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Encyclopedia > Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. When applied to vector spaces of functions (which typically are infinite dimensional) dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in the study of functional analysis. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...


There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

Contents

Algebraic dual space

Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on V, i.e., scalar-valued linear maps on V (in this context, a "scalar" is a member of the base-field F). V* itself becomes a vector space over F under the following definition of addition and scalar multiplication: In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

 (phi + psi )( x ) = phi ( x ) + psi ( x ) ,
 ( a phi ) ( x ) = a phi ( x ) ,

for all φ,ψ in V*, a in F and x in V. In the language of tensors, elements of V are sometimes called contravariant vectors, and elements of V*, covariant vectors, covectors or one-forms. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ... In linear algebra a one-form on a vector space is the same as a linear functional on it. ...


The finite dimensional case

If V is finite-dimensional, then V* has the same dimension as V; if {e1,...,en} is a basis for V, then the associated dual basis {e1,...,en} of V* is given by In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

 mathbf{e}^i (mathbf{e}_j)= left{begin{matrix} 1, & mbox{if }i = j  0, & mbox{if } i ne j end{matrix}right.

In the case of R2, its basis is B={e1=(1,0),e2=(0,1)}. Then, e1, and e2 are one-forms (functions which map a vector to a scalar) such that e1(e1)=1, e1(e2)=0, e2(e1)=0, and e2(e2)=1. (Note: The superscript here is an index, not an exponent.)


Concretely, if we interpret Rn as the space of columns of n real numbers, its dual space is typically written as the space of rows of n real numbers. Such a row acts on Rn as a linear functional by ordinary matrix multiplication. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article gives an overview of the various ways to perform matrix multiplication. ...


If V consists of the space of geometrical vectors (arrows) in the plane, then the elements of the dual V* can be intuitively represented as collections of parallel lines. Such a collection of lines can be applied to a vector to yield a number in the following way: one counts how many of the lines the vector crosses. A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...


The infinite dimensional case

If V is not finite-dimensional but has a basis[1] eα indexed by an infinite set A, then the same construction as in the finite dimensional case yields linearly independent elements eα (αA) of the dual space, but they will not form a basis. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...


Consider, for instance, the space R, whose elements are those sequences of real numbers which have only finitely many non-zero entries, which has a basis indexed by the natural numbers N: for iN, ei is the sequence which is zero apart from the ith term, which is one. The dual space of R is RN, the space of all sequences of real numbers: such a sequence (an) is applied to an element (xn) of R to give the number ∑nanxn, which is a finite sum because there are only finitely many nonzero xn. The dimension of R is countably infinite, whereas RN does not have a countable basis. For other senses of this word, see sequence (disambiguation). ... In mathematics, the dimension of a vector space V is the cardinality (i. ...


This observation generalizes to any[1] infinite dimensional vector space V over any field F: a choice of basis {eα:αA} identifies V with the space (FA)0 of functions f:AF such that fα=f(α) is nonzero for only finitely many αA, where such a function f is identified with the vector

sum_{alphain A} f_alphamathbf{e}_alpha

in V (the sum is finite by the assumption on f and any vV may be written in this way by the definition of a basis).


The dual space of V may then be identified with the space FA of all functions from A to F: a linear functional T on V is uniquely determined by the values θα=T(eα) it takes on the basis of V, and any function θ:AF (with θ(α)=θα) defines linear functional T on V by

Tbiggl(sum_{alphain A} f_alpha e_alphabiggr) = sum_{alphain A} theta_alpha f_alpha.

Again the sum is finite because fα is nonzero for only finitely many α.


Note that (FA)0 may be identified (essentially by definition) with the direct sum of infinitely many copies of F (viewed as a 1-dimensional vector space over itself) indexed by A, i.e., there are linear isomorphisms In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Vcong (mathbb F^A)_0congbigoplus_{alphain A} mathbb{F}.

On the other hand FA is (again by definition), the direct product of infinitely many copies of F indexed by A, and so the identification In mathematics, one can often define a direct product of objects already known, giving a new one. ...

 V^*cong biggl(bigoplus_{alphain A}mathbb{F}biggr)^*cong prod_{alphain A}mathbb{F}^*congprod_{alphain A}mathbb{F}cong mathbb{F}^A

is a special case of a general result relating direct sums (of modules) to direct products. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...


Thus if the basis is infinite, then there are always more vectors in the dual space than the original vector space. This is in marked contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.


Bilinear products and dual spaces

As we saw above, if V is finite-dimensional, then V is isomorphic to V*, but the isomorphism is not natural and depends on the basis of V we started out with. In fact, any isomorphism Φ from V to V* defines a unique non-degenerate bilinear form on V by In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

 langle v,w rangle = (Phi (v))(w) ,

and conversely every such non-degenerate bilinear product on a finite-dimensional space gives rise to an isomorphism from V to V*.


Injection into the double-dual

There is a natural homomorphism Ψ from V into the double dual V**, defined by (Ψ(v))(φ) = φ(v) for all v in V, φ in V*. This map Ψ is always injective[1]; it is an isomorphism if and only if V is finite-dimensional. (Infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.) In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


Transpose of a linear map

If  f: V to W is a linear map, we may define its transpose (or dual) f*: W* to V* by In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

f^* (varphi) = varphi circ f ,

where φ is an element of W*. In that case, f * (φ) is also known as the pullback of φ by f. Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...


The assignment f mapsto , f^* produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg)* = g*f*. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. Note that one can identify (f*)* with f using the natural injection into the double dual. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For functors in computer science, see the function object article. ...


If the linear map f is represented by the matrix A with respect to two bases of V and W, then f* is represented by the transpose matrix tA with respect to the dual bases of W* and V*, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f* is represented by the same matrix acting by the right on row vectors. These points of view are related by the canonical inner product on Rn, which identifies the space of column vectors with the dual space of row vectors. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...


Continuous dual space

See main article Continuous dual space In mathematics it can be shown that any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. In many cases, these two spaces are isomorphic which means that the distinction between their elements is not always apparent. ...


When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field. This gives rise to the notion of the "continuous dual space" which is a linear subspace of the algebraic dual space V*, denoted V ′. For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space. In topological contexts sometimes V* may also be used for just the continuous dual space and the continuous dual may just be called the dual. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics it can be shown that any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. In many cases, these two spaces are isomorphic which means that the distinction between their elements is not always apparent. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...


The continuous dual V ′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space. A norm ||φ|| of a continuous linear functional on V is defined by In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

|phi | = sup { |phi ( x )| : |x| le 1 }

This turns the continuous dual into a normed vector space, indeed into a Banach space so long as the underlying field is complete, which is often included in the definition of the normed vector space. In other words, this dual of a normed space over a complete field is necessarily complete.


Examples

Let 1 < p < ∞ be a real number and consider the Banach space lp of all sequences a = (an) for which In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... For other senses of this word, see sequence (disambiguation). ...

|mathbf{a}|_p = left ( sum_{n=0}^infty |a_n|^p right) ^{1/p}

is finite. Define the number q by 1/p + 1/q = 1. Then the continuous dual of lp is naturally identified with lq: given an element φ ∈ (lp)', the corresponding element of lq is the sequence (φ(en)) where en denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (an) ∈ lq, the corresponding continuous linear functional φ on lp is defined by φ(b) = ∑n an bn for all b = (bn) ∈ lp (see Hölder's inequality). In mathematical analysis, Hölders inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ...


In a similar manner, the continuous dual of l1 is naturally identified with l (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremums norm) and c0 (the sequences converging to zero) are both naturally identified with l1. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...


Further properties

If V is a Hilbert space, then its continuous dual is a Hilbert space which is anti-isomorphic to V. This is the content of the Riesz representation theorem, and gives rise to the bra-ket notation used by physicists in the mathematical formulation of quantum mechanics. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...


In analogy with the case of the algebraic double dual, there is always a naturally defined injective continuous linear operator Ψ : VV ′′ from V into its continuous double dual V ′′. This map is in fact an isometry, meaning ||Ψ(x)|| = ||x|| for all x in V. Spaces for which the map Ψ is a bijection are called reflexive. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... A bijective function. ... This page concerns the reflexivity of a Banach space. ...


The continuous dual can be used to define a new topology on V, called the weak topology. In mathematics, weak topology is an alternative term for initial topology. ...


If the dual of V is separable, then so is the space V itself. The converse is not true; the space l1 is separable, but its dual is l, which is not separable. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...


Notes

  1. ^ a b c Several assertions in this article require the axiom of choice for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that RN has a basis. It is also needed to show that the dual of an infinite dimensional vector space V is nonempty, and hence that the natural map from V to its double dual is injective.

In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

See also


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