Vector space

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A vector space is a mathematical structure formed by a collection of objects, called vectors, that may be scaled and added. In a vector space these two operations adhere to a number of axioms that generalize common properties of tuples of real numbers such as Euclidean vectors in the plane or three-dimensional Euclidean space.

Historically, the first ideas leading to modern vector spaces can be traced back as far as 17^th century's analytic geometry, matrices and systems of linear equations. The more abstract treatment, first formulated by Giuseppe Peano in the late 19^th century, encompasses more general objects than Euclidean space, but much of the theory retains its linear flavor inherent to lines and planes and their higher-dimensional analogs. Vector spaces are a keystone of linear algebra and are well-understood from this point of view, since they are completely specified by a single number called dimension.

Later enhancements of the theory are due to the widespread presence of vector spaces in mathematical analysis, mainly in the guise of function spaces. Analytical problems call for a notion that goes beyond linear algebra by taking into account convergence questions. This is accomplished by considering vector spaces with additional data, mostly spaces endowed with a suitable topology, thus allowing to consider proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces have a richer and more complicated theory.

Vector spaces are applied throughout mathematics, science and engineering. They are used in methods such as Fourier expansion, which is employed in modern sound and image compression routines, or provides the framework to solution techniques of partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors, which in turn allows to examine local properties of manifolds by linearization techniques.

Contents 1 Motivation and definition 1.1 Definition 1.2 Alternative formulations and elementary consequences of the definition 2 History 3 Examples 3.1 Coordinate and function spaces 3.2 Linear equations 3.3 Field extensions 4 Bases and dimension 5 Linear maps and matrices 5.1 Matrices 5.2 Eigenvalues and eigenvectors 6 Basic constructions 6.1 Subspaces and quotient spaces 6.2 Direct product and direct sum 6.3 Tensor product 7 Vector spaces with additional structure 7.1 Normed vector spaces and inner product spaces 7.2 Topological vector spaces 7.2.1 Banach spaces 7.2.2 Hilbert spaces 7.3 Algebras over fields 8 Applications 8.1 Distributions 8.2 Fourier expansion 8.3 Differential geometry 9 Generalizations 9.1 Vector bundles 9.2 Modules 9.3 Affine and projective spaces 10 See also 11 Notes 12 Footnotes 13 References 13.1 Linear algebra 13.2 Analysis 13.3 Historical references 13.4 Further references

[edit] Motivation and definition

The Euclidean plane R², consisting of pairs (x, y) of real numbers is a vector space: any two pairs of real numbers can be added,

(x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂),

and any pair (x, y) can be multiplied by a real number s to yield another vector (sx, sy). Subtraction can be defined from these two operations, since

(x₁, y₁) − (x₂, y₂) = (x₁, y₁) + (−1) · (x₂, y₂).

The notion of a vector space is an extension of this idea, and is more general in several ways. Firstly, instead of the real numbers other fields, such as the complex numbers or finite fields, are allowed.^{[nb 1]} Secondly, the dimension of the space (which is two in the above example) can be an arbitrary non-negative integer or infinite. Another conceptually important point is there is no particular set of vectors in terms of which other vectors are expressed. For example, there is no preference of representing the vector (x, y) as

(x, y) = x · (1, 0) + y · (0, 1)

over

(x, y) = (−1/3·x + 2/3·y) · (−1, 1) + (1/3·x + 1/3·y) · (2, 1).

[edit] Definition

The definition of a vector space requires a field F such as the field of rational, real or complex numbers. A vector space is a set V together with two operations that combine two elements to a third:

• vector addition: any two vectors, i.e., elements of V, v and w can be added to yield a third vector v + w
• scalar multiplication: any vector can be "scaled": multiplied by a scalar, an element of F. The product of a vector v and scalar a is denoted av.

To specify the field F, one speaks of an F-vector space or a vector space over F. For F = R or C, they are also called real and complex vector spaces, respectively. To qualify as a vector space, addition and multiplication have to adhere to a number of requirements called axioms generalizing the situation of Euclidean plane R² or Euclidean space R³.^[1] For distinction, vectors v will be denoted in boldface.^{[nb 2]} In the formulation of the axioms below, let u, v, w be arbitrary vectors in V, and a, b be scalars, respectively.

Axiom	Signification
Associativity of addition	u + (v + w) = (u + v) + w
Commutativity of addition	v + w = w + v
Identity element of addition	There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition	For all v ∈ V, there exists an element w ∈ V, called the additive inverse of v, such that v + w = 0. The additive inverse is also denoted −v.
Distributivity of scalar multiplication with respect to vector addition	a(v + w) = av + aw
Distributivity of scalar multiplication with respect to field addition	(a + b)v = av + bv
Compatibility of scalar multiplication with field multiplication	a(bv) = (ab)v [nb 3]
Identity element of scalar multiplication	1v = v, where 1 denotes the multiplicative identity in F

The space V = R² over the real numbers, with the addition and multiplication as above, is indeed a vector space. Checking the axioms reduces to verifying simple identities such as

(x, y) + (0, 0) = (x, y),

so that (0, 0) is the zero vector of V. The distributive law amounts to

(a + b) · (x, y) = a · (x, y) + b · (x, y).

In contrast to the intuition stemming from R² and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see below.

[edit] Alternative formulations and elementary consequences of the definition

The requirement that vector addition and scalar multiplication be binary operations includes (by definition of binary operations) a property called closure: that u + v and av are in V for all a, u, and v. Some authors mention these properties as separate axioms.

The first four axioms can be subsumed by requiring the set of vectors to be an abelian group under addition, and the rest are equivalent to a ring homomorphism ƒ from the field into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (ƒ(a))(v). This can be seen as the starting point of defining vector spaces without referring to a field.^[2]

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 a vector v are unique. Other properties follow from the distributive law, for example scalar multiplication by zero yields the zero vector and no other scalar multiplication yields the zero vector.

[edit] History

Further information: History of algebra

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians Descartes and Fermat founded the bases of analytic geometry by tying the solutions of an equation with two variables to the determination of a plane curve.^[3] To achieve geometric solutions without using coordinates, Bernhard Bolzano introduced in 1804 certain operations on points, lines and planes, which are predecessors of vectors.^[4] This work was made use of in the conception of barycentric coordinates by August Ferdinand Möbius in 1827.^[5] The foundation of the definition of vectors was the Bellavitis' definition 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 Argand and Hamilton and the inception of quaternions by the latter.^[6] They are elements in R² and R⁴; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations.

In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.^[7] In his work, the concepts of linear independence and dimension, as well as scalar products 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 algebras. Italian mathematician Peano was the first to give the modern definition of vector spaces and linear maps in 1888.^[8]

An important development of vector spaces is due to the construction of function spaces by Henri Lebesgue. This was later formalized by Banach and Hilbert, around 1920.^[9] At this time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces. Also at this time, the first studies concerning infinite-dimensional vector spaces were done.

[edit] Examples

Main article: Examples of vector spaces

[edit] Coordinate and function spaces

The first example of a vector space over a field F is the field itself, equipped with its standard addition and multiplication. This is the case n = 1 of a vector space usually denoted Fⁿ, known as the coordinate space. Here, n is an integer. Its elements are n-tuples:

(a₁, a₂, ..., a_n), where the a_i are elements of F.^[10]

Infinite coordinate sequences, and more generally 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 ƒ and g is given by

(ƒ + g)(w) = ƒ(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 Rⁿ. 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 will still have that property.^[11] Therefore, the set of such functions are vector spaces. They are studied in more 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) = r_nxⁿ + r_n−1xⁿ⁻¹ + ... + r₁x + r₀, where the coefficients r_n, ..., r₀ are in F.^[12]

Power series are similar, except that infinitely many terms are allowed.^[13]

[edit] Linear equations

Main articles: Linear equation, Linear differential equation, and Systems of linear equations

Systems of homogeneous linear equations are closely tied to vector spaces.^[14] 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 = is the matrix containing the coefficients of the given equations, x is the vector (a, b, c), and 0 = (0, 0) is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example

ƒ ''(x) + 2ƒ '(x) + ƒ (x) = 0

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

[edit] Field extensions

A common situation in algebra, and especially in algebraic number theory is a field F containing a smaller field E. By the given multiplication and addition operations of F, F becomes an E-vector space, also called a field extension of E.^[15] For example the complex numbers are a vector space over R. A particularly interesting type of field extension in number theory is Q(α), the extension of the rational numbers Q by a fixed complex number α. Q(α) is the smallest field containing the rationals and a fixed complex number α, and its dimension as a vector space over Q depends on the choice of α.

[edit] Bases and dimension

Main articles: Basis and Dimension

Bases reveal the structure of vector spaces in a concise way. Formally, a basis is a (finite or infinite) set B = {v_i}_{i ∈ I} of vectors indexed by some index set I that spans the whole space, and minimal with this property. The former means that any vector v can be expressed as a finite sum (called linear combination of the basis elements)

v = a₁v_i1 + a₂v_i2 + ... + a_nv_in,

where the a_k are scalars and v_ik (k = 1, ..., n) elements of the basis B. Minimality, on the other hand, is made formal by requiring B to be linearly independent. A set of vectors is said to be linearly independent if none of its elements can be expressed as a linear combination of the remaining ones. Equivalently, an equation

a₁v_i1 + a_i2v₂ + ... + a_nv_in = 0

can only hold if and only if all scalars a₁, ..., a_n equal zero. By definition every vector can be expressed as a finite sum of basis elements. Because of linear independence any such representation is unique.^[16] Vector spaces are sometimes introduced from this coordinatised viewpoint.

Every vector space has a basis. This fact relies on Zorn’s Lemma, an equivalent formulation of the axiom of choice.^[17] Given the other axioms of Zermelo-Fraenkel set theory, the existence of bases is equivalent to the axiom of choice.^[18] 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.^[19] It is called the dimension of the vector space, denoted dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.^[20]

The dimension of the coordinate space Fⁿ is n, since any vector (x₁, x₂, ..., x_n) can be uniquely expressed as a linear combination of n vectors (called coordinate vectors) e₁ = (1, 0, ..., 0), e₂ = (0, 1, 0, ..., 0), to e_n = (0, 0, ..., 0, 1), namely the sum

x₁e₁ + x₂e₂ + ... + x_ne_n,

The dimension of function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.^{[nb 4]} Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of an homogeneous ordinary differential equation equals the degree of the equation.^[21] For example, the above equation has degree 2. The solution space is generated by e^x and xe^x. These two functions are linearly independent over R, so the dimension of this space is two.

The dimension (or degree) of the field extension Q(α) over Q depends on whether or not α is algebraic, in the sense that it satisfies some polynomial equation

q_nαⁿ + q_n−1αⁿ⁻¹ + ... + q₀ = 0, with rational coefficients q_n, ..., q₀.

If it is algebraic the dimension is finite. More precisely, it equals the degree of the minimal polynomial having this number as a root.^[22] For example, the complex numbers are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i² + 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 degree of Q(α) over Q is infinite. For instance, for α = π there is no such equation, since π is transcendental.^[23]

[edit] Linear maps and matrices

Main article: Linear map

As with many algebraic entities, the relation of two vector spaces is expressed by maps between them. In the context of vector spaces, the corresponding concept is called linear map or linear transformation.^[24] They are functions ƒ : V → W that are compatible with the relevant structure—i.e., they preserve sums and scalar multiplication:

ƒ(v + w) = ƒ(v) + ƒ(w) and ƒ(a · v) = a · ƒ(v).

An isomorphism is a linear map ƒ : V → W such that there exists an inverse map g : W → V, which is a map such that the two possible compositions ƒ ∘ g : W → W and g ∘ ƒ : V → V are identity maps. Equivalently, ƒ is both one-to-one (injective) and onto (surjective).^[25] If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces, since all identities holding in V are, via ƒ, transported to similar ones in W, and vice versa via g.

Linear maps V → W between two fixed vector spaces form a vector space Hom_F(V, W), also denoted L(V, W).^[26] The space of linear maps from V to F is called the dual vector space, denoted V^∗.^[27] Via the injective natural map V → V^∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.^[28]

Once a basis of V is chosen, linear maps ƒ : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.^[29] If dim V = dim W, a 1-to-1 correspondence between two fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.^[30] Thus any vector spaces is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional vector spaces over F is isomorphic to Fⁿ.

[edit] Matrices

Main articles: Matrix and Determinant

Matrices are a useful notion to encode linear maps.^[31] 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

,

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 ƒ : V → W is uniquely represented by a matrix via this assignment.^[32]

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.^[33] The determinant of a real square matrix also determines whether the corresponding linear transformation is orientation preserving or not: it is so if and only if the determinant is positive.

[edit] Eigenvalues and eigenvectors

Main articles: Eigenvalues and eigenvectors and eigenspace

Endomorphisms, linear maps ƒ : V → V, are particularly important. In this case, vectors v can be compared with their image under ƒ, ƒ(v). Any nonzero vector v satisfying λv = ƒ(v), where λ is a scalar, is called an eigenvector of ƒ with eigenvalue λ.^{[nb 5][34]} Equivalently, v is an element of kernel of the difference ƒ − λ · Id (the identity map V → V). In the finite-dimensional case, this can be rephrased using determinants: ƒ having eigenvalue λ is the same as

det (ƒ − 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 ƒ.^[35] 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.^{[nb 6]} The spectral theorem describes the infinite-dimensional case; to accomplish this aim, the machinery of functional analysis is needed, see below. The set of all eigenvectors corresponding to a particular eigenvalue (and a particular ƒ) forms a vector space known as the eigenspace corresponding to the eigenvalue (and ƒ) in question.

[edit] 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 concrete definitions given below, they are also characterized by universal properties, which determine an object X by specifying the linear maps from X to any other vector space.

[edit] Subspaces and quotient spaces

Main articles: Linear subspace and Quotient vector space

A nonempty subset W of a vector space V that is closed under addition and scalar multiplication is called a subspace of V.^[36] 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. Expressed in terms of elements, the span is the subspace consisting linear combinations of elements of S.^[37]

The counterpart to subspaces are quotient vector spaces.^[38] 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 7]} This way, the quotient space "forgets" information that is contained in the subspace W.

For any linear map ƒ : V → W, the kernel ker(ƒ ) consists of vectors v that are mapped to 0 in W.^[39] Both kernel and image im(ƒ ) = {ƒ(v), v ∈ V} are linear subspaces of V and W, respectively.^[40] They are related by an elementary but fundamental isomorphism

V / ker(ƒ ) ≅ im(ƒ ).

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.^[41]

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

, where the coefficients a_i are functions in x, too.

In the corresponding map

,

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

[edit] Direct product and direct sum

Main articles: Direct product and Direct sum

The direct product of a family of vector spaces V_i, where i runs through some index set I, consists of the set of all tuples (v_i)_{i ∈ I}, which specify for each index i an element v_i of V_i.^[42] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum (also called coproduct and denoted ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but differ otherwise.

[edit] Tensor product

Main article: Tensor product of vector spaces

The tensor product V ⊗_F W, or simply V ⊗ 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.^[43]

These rules ensure that the map ƒ 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 ƒ equals g: u(v ⊗ w) = g(v, w).^[44] This is called the universal property of the tensor product, an instance of the method to indirectly define objects by specifying maps from or to this object, a technique much used in advanced abstract algebra.

[edit] 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 ad hoc do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not per se adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. The needs of functional analysis require considering additional structures. Much the same way the axiomatic treatment of vector spaces reveals their essential algebraic features, studying vector spaces with additional datums abstractly turns out to be advantageous, too.

A first example of an additional datum is an order ≤, a token by which vectors can be compared.^[45] Rⁿ can be ordered by comparing the vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions

ƒ = ƒ ⁺ − ƒ ⁻,

where ƒ ⁺ denotes the positive part of ƒ and ƒ ⁻ the negative part.^[46]

[edit] Normed vector spaces and inner product spaces

Main articles: Normed vector space and Inner product space

"Measuring" vectors is a frequent need, either 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 |v| and 〈v | w〉, respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm . Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.^[47]

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

〈x | y〉 = x · y = x₁y₁ + ... + x_ny_n.

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

Because of this, two vectors satisfying 〈x | y〉 = 0 are called orthogonal.

An important variant of the standard dot product is used in Minkowski space: R⁴ endowed with the Lorentz inner product

〈x | y〉 = x₁y₁ + x₂y₂ + x₃y₃ − x₄y₄.^[48]

It is crucial to the mathematical treatment of special relativity, where the fourth coordinate—corresponding to time, as opposed to three space-dimensions—is singled out.

[edit] Topological vector spaces

Main article: Topological vector space

Convergence questions are addressed by considering vector spaces V carrying a compatible topology, a structure that allows to talk about elements being close to each other.^[49][50] Compatible here means that addition and scalar multiplication should 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 8]} 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

denotes the limit of the corresponding finite partial sums of the sequence (ƒ_i)_i∈N of elements of V. For example, the ƒ_i could be (real or complex) functions belonging to some function space V, and the series is then 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.

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.^[51] In contrast, the space of all continuous functions on [0,1] with the topology of uniform convergence is complete.^[52] Banach and Hilbert spaces are complete topological 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 are equivalent, giving rise to the same notion of convergence.^[53] 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, it is useful to require maps to be continuous.^[54] For example, the dual space V^∗ consists of continuous functionals V → R (or C). The fundamental Hahn-Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.^[55]

[edit] Banach spaces

Main article: Banach space

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.^[56] A first example is the vector space l^p consisting of infinite vectors with real entries x = (x₁, x₂, ...) whose p-norm (1 ≤ p ≤ ∞) given by

for p < ∞ and

is finite. The topologies on the infinite-dimensional space l^p are inequivalent for different p. E.g. the sequence of vectors x_n = (2⁻ⁿ, 2⁻ⁿ, ..., 2⁻ⁿ, 0, 0, ...), i.e. the first 2ⁿ components are 2⁻ⁿ, the following ones are 0, converges to the zero vector for p = ∞, but does not for p = 1:

, but .

More generally than sequences of real numbers, functions ƒ : Ω → R are endowed with a norm that replaces the above sum by the Lebesgue integral

.^{[nb 9]}

The space of integrable functions on a given domain Ω (for example an interval) satisfying |ƒ |_p < ∞, and equipped with this norm are called Lebesgue spaces, denoted L^p(Ω). These spaces are complete.^[57] (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 10]}) Concretely this means that for any sequence of Lebesgue-integrable functions ƒ₁, ƒ₂, ... with | ƒ_n|_p < ∞, satisfying the condition

there exists a function ƒ(x) belonging to the vector space L^p(Ω) such that

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.^[58]

[edit] Hilbert spaces

Main article: Hilbert space

Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.^[59] A key case is the Hilbert space L²(Ω), with inner product given by

, where denotes the complex conjugate of g(x).^{[60][nb 11]}

By definition, Cauchy sequences in any Hilbert space converge to a limit. Conversely, finding a sequence of functions ƒ_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 ƒ by polynomials.^[61] By the Stone-Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.^[62] 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 (i.e., 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 dimension.^{[nb 12]} Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, together with the Gram-Schmidt process, it also allows to construct a basis of orthogonal vectors.^[63] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space.

The solutions to various important 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, that serve as the axes in a corresponding Hilbert space.^[64] 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 determining a wavefunction.^[65] 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 that acts upon functions in terms of these eigenfunctions and their eigenvalues.^[66]

[edit] Algebras over fields

Main articles: Algebra over a field and Lie algebra

In general, vector spaces do not possess a multiplication operation. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field.^[67] Many algebras stem from functions on some geometrical object: since functions with values in a field can be multiplied, these entities form algebras. The Stone-Weierstrass theorem mentioned above, 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, whose 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.^[68]

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):

• anticommutativity: [x, y] = −[y, x], and
• the Jacobi identity: [x, [y, z]] + [y, [x, z]] + [z, [x, y]] = 0.^[69]

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.^[70] 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 ⊗. 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.^[71]

[edit] Applications

As vector spaces occur in many mathematical circumstances, namely wherever functions with values in some field are involved, they have manifold applications. 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.^[72] Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains, such as group theory.^[73]

[edit] Distributions

Main article: Distribution

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.^[74] The latter space is endowed with a topology that takes into account not only ƒ itself, but also all its higher derivatives. A standard example is the result of integrating a test function ƒ over some domain Ω:

Another example is the Dirac distribution, denoted by δ, which associates to a test function ƒ its value at the origin: δ(ƒ) = ƒ(0). 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 strictly 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 (e.g., using the Lax-Milgram theorem, a consequence of the Riesz representation theorem).^[75]

[edit] Fourier expansion

Main article: Fourier expansion

Resolving a periodic function into sums of trigonometric functions is known as the Fourier expansion, a technique much used in physics and engineering.^[76] By the Stone-Weierstrass theorem (see above), any continuous function ƒ(x) on a bounded, closed interval (equivalently, any periodic function) can be written as

.

The coefficients a_m and b_m are called Fourier coefficients of ƒ.

In 1822, Fourier first used this technique to solve the heat equation on a special domain.^[77] More recently, Fourier series have been developed for a variety of applications such as the discrete Fourier Transform for high-speed multiplications of large integers.^[78] The closely related discrete cosine transform is used in JPEG image compression.^[79]

[edit] Differential geometry

Main articles: Tangent space and Cotangent space

The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. Even in a three-dimensional Euclidean space, it is impossible to prescribe a basis of the tangent space in a natural way, and so it is most easily conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the best linear approximation, or linearization, of a surface at a point.^{[nb 13]}

As the notion of a surface is generalized to higher dimensions as a differentiable manifold, so the tangent plane generalizes to the tangent space, which is the linearization of a differentiable manifold at a point.^[80]

The dual of the tangent space is called cotangent space. Differential forms are elements of the exterior algebra of the cotangent space. They generalize the "dx" in standard integration

Further vector space constructions, in particular tensors are widely used in geometry and beyond. Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.^[81] 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 curvature of space-time.^[82][83] 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.^[84]

The cotangent space is also used in commutative algebra and algebraic geometry to define regular local rings, an algebraic adaptation of smoothness in differential geometry, by comparing the dimension of the cotangent space of the ring (defined in a purely algebraic manner) to the Krull dimension of the ring.^[85]

[edit] Generalizations

[edit] Vector bundles

Main articles: Vector bundle and Tangent bundle

A family of vector spaces, parametrised continuously by some topological space X, is a vector bundle.