Editing Vector space (section)

==Vector spaces with additional structure==
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field 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>{{sfn|Rudin|1991|loc=p.3}}

A vector space may be given a [[partial order]] <math>\,\leq,\,</math> under which some vectors can be compared.{{sfn|Schaefer|Wolff|1999|loc = pp. 204–205}} For example, <math>n</math>-dimensional real space <math>\mathbf{R}^n</math> 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
<math display=block>f = f^+ - f^-.</math>
where <math>f^+</math> denotes the positive part of <math>f</math> and <math>f^-</math> the negative part.{{sfn|Bourbaki|2004|loc=ch. 2, p. 48}}

===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 {{nowrap|<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 {{nowrap|<math display="inline">|\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.{{sfn|Roman|2005|loc=ch. 9}}

Coordinate space <math>F^n</math> can be equipped with the standard [[dot product]]:
<math display=block>\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n.</math>
In <math>\mathbf{R}^2,</math> this reflects the common notion of the angle between two vectors <math>\mathbf{x}</math> and <math>\mathbf{y},</math> by the [[law of cosines]]:
<math display=block>\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]]: <math>\mathbf{R}^4</math> endowed with the Lorentz product{{sfn|Naber|2003|loc=ch. 1.2}}
<math display=block>\lang \mathbf x | \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4.</math>
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]]. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written <math display=block>\lang \mathbf x | \mathbf y \rang =  - x_0 y_0+x_1 y_1 + x_2 y_2 + x_3 y_3.</math>

===Topological vector spaces===
{{Main|Topological vector space}}
Convergence questions are treated by considering vector spaces <math>V</math> carrying a compatible [[topological space|topology]], a structure that allows one to talk about elements being [[neighborhood (topology)|close to each other]].{{sfnm
 | 1a1 = Treves | 1y = 1967 
 | 2a1 = Bourbaki | 2y = 1987
}} Compatible here means that addition and scalar multiplication have to be [[continuous map]]s. Roughly, if <math>\mathbf{x}</math> and <math>\mathbf{y}</math> in <math>V</math>, and <math>a</math> in <math>F</math> vary by a bounded amount, then so do <math>\mathbf{x} + \mathbf{y}</math> and <math>a \mathbf{x}.</math><ref group=nb>This requirement implies that the topology gives rise to a [[uniform structure]], {{harvtxt|Bourbaki|1989}}, loc = ch. II.</ref> To make sense of specifying the amount a scalar changes, the field <math>F</math> also has to carry a topology in this context; a common choice is the reals or the complex numbers.

In such ''topological vector spaces'' one can consider [[series (mathematics)|series]] of vectors. The [[infinite sum]]
<math display=block>\sum_{i=1}^\infty f_i ~=~ \lim_{n \to \infty} f_1 + \cdots + f_n</math>
denotes the [[limit of a sequence|limit]] of the corresponding finite partial sums of the sequence <math>f_1, f_2, \ldots</math> of elements of <math>V.</math> For example, the <math>f_i</math> could be (real or complex) functions belonging to some [[function space]] <math>V,</math> 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.{{sfn|Schaefer|Wolff|1999|loc=p. 7}}

[[Image:Vector norms2.svg|class=skin-invert-image|thumb|right|250px|[[Unit ball|Unit "spheres"]] in <math>\mathbf{R}^2</math> consist of plane vectors of norm 1.  Depicted are the unit spheres in different [[Lp norm|<math>p</math>-norm]]s, for <math>p = 1, 2,</math> and <math>\infty.</math> 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 <math>[0, 1],</math> equipped with the [[topology of uniform convergence]] is not complete because any continuous function on <math>[0, 1]</math> 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 <math>[0, 1]</math> 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 <math>\mathbf{v}_n</math> converges to <math>\mathbf{v}</math> if and only if
<math display=block>\lim_{n \to \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]]—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.{{sfn|Choquet|1966|loc=Proposition III.7.2}} The image at the right shows the equivalence of the <math>1</math>-norm and <math>\infty</math>-norm on <math>\mathbf{R}^2:</math> 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 \to W,</math> maps between topological vector spaces are required to be continuous.{{sfn|Treves|1967|loc=p. 34–36}} In particular, the <span id=label2>(topological) dual space <math>V^*</math> consists of continuous functionals <math>V \to \mathbf{R}</math> (or to <math>\mathbf{C}</math>). The fundamental [[Hahn–Banach theorem]] is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.{{sfn|Lang|1983|loc=Cor. 4.1.2, p. 69}}</span>

====Banach spaces====
{{Main|Banach space}}
''[[Banach space]]s'', introduced by [[Stefan Banach]], are complete normed vector spaces.{{sfn|Treves|1967|loc=ch. 11}}

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 display=block>\|\mathbf{x}\|_\infty := \sup_i |x_i| \qquad \text{ for } p = \infty, \text{ and }</math>
<math display=block>\|\mathbf{x}\|_p := \left(\sum_i |x_i|^p\right)^\frac{1}{p} \qquad \text{ for } p < \infty.</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 display=block>\|\mathbf{x}_n\|_\infty = \sup (2^{-n}, 0) = 2^{-n} \to 0,</math> 
but 
<math display=block>\|\mathbf{x}_n\|_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 : \Omega \to \Reals</math> are endowed with a norm that replaces the above sum by the [[Lebesgue integral]] 
<math display=block>\|f\|_p := \left(\int_{\Omega} |f(x)|^p \, {d\mu(x)}\right)^\frac{1}{p}.</math>

The space of [[integrable function]]s on a given [[domain of a function|domain]] <math>\Omega</math> (for example an interval) satisfying <math>\|f\|_p < \infty,</math> and equipped with this norm are called [[Lp space|Lebesgue spaces]], denoted <math>L^{\;\!p}(\Omega).</math><ref group="nb">The [[triangle inequality]] for <math>\|f + g\|_p \leq \|f\|_p + \|g\|_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.{{sfn|Treves|1967|loc=Theorem 11.2, p. 102}} (If one uses the [[Riemann integral]] instead, the space is {{em|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.", {{harvtxt|Dudley|1989}}, §5.3, p. 125.</ref>) Concretely this means that for any sequence of Lebesgue-integrable functions <math>f_1, f_2, \ldots, f_n, \ldots</math> with <math>\|f_n\|_p < \infty,</math> satisfying the condition
<math display=block>\lim_{k,\ n \to \infty} \int_{\Omega} \left|f_k(x) - f_n(x)\right|^p \, {d\mu(x)} = 0</math>
there exists a function <math>f(x)</math> belonging to the vector space <math>L^{\;\!p}(\Omega)</math> such that
<math display=block>\lim_{k \to \infty} \int_{\Omega} \left|f(x) - f_k(x)\right|^p \, {d\mu(x)} = 0.</math>

Imposing boundedness conditions not only on the function, but also on its [[derivative]]s leads to [[Sobolev space]]s.{{sfn|Evans|1998|loc = ch. 5}}
{{Clear}}

====Hilbert spaces====
{{Main|Hilbert space}}
[[Image:Periodic identity function.gif|class=skin-invert-image|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]].{{sfn|Treves|1967|loc=ch. 12}} The Hilbert space <math>L^2(\Omega),</math> with inner product given by
<math display=block>\langle f\ , \ g \rangle = \int_\Omega f(x) \overline{g(x)} \, dx,</math>
where <math>\overline{g(x)}</math> denotes the [[complex conjugate]] of <math>g(x),</math>{{sfn|Dennery|Krzywicki|1996|loc = p.190}}<ref group=nb>For <math>p \neq 2,</math> <math>L^p(\Omega)</math> 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 <math>f_n</math> with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the [[Taylor approximation]], established an approximation of [[differentiable function]]s <math>f</math> by polynomials.{{sfn|Lang|1993|loc = Th. XIII.6, p. 349}} By the [[Stone–Weierstrass theorem]], every continuous function on <math>[a, b]</math> can be approximated as closely as desired by a polynomial.{{sfn|Lang|1993|loc = Th. III.1.1}} A similar approximation technique by [[trigonometric function]]s is commonly called [[Fourier expansion]], and is much applied in engineering. 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 <math>H,</math> 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 <math>H,</math> 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 of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is 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]].{{sfn|Choquet|1966|loc = Lemma III.16.11}} 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.{{sfn|Kreyszig|1999|loc=Chapter 11}} 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.{{sfn|Griffiths|1995|loc=Chapter 1}} 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>{{sfn|Lang|1993|loc =ch. XVII.3}}

===Algebras over fields===
{{Main|Algebra over a field|Lie algebra}}
[[Image:Rectangular hyperbola.svg|class=skin-invert-image|right|thumb|250px|A [[hyperbola]], given by the equation <math>x \cdot y = 1.</math> The [[coordinate ring]] of functions on this hyperbola is given by <math>\mathbf{R}[x, y] / (x \cdot y - 1),</math> an infinite-dimensional vector space over <math>\mathbf{R}.</math>]]
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'' (or ''F''-algebra if the field ''F'' is specified).{{sfn|Lang|2002|loc=ch. III.1, p. 121}}

For example, the set of all [[polynomial]]s <math>p(t)</math> forms an algebra known as the [[polynomial ring]]: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their [[quotient ring|quotients]] form the basis of [[algebraic geometry]], because they are [[coordinate ring|rings of functions of algebraic geometric objects]].{{sfn|Eisenbud|1995|loc=ch. 1.6}}

Another crucial example are ''[[Lie algebra]]s'', which are neither commutative nor associative, but the failure to be so is limited by the constraints (<math>[x, y]</math> denotes the product of <math>x</math> and <math>y</math>):
* <math>[x, y] = - [y, x]</math> ([[anticommutativity]]), and
* <math>[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0</math> ([[Jacobi identity]]).{{sfn|Varadarajan|1974}}
Examples include the vector space of <math>n</math>-by-<math>n</math> matrices, with <math>[x, y] = x y - y x,</math> the [[commutator]] of two matrices, and <math>\mathbf{R}^3,</math> endowed with the [[cross product]].

The [[tensor algebra]] <math>\operatorname{T}(V)</math> is a formal way of adding products to any vector space <math>V</math> to obtain an algebra.{{sfn|Lang|2002|loc=ch. XVI.7}} As a vector space, it is spanned by symbols, called simple [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_n,</math>
where the [[rank of a tensor|degree]] <math>n</math> 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 in the above section on [[#Tensor product|tensor products]]. In general, there are no relations between <math>\mathbf{v}_1 \otimes \mathbf{v}_2</math> and <math>\mathbf{v}_2 \otimes \mathbf{v}_1.</math> Forcing two such elements to be equal leads to the [[symmetric algebra]], whereas forcing <math>\mathbf{v}_1 \otimes \mathbf{v}_2 = - \mathbf{v}_2 \otimes \mathbf{v}_1</math> yields the [[exterior algebra]].{{sfn|Lang|2002|loc=ch. XVI.8}}