p}\right)^\frac{1}{p}</math> for <math>p < \infty</math>&nbsp;&nbsp; and&nbsp;&nbsp; <math>\left\Vert\mathbf x\right\Vert_{\infty} := \text{sup}_{i} \left\vert x_{i}\right\vert</math>.
<!---- "is finite." - ?!  ----> 

The topologies on the infinite-dimensional space 
<math>\ell^{p}</math> 
are inequivalent for different <math>p</math>. For example, the sequence of vectors 
<math>\mathbf{x}_{n} = \left(2^{-n}, 2^{-n}, \ldots, 2^{-n}, 0, 0, \ldots\right)</math>, 
in which the first <math>2^{n}</math> components are <math>2^{-n}</math> and the following ones are <math>0</math>, converges to the [[zero vector]] for 
<math>p = \infty</math>, 
but does not for 
<math>p = 1</math>:
:<math>
    \left\Vert\mathbf{x}_{n}\right\Vert_{_\infty} = \sup (2^{-n}, 0) 
  = 2^{-n} \rightarrow 0
</math>, but <math>
    \left\Vert\mathbf{x}_{n}\right\Vert_{1} 
  = \sum_{i=1}^{2^n} 2^{-n} = 2^n \cdot 2^{-n} = 1.
</math>

More generally than sequences of real numbers, functions 
<math>f\colon \Omega \to \mathbb{R}</math>
are endowed with a norm that replaces the above sum by the [[Lebesgue integral]]
:<math>
  \left\Vert{f}\right\Vert_{p} := \left(
    \int_{\Omega} \left\vert{f}\left(x\right)\right\vert^{p} \, {d\mu\left(x\right)} 
  \right)^\frac{1}{p}.
</math>

The space of [[integrable function]]s on a given [[domain (mathematics)|domain]] 
<math>\Omega</math> 
(for example an interval) satisfying 
<math>\left\Vert{f}\right\Vert_{p} < \infty</math>,
and equipped with this norm are called [[Lp space|Lebesgue spaces]], denoted <math>L^{\;\!p}\left(\Omega\right)</math>.<ref group=nb> The [[triangle inequality]] for 
<math>\left\Vert{f + g}\right\Vert_{p} \leq \left\Vert{f}\right\Vert_{p} + \left\Vert{g}\right\Vert_{p}</math>
is provided by the [[Minkowski inequality]]. For technical reasons, in the context of functions one has to identify functions that agree [[almost everywhere]] to get a norm, and not only a [[seminorm]].</ref>  

These spaces are complete.<ref>{{Harvard citations|last = Treves|year = 1967|nb = yes|loc=Theorem 11.2, p. 102}}</ref> (If one uses the [[Riemann integral]] instead, the space is ''not'' complete, which may be seen as a justification for Lebesgue's integration theory.<ref group=nb>
"Many functions in 
<math>L^{2}</math> 
of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the 
<math>L^{2}</math>
norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", {{Harvard citations|last = Dudley|year = 1989|nb = yes|loc=§5.3, p. 125}}
</ref>) Concretely this means that for any sequence of Lebesgue-integrable functions&nbsp;&nbsp; 
<math>f_{1}, f_{2}, \ldots, f_{n},\ldots</math>&nbsp;&nbsp;
with
<math>\left\Vert{f}_{n}\right\Vert_{p}<\infty</math>, 
satisfying the condition
:<math>\lim_{k,\ n \to \infty}\int_{\Omega} 
  \left\vert{f}_{k}(x) - {f}_{n}(x)\right\vert^{p} \, 
  {d\mu\left(x\right)} = 0
</math>

there exists a function <math>{f}\left(x\right)</math> belonging to the vector space <math>L^{\;\!p}\left(\Omega\right)</math> such that
:<math>\lim_{k \to \infty}\int_{\Omega} 
  \left\vert{f}\left(x\right) - {f}_{k}\left(x\right)\right\vert^{p} \, 
  {d\mu\left(x\right)} = 0.
</math>

Imposing boundedness conditions not only on the function, but also on its [[derivative]]s leads to [[Sobolev space]]s.<ref>{{Harvard citations|last = Evans|year = 1998|loc = ch. 5|nb = yes}}</ref>
{{Clear}}

====Hilbert spaces====
{{Main|Hilbert space}}
[[Image:Periodic identity function.gif|right|thumb|400px|The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).]]
Complete inner product spaces are known as ''Hilbert spaces'', in honor of [[David Hilbert]].<ref>{{Harvard citations|last = Treves|year = 1967|nb = yes|loc=ch. 12}}</ref>
The Hilbert space ''L''<sup>2</sup>(Ω), with inner product given by
:<math> \langle f\ , \ g \rangle = \int_\Omega f(x) \overline{g(x)} \, dx,</math>
where <math>\overline{g(x)}</math> denotes the [[complex conjugate]] of ''g''(''x''),<ref name=Dennery>{{Harvard citations|last = Dennery|year = 1996|loc = p.190|nb = yes}}</ref><ref group=nb>For ''p'' ≠2, ''L''<sup>''p''</sup>(Ω) is not a Hilbert space.</ref> is a key case.

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions ''f''<sub>''n''</sub> with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the [[Taylor approximation]], established an approximation of [[differentiable function]]s ''f'' by polynomials.<ref>{{Harvard citations|last = Lang|year = 1993|loc = Th. XIII.6, p. 349|nb = yes}}</ref> By the [[Stone–Weierstrass theorem]], every continuous function on {{math|[''a'', ''b'']}} can be approximated as closely as desired by a polynomial.<ref>{{Harvard citations|last = Lang|year = 1993|loc = Th. III.1.1|nb = yes}}</ref> A similar approximation technique by [[trigonometric function]]s is commonly called [[Fourier expansion]], and is much applied in engineering, see [[#labelFourier|below]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space ''H'', in the sense that the ''[[closure (topology)|closure]]'' of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a ''basis'' of ''H'', its cardinality is known as the [[Hilbert space dimension]].<ref group=nb>A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra [[#label1|above]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. -->  For distinction, the latter is then called a [[Hamel basis]].</ref> Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the [[Gram–Schmidt process]], it enables one to construct a [[orthogonal basis|basis of orthogonal vectors]].<ref>{{Harvard citations|last = Choquet|year = 1966|loc = Lemma III.16.11|nb = yes}}</ref> Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional [[Euclidean space]].

The solutions to various [[differential equation]]s can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical properties are used as basis functions, often orthogonal.<ref>{{harvs|last=Kreyszig|year=1999|loc=Chapter 11|nb=yes}}</ref> As an example from physics, the time-dependent [[Schrödinger equation]] in [[quantum mechanics]] describes the change of physical properties in time by means of a [[partial differential equation]], whose solutions are called [[wavefunction]]s.<ref>{{harvs|last=Griffiths|year=1995|loc=Chapter 1|nb=yes}}</ref> Definite values for physical properties such as energy, or momentum, correspond to [[eigenvalue]]s of a certain (linear) [[differential operator]] and the associated wavefunctions are called [[eigenstate]]s. The <span id=labelSpectralTheorem>[[spectral theorem]] decomposes a linear [[compact operator]] acting on functions in terms of these eigenfunctions and their eigenvalues.</span><ref>{{Harvard citations|last = Lang|year = 1993|loc =ch. XVII.3|nb = yes}}</ref>
{{Clear}}

===Algebras over fields===
{{Main|Algebra over a field|Lie algebra}}
[[Image:Rectangular hyperbola.svg|right|thumb|250px|A [[hyperbola]], given by the equation {{math|1=''x'' ⋅ ''y'' = 1}}. The [[coordinate ring]] of functions on this hyperbola is given by {{math|'''R'''[''x'', ''y''] / (''x'' &middot; ''y'' − 1)}}, an infinite-dimensional vector space over {{math|'''R'''}}.]]
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional [[bilinear operator]] defining the multiplication of two vectors is an ''algebra over a field''.<ref>{{Harvard citations|last = Lang|year = 2002|nb = yes|loc=ch. III.1, p. 121}}</ref> Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on [[Banach algebra]]s which are both Banach spaces and algebras.

[[Commutative algebra]] makes great use of [[polynomial ring|rings of polynomials]] in one or several variables, introduced [[#labelPolynomialRing|above]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> Their multiplication is both [[commutative]] and [[associative]]. These rings and their [[quotient ring|quotients]] form the basis of [[algebraic geometry]], because they are [[coordinate ring|rings of functions of algebraic geometric objects]].<ref>{{Harvard citations|last = Eisenbud|year = 1995|nb = yes|loc=ch. 1.6}}</ref>

Another crucial example are ''Lie algebras'', which are neither commutative nor associative, but the failure to be so is limited by the constraints ({{math|[''x'', ''y'']}} denotes the product of {{math|''x''}} and {{math|''y''}}):
*{{math|1=[''x'', ''y''] = −[''y'', ''x'']}} ([[anticommutativity]]), and
*{{math|1=[''x'', [''y'', ''z'']] + [''y'', [''z'', ''x'']] + [''z'', [''x'', ''y'']] = 0}} ([[Jacobi identity]]).<ref>{{Harvard citations|last = Varadarajan|year = 1974|nb = yes}}</ref>
Examples include the vector space of ''n''-by-''n'' matrices, with {{math|1=[''x'', ''y''] = ''xy'' − ''yx''}}, the [[commutator]] of two matrices, and {{math|'''R'''<sup>3</sup>}}, endowed with the [[cross product]].

The [[tensor algebra]] T(''V'') is a formal way of adding products to any vector space ''V'' to obtain an algebra.<ref>{{Harvard citations|last = Lang|year = 2002|nb = yes|loc=ch. XVI.7}}</ref> As a vector space, it is spanned by symbols, called simple [[tensor]]s
:{{math|'''v'''<sub>1</sub> ⊗ '''v'''<sub>2</sub> ⊗ ... ⊗ '''v'''<sub>''n''</sub>}}, where the [[rank of a tensor|degree]] {{math|''n''}} varies.
The multiplication is given by concatenating such symbols, imposing the [[distributive law]] under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced [[#Tensor product|above]].{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> In general, there are no relations between {{math|'''v'''<sub>1</sub> ⊗ '''v'''<sub>2</sub>}} and {{math|'''v'''<sub>2</sub> ⊗ '''v'''<sub>1</sub>}}. Forcing two such elements to be equal leads to the [[symmetric algebra]], whereas forcing {{math|1='''v'''<sub>1</sub> ⊗ '''v'''<sub>2</sub> = − '''v'''<sub>2</sub> ⊗ '''v'''<sub>1</sub>}} yields the [[exterior algebra]].<ref>{{Harvard citations|last = Lang|year = 2002|nb = yes|loc=ch. XVI.8}}</ref>

When a field, {{math|''F''}} is explicitly stated, a common term used is {{math|''F''}}-algebra.

==Applications==
Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in [[optimization (mathematics)|optimization]]. The [[minimax theorem]] of [[game theory]] stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods.<ref>{{Harvard citations|last = Luenberger|year = 1997|nb = yes|loc=§7.13}}</ref> [[Representation theory]] fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as [[group theory]].<ref>See [[representation theory]] and [[group representation]].</ref>

===Distributions===
{{Main|Distribution (mathematics)|l1=Distribution}}
A ''distribution'' (or ''generalized function'') is a linear map assigning a number to each [[test function|"test" function]], typically a [[smooth function]] with [[compact support]], in a continuous way: in the [[#label2|above]]{{huh|date=February 2019}}<!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --> terminology the space of distributions is the (continuous) dual of the test function space.<ref>{{Harvard citations|last = Lang|year = 1993|loc =Ch. XI.1|nb = yes}}</ref> The latter space is endowed with a topology that takes into account not only ''f'' itself, but also all its higher derivatives. A standard example is the result of integrating a test function ''f'' over some domain Ω:
:<math>I(f) = \int_\Omega f(x)\,dx.</math>
When {{math|1=Ω = {''p''},}} the set consisting of a single point, this reduces to the [[Dirac distribution]], denoted by δ, which associates to a test function ''f'' its value at the {{math|1=''p'': δ(''f'') = ''f''(''p'')}}. Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore, the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, [[Green's function]]s and [[fundamental solution]]s are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (for example, using the [[Lax–Milgram theorem]], a consequence of the [[Riesz representation theorem]]).<ref>{{Harvard citations|last = Evans|year = 1998|loc =Th. 6.2.1|nb = yes}}</ref>

===Fourier analysis===
{{Main|Fourier analysis}}
[[Image:Heat eqn.gif|thumb|right|200px|The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder regions than red).]]

Resolving a [[periodic function]] into a sum of [[trigonometric functions]] forms a <span id=labelFourier>''[[Fourier series]]''</span>, a technique much used in physics and engineering.<ref group=nb>Although the Fourier series is periodic, the technique can be applied to any ''L''<sup>2</sup> function on an interval by considering the function to be continued periodically outside the interval. See {{Harvard citations|last = Kreyszig|year = 1988|loc=p. 601|nb = yes}}</ref><ref>{{Harvard citations|last = Folland|year = 1992|loc = p. 349 ''ff''|nb = yes}}</ref> The underlying vector space is usually the [[Hilbert space]] ''L''<sup>2</sup>(0, 2π), for which the functions sin ''mx'' and cos ''mx'' (''m'' an integer) form an orthogonal basis.<ref>{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=p. 150|nb=yes}}</ref> The [[Fourier expansion]] of an ''L''<sup>2</sup> function ''f'' is
:<math>
\frac{a_0}{2} + \sum_{m=1}^{\infty}\left[a_m\cos\left(mx\right)+b_m\sin\left(mx\right)\right].
</math>

The coefficients ''a''<sub>''m''</sub> and ''b''<sub>''m''</sub> are called [[Fourier coefficient]]s of ''f'', and are calculated by the formulas<ref name="ReferenceA">{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=§4.5|nb=yes}}</ref>
:<math>a_m = \frac{1}{\pi} \int_0^{2 \pi} f(t) \cos (mt) \, dt</math>, <math>b_m = \frac{1}{\pi} \int_0^{2 \pi} f(t) \sin (mt) \, dt.</math>

In physical terms the function is represented as a [[Superposition principle|superposition]] of [[sine waves]] and the coefficients give information about the function's [[frequency spectrum]].<ref>{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=p. 57|nb=yes}}</ref> A complex-number form of Fourier series is also commonly used.<ref name="ReferenceA"/> The concrete formulae above are consequences of a more general [[duality (mathematics)|mathematical duality]] called [[Pontryagin duality]].<ref>{{Harvard citations|last1=Loomis  | year=1953|loc=Ch. VII|nb=yes}}</ref> Applied to the [[group (mathematics)|group]] '''R''', it yields the classical Fourier transform; an application in physics are [[reciprocal lattice]]s, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a [[lattice (group)|lattice]] encoding positions of [[atom]]s in [[crystal]]s.<ref>{{Harvard citations|last1=Ashcroft | last2=Mermin | year=1976|loc=Ch. 5|nb=yes}}</ref>

Fourier series are used to solve [[boundary value problem]]s in [[partial differential equation]]s.<ref>{{Harvard citations|last = Kreyszig|year = 1988|loc=p. 667|nb = yes}}</ref> In 1822, [[Joseph Fourier|Fourier]] first used this technique to solve the [[heat equation]].<ref>{{Harvard citations|last = Fourier|year = 1822|nb = yes}}</ref> A discrete version of the Fourier series can be used in [[Sampling (signal processing)|sampling]] applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points.<ref>{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=p. 67|nb=yes}}</ref> The set of coefficients is known as the [[discrete Fourier transform]] (DFT) of the given sample sequence. The DFT is one of the key tools of [[digital signal processing]], a field whose applications include [[radar]], [[speech encoding]], [[image compression]].<ref>{{Harvard citations|last1 = Ifeachor|last2 = Jervis|year=2002|loc=pp. 3–4, 11|nb=yes}}</ref> The [[JPEG]] image format is an application of the closely related [[discrete cosine transform]].<ref>{{Harvard citations|last = Wallace|year = Feb 1992|nb = yes}}</ref>

The [[fast Fourier transform]] is an algorithm for rapidly computing the discrete Fourier transform.<ref>{{Harvard citations|last1 = Ifeachor|last2 = Jervis|year=2002|loc=p. 132|nb=yes}}</ref> It is used not only for calculating the Fourier coefficients but, using the [[convolution theorem]], also for computing the [[convolution]] of two finite sequences.<ref>{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=§10.2|nb=yes}}</ref> They in turn are applied in [[digital filter]]s<ref>{{Harvard citations|last1 = Ifeachor|last2 = Jervis|year=2002|loc=pp. 307–310|nb=yes}}</ref> and as a rapid [[multiplication algorithm]] for polynomials and large integers ([[Schönhage–Strassen algorithm]]).<ref>{{Harvard citations|last1 = Gasquet|last2 = Witomski|year=1999|loc=§10.3|nb=yes}}</ref><ref>{{Harvard citations|last1=Schönhage|last2=Strassen|year=1971|nb=yes}}</ref>

===Differential geometry===
{{Main|Tangent space}}
[[Image:Image Tangent-plane.svg|right|thumb|200px|The tangent space to the [[2-sphere]] at some point is the infinite plane touching the sphere in this point.]]
The [[tangent plane]] to a surface at a point is naturally a vector space whose origin is identified with the point of contact.  The tangent plane is the best [[linear approximation]], or [[linearization]], of a surface at a point.<ref group=nb>That is to say {{harv|BSE-3|2001}}, the plane passing through the point of contact ''P'' such that the distance from a point ''P''<sub>1</sub> on the surface to the plane is [[little o|infinitesimally small]] compared to the distance from ''P''<sub>1</sub> to ''P'' in the limit as ''P''<sub>1</sub> approaches ''P'' along the surface.</ref> Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The ''tangent space'' is the generalization to higher-dimensional [[differentiable manifold]]s.<ref name="ReferenceB">{{Harvard citations|last = Spivak|year = 1999|loc = ch. 3|nb = yes}}</ref>

[[Riemannian manifold]]s are manifolds whose tangent spaces are endowed with a [[Riemannian metric|suitable inner product]].<ref>{{Harvard citations|last = Jost|year = 2005|nb = yes}}. See also [[Lorentzian manifold]].</ref> Derived therefrom, the [[Riemann curvature tensor]] encodes all [[curvature (mathematics)|curvatures]] of a manifold in one object, which finds applications in [[general relativity]], for example, where the [[Einstein curvature tensor]] describes the matter and energy content of [[space-time]].<ref>{{Harvard citations|last1 = Misner|last2=Thorne|last3=Wheeler|year = 1973|loc = ch. 1.8.7, p. 222 and ch. 2.13.5, p. 325|nb = yes}}</ref><ref>{{Harvard citations|last = Jost|year = 2005|loc = ch. 3.1|nb = yes}}</ref> The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify [[compact Lie group]]s.<ref>{{Harvard citations|last = Varadarajan|year = 1974|loc = ch. 4.3, Theorem 4.3.27|nb = yes}}</ref>

==Generalizations==

===Vector bundles===
{{Main|Vector bundle|Tangent bundle}}
[[Image:Mobius strip illus.svg|thumb|249px|right|A Möbius strip. Locally, it [[homeomorphism|looks like]] {{math|''U'' × '''R'''}}.]]
A ''vector bundle'' is a family of vector spaces parametrized continuously by a [[topological space]] ''X''.<ref name="ReferenceB"/> More precisely, a vector bundle over ''X'' is a topological space ''E'' equipped with a continuous map 
:π&thinsp;: ''E'' → ''X''
such that for every ''x'' in ''X'', the [[fiber (mathematics)|fiber]] π<sup>−1</sup>(''x'') is a vector space. The case dim {{math|1=''V'' = 1}} is called a [[line bundle]]. For any vector space ''V'', the projection {{math|''X'' × ''V'' → ''X''}} makes the product {{math|''X'' × ''V''}} into a [[trivial bundle|"trivial" vector bundle]]. Vector bundles over ''X'' are required to be [[locally]] a product of ''X'' and some (fixed) vector space ''V'': for every ''x'' in ''X'', there is a [[neighborhood (topology)|neighborhood]] ''U'' of ''x'' such that the restriction of π to π<sup>−1</sup>(''U'') is isomorphic<ref group=nb>That is, there is a [[homeomorphism]] from π<sup>−1</sup>(''U'') to {{math|''V'' × ''U''}} which restricts to linear isomorphisms between fibers.</ref> to the trivial bundle {{math|''U'' × ''V'' → ''U''}}. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space ''X'') be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle {{math|''X'' × ''V''}}). For example, the [[Möbius strip]] can be seen as a line bundle over the circle ''S''<sup>1</sup> (by [[homeomorphism#Examples|identifying open intervals with the real line]]). It is, however, different from the [[cylinder (geometry)|cylinder]] {{math|''S''<sup>1</sup> × '''R'''}}, because the latter is [[orientable manifold|orientable]] whereas the former is not.<ref>{{Harvard citations|last = Kreyszig|year = 1991|nb = yes|loc=§34, p. 108}}</ref>

Properties of certain vector bundles provide information about the underlying topological space. For example, the [[tangent bundle]] consists of the collection of [[tangent space]]s parametrized by the points of a differentiable manifold. The tangent bundle of the circle ''S''<sup>1</sup> is globally isomorphic to {{math|''S''<sup>1</sup> × '''R'''}}, since there is a global nonzero [[vector field]] on ''S''<sup>1</sup>.<ref group=nb>A line bundle, such as the tangent bundle of ''S''<sup>1</sup> is trivial if and only if there is a [[section (fiber bundle)|section]] that vanishes nowhere, see {{Harvard citations|last = Husemoller|year = 1994|nb = yes|loc=Corollary 8.3}}. The sections of the tangent bundle are just [[vector field]]s.</ref> In contrast, by the [[hairy ball theorem]], there is no (tangent) vector field on the [[2-sphere]] ''S''<sup>2</sup> which is everywhere nonzero.<ref>{{harvard citations|last1=Eisenberg|last2=Guy|year=1979|nb=yes}}</ref> [[K-theory]] studies the isomorphism classes of all vector bundles over some topological space.<ref>{{Harvard citations|last = Atiyah|year = 1989|nb = yes}}</ref> In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real [[division algebra]]s: '''R''', '''C''', the [[quaternion]]s '''H''' and the [[octonion]]s '''O'''.

The [[cotangent bundle]] of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the [[cotangent space]]. [[Section (fiber bundle)|Sections]] of that bundle are known as [[differential form|differential one-form]]s.

===Modules===
{{Main|Module (mathematics)|l1=Module}}
''Modules'' are to [[ring (mathematics)|rings]] what vector spaces are to fields: the same axioms, applied to a ring ''R'' instead of a field ''F'', yield modules.<ref>{{Harvard citations|last = Artin|year = 1991|nb = yes|loc=ch. 12}}</ref> The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have [[multiplicative inverse]]s. For example, modules need not have bases, as the '''Z'''-module (that is, [[abelian group]]) [[Modular arithmetic|'''Z'''/2'''Z''']] shows; those modules that do (including all vector spaces) are known as [[free module]]s. Nevertheless, a vector space can be compactly defined as a [[Module (mathematics)|module]] over a [[Ring (mathematics)|ring]] which is a [[Field (mathematics)|field]], with the elements being called vectors.  Some authors use the term ''vector space'' to mean modules over a [[division ring]].<ref>Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007.</ref>  The algebro-geometric interpretation of commutative rings via their [[spectrum of a ring|spectrum]] allows the development of concepts such as [[locally free module]]s, the algebraic counterpart to vector bundles.

===Affine and projective spaces===
{{Main|Affine space|Projective space}}
[[Image:Affine subspace.svg|thumb|right|200px|An [[affine space|affine plane]] (light blue) in '''R'''<sup>3</sup>. It is a two-dimensional subspace shifted by a vector '''x''' (red).]] 
Roughly, ''affine spaces'' are vector spaces whose origins are not specified.<ref>{{Harvard citations|last = Meyer|year = 2000|nb = yes|loc=Example 5.13.5, p. 436}}</ref> More precisely, an affine space is a set with a [[transitive group action|free transitive]] vector space [[Group action (mathematics)|action]]. In particular, a vector space is an affine space over itself, by the map
:{{math|''V'' × ''V'' → ''V'', ('''v''', '''a''') ↦ '''a''' + '''v'''}}.
If ''W'' is a vector space, then an affine subspace is a subset of ''W'' obtained by translating a linear subspace ''V'' by a fixed vector {{math|'''x''' ∈ ''W''}}; this space is denoted by {{math|'''x''' + ''V''}} (it is a [[coset]] of ''V'' in ''W'') and consists of all vectors of the form {{math|'''x''' + '''v'''}} for {{math|'''v''' ∈ ''V''.}} An important example is the space of solutions of a system of inhomogeneous linear equations
:''A'''''x''' = '''b'''
generalizing the homogeneous case {{math|1='''b''' = 0}} [[#equation3|above]].{{huh|date=February 2019}}<!-- b doesn't appear in equation3?? --><!-- Internal cross-references need to work for print versions of this article, not just the web version, so this should identify its referent in more specific words. --><ref>{{Harvard citations|last = Meyer|year = 2000|nb = yes|loc=Exercise 5.13.15–17, p. 442}}</ref> The space of solutions is the affine subspace {{math|'''x''' + ''V''}} where '''x''' is a particular solution of the equation, and ''V'' is the space of solutions of the homogeneous equation (the [[nullspace]] of ''A'').

The set of one-dimensional subspaces of a fixed finite-dimensional vector space ''V'' is known as ''projective space''; it may be used to formalize the idea of [[parallel (geometry)|parallel]] lines intersecting at infinity.<ref>{{Harvard citations|last = Coxeter|year = 1987|nb = yes}}</ref> [[Grassmannian manifold|Grassmannians]] and [[flag manifold]]s generalize this by parametrizing linear subspaces of fixed dimension ''k'' and [[flag (linear algebra)|flags]] of subspaces, respectively.

==See also==
*[[Vector (mathematics and physics)]], for a list of various kinds of vectors <!-- This item is deliberately left apart -->

{{Col-begin}}
{{Col-1-of-3}}
*[[Cartesian coordinate system]]
*[[Graded vector space]]
{{Col-2-of-3}}
*[[Metric space]]
*[[P-vector]]
*[[Riesz–Fischer theorem]]
{{Col-3-of-3}}
*[[Space (mathematics)]]
*[[Ordered vector space]]
{{col-end}}

== Notes ==
{{reflist|group=nb|3}}

==Citations==
{{Reflist|20em}}

==References==

===Algebra===
* {{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=978-0-89871-510-1 | year=1991}}
* {{Citation | last1=Blass | first1=Andreas | title=Axiomatic set theory (Boulder, Colorado, 1983) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Contemporary Mathematics | mr=763890 | year=1984 | volume=31 | chapter=Existence of bases implies the axiom of choice | pages=31–33}}
* {{Citation | last1=Brown | first1=William A. | title=Matrices and vector spaces | publisher=M. Dekker | location=New York | isbn=978-0-8247-8419-5 | year=1991}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987}}
* {{Lang Algebra}}
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=Algebra | edition=3rd | pages=193–222 | isbn=978-0-8218-1646-2 | year=1999}}
* {{Citation | last1=Meyer | first1=Carl D. | title=Matrix Analysis and Applied Linear Algebra | url=http://www.matrixanalysis.com/ | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-454-8 | year=2000}}
* {{Citation | last1=Roman | first1=Steven | title=Advanced Linear Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-24766-3 | year=2005 | volume=135}}
* {{Citation | last1=Spindler | first1=Karlheinz | title=Abstract Algebra with Applications: Volume 1: Vector spaces and groups
 | publisher=CRC | isbn=978-0-8247-9144-5 | year=1993}}
* {{Citation | last1=van der Waerden | first1=Bartel Leendert | author1-link=Bartel Leendert van der Waerden | title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=9th | isbn=978-3-540-56799-8 | year=1993|language=de}}

===Analysis===
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Topological vector spaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Elements of mathematics | isbn=978-3-540-13627-9 | year=1987}}
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Integration I | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-41129-1 | year=2004}}
* {{Citation | last1=Braun | first1=Martin | title=Differential equations and their applications: an introduction to applied mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-97894-9 | year=1993}}
* {{springer|last=BSE-3|title=Tangent plane|id=T/t092180}}
* {{Citation | last1=Choquet | first1=Gustave | author1-link=Gustave Choquet | title=Topology | publisher=[[Academic Press]] | location=Boston, MA | year=1966}}
* {{Citation | last1=Dennery | first1=Philippe | last2=Krzywicki | first2=Andre | title=Mathematics for Physicists | publisher=Courier Dover Publications | isbn=978-0-486-69193-0 | year=1996}}
* {{Citation | last1=Dudley | first1=Richard M. | title=Real analysis and probability | publisher=Wadsworth & Brooks/Cole Advanced Books & Software | location=Pacific Grove, CA | series=The Wadsworth & Brooks/Cole Mathematics Series | isbn=978-0-534-10050-6 | year=1989}}
* {{Citation | last1=Dunham | first1=William | title=The Calculus Gallery | publisher=[[Princeton University Press]] | isbn=978-0-691-09565-3 | year=2005}}
* {{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}
* {{Citation | last1=Folland | first1=Gerald B. | title=Fourier Analysis and Its Applications | publisher=Brooks-Cole | isbn=978-0-534-17094-3 | year=1992}}
* {{Citation | last = Gasquet | first = Claude | last2 = Witomski | first2 = Patrick | publication-date = 1999 | title = Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets | series = Texts in Applied Mathematics | location = New York | publisher = Springer-Verlag | isbn = 978-0-387-98485-8 | year = 1999}}
* {{Citation | last = Ifeachor | first = Emmanuel C. | last2 = Jervis | first2 = Barrie W. | publication-date = 2002 | title = Digital Signal Processing: A Practical Approach | edition = 2nd | location = Harlow, Essex, England | publisher = Prentice-Hall | isbn = 978-0-201-59619-9 | year = 2001}}
* {{Citation | last = Krantz | first = Steven G. | publication-date = 1999 | title = A Panorama of Harmonic Analysis | series = Carus Mathematical Monographs | location = Washington, DC | publisher = Mathematical Association of America | isbn = 978-0-88385-031-2 | year = 1999}}
* {{Citation | last = Kreyszig | first = Erwin | author-link = Erwin Kreyszig | publication-date = 1988 | title = Advanced Engineering Mathematics | edition = 6th | location = New York | publisher = John Wiley & Sons | isbn = 978-0-471-85824-9 | year = 1988}}
*{{Citation | last1=Kreyszig | first1=Erwin | author1-link=Erwin Kreyszig | title=Introductory functional analysis with applications | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-50459-7 | mr=992618 | year=1989}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Real analysis | publisher=[[Addison-Wesley]] | isbn=978-0-201-14179-5 | year=1983}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Real and functional analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94001-4 | year=1993}}
* {{Citation | last1=Loomis | first1=Lynn H. | title=An introduction to abstract harmonic analysis | publisher=D. Van Nostrand Company, Inc. | location=Toronto-New York–London | year=1953 | pages=x+190| hdl=2027/uc1.b4250788 | hdl-access=free }}
* {{Citation | author1-link = Helmut Schaefer | last1=Schaefer | first1=Helmut H. | last2=Wolff | first2=M.P. | title=Topological vector spaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98726-2 | year=1999}}
* {{Citation | last1=Treves | first1=François | title=Topological vector spaces, distributions and kernels | publisher=[[Academic Press]] | location=Boston, MA | year=1967}}

===Historical references===
* {{Citation | last1=Banach | first1=Stefan | author1-link=Stefan Banach | title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations) | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf | year=1922 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=3| pages=133–181 |language=fr| doi=10.4064/fm-3-1-133-181 }}
* {{Citation | last1=Bolzano | first1=Bernard | author1-link=Bernard Bolzano | title=Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) | url=http://dml.cz/handle/10338.dmlcz/400338 | year=1804|language=de}}
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Éléments d'histoire des mathématiques (Elements of history of mathematics) | publisher=Hermann | location=Paris | year=1969|language=fr}}
* {{Citation | last1=Dorier | first1=Jean-Luc | title=A general outline of the genesis of vector space theory | mr=1347828 | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=227–261 | doi=10.1006/hmat.1995.1024| url=http://archive-ouverte.unige.ch/unige:16642 | doi-access=free }}
* {{Citation | last1=Fourier | first1=Jean Baptiste Joseph | author1-link=Joseph Fourier | title=Théorie analytique de la chaleur | url=https://books.google.com/?id=TDQJAAAAIAAJ | publisher=Chez Firmin Didot, père et fils | year=1822|language=fr}}
* {{Citation | last1=Grassmann | first1=Hermann | author1-link=Hermann Grassmann | title=Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik | url=https://books.google.com/?id=bKgAAAAAMAAJ&pg=PA1 | year=1844 | publisher=O. Wigand|language=de}}, reprint: {{Citation | title=Extension Theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2031-5 | year=2000 | author=Hermann Grassmann. Translated by Lloyd C. Kannenberg. | editor1-last=Kannenberg | editor1-first=L.C.}}
* {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=Lectures on Quaternions | url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 | publisher=Royal Irish Academy | year=1853}}
* {{Citation | last1=Möbius | first1=August Ferdinand | author1-link=August Ferdinand Möbius | title=Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) | url=http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 | archive-url=https://web.archive.org/web/20061123192612/http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 | url-status=dead | archive-date=2006-11-23 | year=1827|language=de}}
* {{Citation | last1=Moore | first1=Gregory H. | title=The axiomatization of linear algebra: 1875–1940 | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=262–303 | doi=10.1006/hmat.1995.1025| doi-access=free }}
* {{Citation | last1=Peano | first1=Giuseppe | author1-link=Giuseppe Peano | title=Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva | year=1888 | location=Turin|language=it}}

===Further references===
* {{Citation | last1=Ashcroft | first1=Neil | last2=Mermin | first2=N. David | author1-link=Neil Ashcroft | author2-link=N. David Mermin | title=Solid State Physics | publisher=Thomson Learning | location=Toronto | isbn=978-0-03-083993-1 | year=1976 | url-access=registration | url=https://archive.org/details/solidstatephysic00ashc }}
* {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=K-theory | publisher=[[Addison-Wesley]] | edition=2nd | series=Advanced Book Classics | isbn=978-0-201-09394-0 | mr=1043170 | year=1989}}
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Elements of Mathematics : Algebra I Chapters 1-3 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-64243-5 | year=1998}}
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=General Topology. Chapters 1-4 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-64241-1 | year=1989}}
* {{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Projective Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-96532-1 | year=1987}}
* {{Citation | last1=Eisenberg | first1=Murray | last2=Guy | first2=Robert | title=A proof of the hairy ball theorem | year=1979 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | volume=86 | issue=7 | pages=572–574 | doi=10.2307/2320587 | jstor=2320587 }}
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94269-8 | mr=1322960 | year=1995 | volume=150}}
* {{Citation | last1=Goldrei | first1=Derek | title=Classic Set Theory: A guided independent study | publisher=[[Chapman and Hall]] | location=London | edition=1st | isbn=978-0-412-60610-6 | year=1996}}
* {{Citation | last=Griffiths | first=David J. | author-link=David J. Griffiths |title=Introduction to Quantum Mechanics | year=1995 |publisher=[[Prentice Hall]] |location=Upper Saddle River, NJ |isbn=978-0-13-124405-4}}
* {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=Finite-dimensional vector spaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90093-3 | year=1974}}
* {{Citation | last1=Halpern | first1=James D. | title=Bases in Vector Spaces and the Axiom of Choice |date=Jun 1966 |journal=[[Proceedings of the American Mathematical Society]] | volume=17 | issue=3 | pages=670–673 | doi=10.2307/2035388 | jstor=2035388 | doi-access=free }}
* {{Citation | last1=Hughes-Hallett | first1=Deborah | last2=McCallum | first2=William G. | last3=Gleason | first3=Andrew M. | title=Calculus : Single and Multivariable | date=2013 | publisher=[[John Wiley & Sons]] | isbn=978-0470-88861-2 | edition=6}}
* {{Citation | last1=Husemoller | first1=Dale | title=Fibre Bundles | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-94087-8 | year=1994}}
* {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=4th | isbn=978-3-540-25907-7 | year=2005}}
* {{Citation | last1=Kreyszig | first1=Erwin | author1-link=Erwin Kreyszig | title=Differential geometry | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-66721-8 | year=1991 | pages=xiv+352}}
* {{Citation |last=Kreyszig |first=Erwin |title=Advanced Engineering Mathematics |edition=8th |year=1999 |publisher=[[John Wiley & Sons]] | location=New York |isbn=978-0-471-15496-9}}
* {{Citation | last1=Luenberger | first1=David | title=Optimization by vector space methods | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-18117-0 | year=1997}}
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=Categories for the Working Mathematician | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998| title-link=Categories for the Working Mathematician }}
* {{Citation | last1=Misner | first1=Charles W. | author1-link=Charles W. Misner | last2=Thorne | first2=Kip | author2-link=Kip Thorne | last3=Wheeler | first3=John Archibald | author3-link=John Archibald Wheeler | title=Gravitation | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973| title-link=Gravitation (book) }}
* {{Citation | last1=Naber | first1=Gregory L. | title=The geometry of Minkowski spacetime | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-43235-9 | mr=2044239 | year=2003}}
* {{Citation | last1=Schönhage | first1=A. | author1-link=Arnold Schönhage | last2=Strassen | first2=Volker | author2-link=Volker Strassen | title=Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers) | year=1971 | journal=Computing | issn=0010-485X | volume=7 | issue=3–4 | pages=281–292 | doi=10.1007/bf02242355|language=de}}
* {{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=A Comprehensive Introduction to Differential Geometry (Volume Two) | publisher=Publish or Perish | location=Houston, TX | year=1999}}
* {{Citation | last1=Stewart | first1=Ian | authorlink=Ian Stewart (mathematician) | title=Galois Theory | year=1975 | publisher=[[Chapman and Hall]] | isbn=978-0-412-10800-6 | location=London | series=[[Chapman and Hall]] Mathematics Series }}
* {{Citation | last1=Varadarajan | first1=V. S. | title=Lie groups, Lie algebras, and their representations | publisher=[[Prentice Hall]] | isbn=978-0-13-535732-3 | year=1974}}
* {{Citation | last1=Wallace | first1=G.K. | title=The JPEG still picture compression standard | year=Feb 1992 | journal=IEEE Transactions on Consumer Electronics | issn=0098-3063 | volume=38 | issue=1 | pages=xviii–xxxiv | doi=10.1109/30.125072| url=http://www.csc.ncsu.edu/faculty/rhee/export/papers/TheJPEGStillPictureCompressionStandard.pdf | citeseerx=10.1.1.318.4292 }}
* {{Weibel IHA}}

== External links ==
{{Wikibooks|Linear Algebra|Definition and Examples of Vector Spaces|Real vector spaces}}
{{Wikibooks|Linear Algebra|Vector spaces}}
* {{springer|title=Vector space|id=p/v096520}}

{{linear algebra}}

{{good article}}

{{DEFAULTSORT:Vector Space}}
[[Category:Concepts in physics]]
[[Category:Group theory]]
[[Category:Mathematical structures]]
[[Category:Vectors (mathematics and physics)]]
[[Category:Vector spaces]]