Vector space

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and 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 v + w.[1] 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 a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction instead.

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

A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows:

${\displaystyle (x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})}$

and

${\displaystyle a(x,y)=(ax,ay)}$ .

The first example above reduces to this one, if the arrows are represented by the pair of Cartesian coordinates of their endpoints.

In this article, vectors are represented in boldface to distinguish them from scalars.[nb 1]

A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. In the following, V × V denotes the Cartesian product of V with itself, and → denotes a mapping from one set to another.

• The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (The resultant vector is also an element of the set V.)
• The second operation, called scalar multiplication · : F × V → V， takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set 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 V are commonly called vectors. Elements of F are commonly called scalars. Common symbols for denoting vector spaces include U, V, and W.[1]

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 pairs of real numbers, respectively.

To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[2] These are listed in the table below, where u, v and w denote arbitrary vectors in V, and a and b denote scalars in F.[3][4]

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:

(x_v, y_v) + (x_w, y_w) = (x_w, y_w) + (x_v, y_v).

Likewise, in the geometric example of vectors as arrows, 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 verified 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

${\displaystyle {\begin{aligned}\mathbf {v} -\mathbf {w} &=\mathbf {v} +(-\mathbf {w} )\\{\frac {\mathbf {v} }{a}}&={\frac {1}{a}}\mathbf {v} \end{aligned}}}$.

When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers 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 F. The notion is then known as an F-vector space or a vector space over F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations.[nb 3] For example, rational numbers form a field.

In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, in general vector spaces, there is no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see § Vector spaces with additional structure below for more.

Vector addition and scalar multiplication are operations, satisfying the closure property: u + v and av are in V for all a in F, and u, v in V. Some older sources mention these properties as separate axioms.[5]

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 F-module structure. In other words, there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (f(a))(v).[6]

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 0 of V and the additive inverse −v of any vector v are unique. Further properties follow by employing also the distributive law for the scalar multiplication, for example av equals 0 if and only if a equals 0 or v equals 0.

The simplest example of a vector space over a field F is the field itself, equipped with its standard addition and multiplication. More generally, all n-tuples (sequences of length n)

(a₁, a₂, ..., a_n)

of elements of F form a vector space that is usually denoted Fⁿ and called a coordinate space.[15] The case n = 1 is the above-mentioned simplest example, in which the field F is also regarded as a vector space over itself. The case F = R and n = 2 was discussed in the introduction above.

The set of complex numbers C, that is, numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ⋅ (x + iy) = (c ⋅ x) + i(c ⋅ y) for real numbers x, y, a, b and 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 x + i y as representing the ordered pair (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 extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition operations of F.[16] For example, the complex numbers are a vector space over R, and the field extension ${\displaystyle \mathbf {Q} (i{\sqrt {5}})}$ is a vector space over Q.

Addition of functions: The sum of the sine and the exponential function is ${\displaystyle \sin +\exp$ :\mathbb {R} \to \mathbb {R} } with ${\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}$

Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (f + g) given by

(f + g)(w) = f(w) + g(w),

and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the real line or an interval, or other subsets of R. Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.[17] Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of functional analysis, see below. Algebraic constraints also yield vector spaces: the vector space F[x] is given by polynomial functions:

f(x) = r₀ + r₁x + ... + r_n−1xⁿ⁻¹ + r_nxⁿ, where the coefficients r₀, ..., r_n are in F.[18]

Systems of homogeneous linear equations are closely tied to vector spaces.[19] For example, the solutions of

a	+	3b	+	c	= 0
4a	+	2b	+	2c	= 0

are given by triples with arbitrary a, b = a/2, and c = −5a/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. Matrices can be used to condense multiple linear equations as above into one vector equation, namely

Ax = 0,

where A = ${\displaystyle {\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}$ is the matrix containing the coefficients of the given equations, x is the vector (a, b, c), Ax denotes the matrix product, and 0 = (0, 0) is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example,

f′′(x) + 2f′(x) + f(x) = 0

yields f(x) = a e^−x + bx e^−x, where a and b are arbitrary constants, and e^x is the natural exponential function.

A typical matrix

Matrices are a useful notion to encode linear maps.[34] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix A gives rise to a linear map from Fⁿ to F^m, by the following

${\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right)}$, where ${\displaystyle \sum }$ denotes summation,

or, using the matrix multiplication of the matrix A with the coordinate vector x:

x ↦ Ax.

Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.[35]

The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r₁, r₂, and r₃.

The determinant det (A) of a square matrix 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.[36] The linear transformation of Rⁿ corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

Endomorphisms, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.[nb 5][37] Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to

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 λ, called the characteristic polynomial of f.[38] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space 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.[39][nb 6] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.

A line passing through the origin (blue, thick) in R³ is a linear subspace. It is the intersection of two 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.[40][nb 7] 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 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 combinations of elements of S.[41]

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 n, a vector hyperplane is thus a subspace of dimension n – 1.

The counterpart to subspaces are quotient vector spaces.[42] Given any subspace W ⊂ V, the quotient space V/W ("V modulo W") is defined as follows: as a set, it consists of v + W = {v + w : w ∈ W}, where v is an arbitrary vector in V. The sum of two such elements v₁ + W and v₂ + W is (v₁ + v₂) + W, and scalar multiplication is given by a · (v + W) = (a · v) + W. The key point in this definition is that v₁ + W = v₂ + W if and only if the difference of v₁ and v₂ lies in W.[nb 8] This way, the quotient space "forgets" information that is contained in the subspace W.

The kernel ker(f) of a linear map f : V → W consists of vectors v that are mapped to 0 in W.[43] The kernel and the image im(f) = {f(v) : v ∈ V} are subspaces of V and W, respectively.[44] 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) that behaves much like the category of abelian groups.[45] 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 groups.

An important example is the kernel of a linear map x ↦ Ax for some fixed matrix A, as above. The kernel of this map is the subspace of vectors x such that Ax = 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

${\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0}$, where the coefficients a_i are functions in x, too.

In the corresponding map

${\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}}}$,

the derivatives of the function f appear linearly (as opposed to f′′(x)², for example). Since differentiation is a linear procedure (that is, (f + g)′ = f′ + g′ and (c·f)′ = c·f′ for a constant c) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D(f) = 0 form a vector space (over R or C).

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 ${\displaystyle \textstyle {\prod _{i\in I}V_{i}}}$ of a family of vector spaces V_i consists of the set of all tuples (v_i)_{i ∈ I}, which specify for each index i in some index set I an element v_i of V_i.[46] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum ${\displaystyle \oplus _{i\in I}V_{i}}$ (also called coproduct and denoted ${\displaystyle \textstyle {\coprod _{i\in I}V_{i}}}$), 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.

The tensor product V ⊗_F W, or simply 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 g : V × W → X is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map 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 tensors

v₁ ⊗ w₁ + v₂ ⊗ w₂ + ... + v_n ⊗ w_n,

subject to the rules

a · (v ⊗ w) = (a · v) ⊗ w = v ⊗ (a · w), where a is a scalar,

(v₁ + v₂) ⊗ w = v₁ ⊗ w + v₂ ⊗ w, and

v ⊗ (w₁ + w₂) = v ⊗ w₁ + v ⊗ w₂.[47]

Commutative diagram depicting the universal property of the tensor product.

These rules ensure that the map f from the V × W to V ⊗ W that maps a tuple (v, w) to v ⊗ w is bilinear. The universality states that given any vector space X and any bilinear map g : V × W → X, there exists a unique map u, shown in the diagram with a dotted arrow, whose composition with f equals g: u(v ⊗ w) = g(v, w).[48] 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.

"Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted ${\displaystyle |\mathbf {v} |}$ and ${\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle }$, respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm ${\displaystyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}}$. Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.[51]

Coordinate space Fⁿ can be equipped with the standard dot product:

${\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}$

In R², this reflects the common notion of the angle between two vectors x and y, by the law of cosines:

${\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}$

Because of this, two vectors satisfying ${\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}$ are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R⁴ endowed with the Lorentz product

${\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}$[52]

In contrast to the standard dot product, it is not positive definite: ${\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }$ also takes negative values, for example for ${\displaystyle \mathbf {x} =(0,0,0,1)}$. Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity.

Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other.[53][54] Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in V, and a in F vary by a bounded amount, then so do x + y and ax.[nb 9] 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 of vectors. The infinite sum

${\displaystyle \sum _{i=0}^{\infty }f_{i}}$

denotes the limit of the corresponding finite partial sums of the sequence (f_i)_i∈N of elements of V. For example, the f_i could be (real or complex) functions belonging to some function space V, in which case the series is a function series. The 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.

Unit "spheres" in R² consist of plane vectors of norm 1. Depicted are the unit spheres in different p-norms, 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 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.[55] In contrast, the space of all continuous functions on [0,1] with the same topology is complete.[56] A norm gives rise to a topology by defining that a sequence of vectors v_n converges to v if and only if

${\displaystyle {\text{lim}}_{n\rightarrow \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}$

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.[57] The image at the right shows the equivalence of the 1-norm and ∞-norm on R²: 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 functionals) V → W, maps between topological vector spaces are required to be continuous.[58] In particular, the (topological) dual space V^∗ consists of continuous functionals V → R (or to C). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[59]

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[60]

A first example is the vector space ${\displaystyle \ell ^{p}}$ consisting of infinite vectors with real entries ${\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}$ whose ${\displaystyle p}$-norm ${\displaystyle (1\leq {p}\leq \infty )}$ given by

${\displaystyle \left\Vert \mathbf {x} \right\Vert _{p}:=\left(\sum _{i}\left\vert x_{i}\right\vert ^{p}\right)^{\frac {1}{p}}}$ for ${\displaystyle p<\infty }$   and   ${\displaystyle \left\Vert \mathbf {x} \right\Vert _{\infty }:={\text{sup}}_{i}\left\vert x_{i}\right\vert }$.

The topologies on the infinite-dimensional space ${\displaystyle \ell ^{p}}$ are inequivalent for different ${\displaystyle p}$. For example, the sequence of vectors ${\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right)}$, in which the first ${\displaystyle 2^{n}}$ components are ${\displaystyle 2^{-n}}$ and the following ones are ${\displaystyle 0}$, converges to the zero vector for ${\displaystyle p=\infty }$, but does not for ${\displaystyle p=1}$:

${\displaystyle \left\Vert \mathbf {x} _{n}\right\Vert _{_{\infty }}=\sup(2^{-n},0)=2^{-n}\rightarrow 0}$, but ${\displaystyle \left\Vert \mathbf {x} _{n}\right\Vert _{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}$

More generally than sequences of real numbers, functions ${\displaystyle f\colon \Omega \to \mathbb {R} }$ are endowed with a norm that replaces the above sum by the Lebesgue integral

${\displaystyle \left\Vert {f}\right\Vert _{p}:=\left(\int _{\Omega }\left\vert {f}\left(x\right)\right\vert ^{p}\,{d\mu \left(x\right)}\right)^{\frac {1}{p}}.}$

The space of integrable functions on a given domain ${\displaystyle \Omega }$ (for example an interval) satisfying ${\displaystyle \left\Vert {f}\right\Vert _{p}<\infty }$, and equipped with this norm are called Lebesgue spaces, denoted ${\displaystyle L^{\;\!p}\left(\Omega \right)}$.[nb 10]

These spaces are complete.[61] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.[nb 11]) Concretely this means that for any sequence of Lebesgue-integrable functions   ${\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }$   with ${\displaystyle \left\Vert {f}_{n}\right\Vert _{p}<\infty }$, satisfying the condition

${\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left\vert {f}_{k}(x)-{f}_{n}(x)\right\vert ^{p}\,{d\mu \left(x\right)}=0}$

there exists a function ${\displaystyle {f}\left(x\right)}$ belonging to the vector space ${\displaystyle L^{\;\!p}\left(\Omega \right)}$ such that

${\displaystyle \lim _{k\to \infty }\int _{\Omega }\left\vert {f}\left(x\right)-{f}_{k}\left(x\right)\right\vert ^{p}\,{d\mu \left(x\right)}=0.}$

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[62]

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.[63] The Hilbert space L²(Ω), with inner product given by

${\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}$

where ${\displaystyle {\overline {g(x)}}}$ denotes the complex conjugate of g(x),[64][nb 12] is a key case.

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions f_n 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 functions f by polynomials.[65] By the Stone–Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.[66] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. 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 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.[nb 13] 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 basis of orthogonal vectors.[67] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space.

The solutions to various differential equations 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.[68] 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 wavefunctions.[69] Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[70]

A hyperbola, given by the equation x ⋅ y = 1. The coordinate ring of functions on this hyperbola is given by R[x, y] / (x · y − 1), an infinite-dimensional vector space over 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.[71] 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 algebras which are both Banach spaces and algebras.

Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects.[72]

Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ([x, y] denotes the product of x and y):

• [x, y] = −[y, x] (anticommutativity), and
• [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (Jacobi identity).[73]

Examples include the vector space of n-by-n matrices, with [x, y] = xy − yx, the commutator of two matrices, and R³, 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.[74] As a vector space, it is spanned by symbols, called simple tensors

v₁ ⊗ v₂ ⊗ ⋯ ⊗ v_n, where the degree 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 above. In general, there are no relations between v₁ ⊗ v₂ and v₂ ⊗ v₁. Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing v₁ ⊗ v₂ = − v₂ ⊗ v₁ yields the exterior algebra.[75]

When a field, F is explicitly stated, a common term used is F-algebra.

A distribution (or generalized function) is a linear map assigning a number to each "test" function, typically a smooth function with compact support, in a continuous way: in the above terminology the space of distributions is the (continuous) dual of the test function space.[78] 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:

${\displaystyle I(f)=\int _{\Omega }f(x)\,dx.}$

When 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 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 functions and fundamental solutions 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).[79]

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 Fourier series, a technique much used in physics and engineering.[nb 14][80] The underlying vector space is usually the Hilbert space L²(0, 2π), for which the functions sin mx and cos mx (m an integer) form an orthogonal basis.[81] The Fourier expansion of an L² function f is

${\displaystyle {\frac {a_{0}}{2}}+\sum _{m=1}^{\infty }\left[a_{m}\cos \left(mx\right)+b_{m}\sin \left(mx\right)\right].}$

The coefficients a_m and b_m are called Fourier coefficients of f, and are calculated by the formulas[82]

${\displaystyle a_{m}={\frac {1}{\pi }}\int _{0}^{2\pi }f(t)\cos(mt)\,dt}$, ${\displaystyle b_{m}={\frac {1}{\pi }}\int _{0}^{2\pi }f(t)\sin(mt)\,dt.}$

In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function's frequency spectrum.[83] A complex-number form of Fourier series is also commonly used.[82] The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality.[84] Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals.[85]

Fourier series are used to solve boundary value problems in partial differential equations.[86] In 1822, Fourier first used this technique to solve the heat equation.[87] A discrete version of the Fourier series can be used in 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.[88] 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.[89] The JPEG image format is an application of the closely related discrete cosine transform.[90]

The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform.[91] It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences.[92] They in turn are applied in digital filters[93] and as a rapid multiplication algorithm for polynomials and large integers (Schönhage–Strassen algorithm).[94][95]

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.[nb 15] 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 manifolds.[96]

Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.[97] Derived therefrom, the Riemann curvature tensor encodes all 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.[98][99] 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 groups.[100]

A Möbius strip. Locally, it looks like U × R.

A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[96] More precisely, a vector bundle over X is a topological space E equipped with a continuous map

π : E → X

such that for every x in X, the fiber π⁻¹(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × V → X makes the product X × V into a "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 U of x such that the restriction of π to π⁻¹(U) is isomorphic[nb 16] to the trivial bundle 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 X × V). For example, the Möbius strip can be seen as a line bundle over the circle S¹ (by identifying open intervals with the real line). It is, however, different from the cylinder S¹ × R, because the latter is orientable whereas the former is not.[101]

Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S¹ is globally isomorphic to S¹ × R, since there is a global nonzero vector field on S¹.[nb 17] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S² which is everywhere nonzero.[102] K-theory studies the isomorphism classes of all vector bundles over some topological space.[103] In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions 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. Sections of that bundle are known as differential one-forms.

Modules are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules.[104] The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors. Some authors use the term vector space to mean modules over a division ring.[105] The algebro-geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles.

An affine plane (light blue) in R³. It is a two-dimensional subspace shifted by a vector x (red).

Roughly, affine spaces are vector spaces whose origins are not specified.[106] More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map

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 x ∈ W; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for v ∈ V. An important example is the space of solutions of a system of inhomogeneous linear equations

Ax = b

generalizing the homogeneous case b = 0 above.[107] The space of solutions is the affine subspace 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 lines intersecting at infinity.[108] Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1
• Blass, Andreas (1984), "Existence of bases implies the axiom of choice", Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, 31, Providence, R.I.: American Mathematical Society, pp. 31–33, MR 0763890
• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Mac Lane, Saunders (1999), Algebra (3rd ed.), pp. 193–222, ISBN 978-0-8218-1646-2
• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8
• Roman, Steven (2005), Advanced Linear Algebra, Graduate Texts in Mathematics, 135 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-24766-3
• Spindler, Karlheinz (1993), Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN 978-0-8247-9144-5
• van der Waerden, Bartel Leendert (1993), Algebra (in German) (9th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56799-8

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13627-9
• Bourbaki, Nicolas (2004), Integration I, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41129-1
• Braun, Martin (1993), Differential equations and their applications: an introduction to applied mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97894-9
• BSE-3 (2001) [1994], "Tangent plane", Encyclopedia of Mathematics, EMS Press
• Choquet, Gustave (1966), Topology, Boston, MA: Academic Press
• Dennery, Philippe; Krzywicki, Andre (1996), Mathematics for Physicists, Courier Dover Publications, ISBN 978-0-486-69193-0
• Dudley, Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-10050-6
• Dunham, William (2005), The Calculus Gallery, Princeton University Press, ISBN 978-0-691-09565-3
• Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9
• Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, ISBN 978-0-534-17094-3
• Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets, Texts in Applied Mathematics, New York: Springer-Verlag, ISBN 978-0-387-98485-8
• Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001), Digital Signal Processing: A Practical Approach (2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002), ISBN 978-0-201-59619-9
• Krantz, Steven G. (1999), A Panorama of Harmonic Analysis, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America, ISBN 978-0-88385-031-2
• Kreyszig, Erwin (1988), Advanced Engineering Mathematics (6th ed.), New York: John Wiley & Sons, ISBN 978-0-471-85824-9
• Kreyszig, Erwin (1989), Introductory functional analysis with applications, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50459-7, MR 0992618
• Lang, Serge (1983), Real analysis, Addison-Wesley, ISBN 978-0-201-14179-5
• Lang, Serge (1993), Real and functional analysis, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94001-4
• Loomis, Lynn H. (1953), An introduction to abstract harmonic analysis, Toronto-New York–London: D. Van Nostrand Company, Inc., pp. x+190, hdl:2027/uc1.b4250788
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Treves, François (1967), Topological vector spaces, distributions and kernels, Boston, MA: Academic Press

• Banach, Stefan (1922), "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathematicae (in French), 3: 133–181, doi:10.4064/fm-3-1-133-181, ISSN 0016-2736
• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)
• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann
• Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", Historia Mathematica, 22 (3): 227–261, doi:10.1006/hmat.1995.1024, MR 1347828
• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et fils
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), O. Wigand, reprint: Grassmann, Hermann (2000), Kannenberg, L.C. (ed.), Extension Theory, translated by Kannenberg, Lloyd C., Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5
• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy
• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German), archived from the original on 2006-11-23
• Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940", Historia Mathematica, 22 (3): 262–303, doi:10.1006/hmat.1995.1025
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin
• Peano, G. (1901) Formulario mathematico: vct axioms via Internet Archive

• Ashcroft, Neil; Mermin, N. David (1976), Solid State Physics, Toronto: Thomson Learning, ISBN 978-0-03-083993-1
• Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR 1043170
• Bourbaki, Nicolas (1998), Elements of Mathematics : Algebra I Chapters 1-3, Berlin, New York: Springer-Verlag, ISBN 978-3-540-64243-5
• Bourbaki, Nicolas (1989), General Topology. Chapters 1-4, Berlin, New York: Springer-Verlag, ISBN 978-3-540-64241-1
• Coxeter, Harold Scott MacDonald (1987), Projective Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96532-1
• Eisenberg, Murray; Guy, Robert (1979), "A proof of the hairy ball theorem", The American Mathematical Monthly, 86 (7): 572–574, doi:10.2307/2320587, JSTOR 2320587
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
• Goldrei, Derek (1996), Classic Set Theory: A guided independent study (1st ed.), London: Chapman and Hall, ISBN 978-0-412-60610-6
• Griffiths, David J. (1995), Introduction to Quantum Mechanics, Upper Saddle River, NJ: Prentice Hall, ISBN 978-0-13-124405-4
• Halmos, Paul R. (1974), Finite-dimensional vector spaces, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3
• Halpern, James D. (Jun 1966), "Bases in Vector Spaces and the Axiom of Choice", Proceedings of the American Mathematical Society, 17 (3): 670–673, doi:10.2307/2035388, JSTOR 2035388
• Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013), Calculus : Single and Multivariable (6 ed.), John Wiley & Sons, ISBN 978-0470-88861-2
• Husemoller, Dale (1994), Fibre Bundles (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94087-8
• Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7
• Kreyszig, Erwin (1991), Differential geometry, New York: Dover Publications, pp. xiv+352, ISBN 978-0-486-66721-8
• Kreyszig, Erwin (1999), Advanced Engineering Mathematics (8th ed.), New York: John Wiley & Sons, ISBN 978-0-471-15496-9
• Luenberger, David (1997), Optimization by vector space methods, New York: John Wiley & Sons, ISBN 978-0-471-18117-0
• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
• Misner, Charles W.; Thorne, Kip; Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0
• Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR 2044239
• Schönhage, A.; Strassen, Volker (1971), "Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)", Computing (in German), 7 (3–4): 281–292, doi:10.1007/bf02242355, ISSN 0010-485X, S2CID 9738629
• Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry (Volume Two), Houston, TX: Publish or Perish
• Stewart, Ian (1975), Galois Theory, Chapman and Hall Mathematics Series, London: Chapman and Hall, ISBN 978-0-412-10800-6
• Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations, Prentice Hall, ISBN 978-0-13-535732-3
• Wallace, G.K. (Feb 1992), "The JPEG still picture compression standard" (PDF), IEEE Transactions on Consumer Electronics, 38 (1): xviii–xxxiv, CiteSeerX 10.1.1.318.4292, doi:10.1109/30.125072, ISSN 0098-3063, archived from the original (PDF) on 2007-01-13, retrieved 2017-10-25
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.