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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the 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. 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 In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions This article assumes some familiarity with Analytic geometry and the concept of a limit. Consequently, the dual space is an important concept in the study of functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article

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.

1 Algebraic dual space
2 Continuous dual space
- 2.1 Examples
- 2.2 Further properties
3 Notes
4 References
5 See also

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. 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 This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional e. , scalar-valued linear maps on V (in this context, a "scalar" is a member of the base-field F). In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that V* itself becomes a vector space over F under the following definition of 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.

Elements of the algebraic dual space V* are sometimes called covectors or one-forms. In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space In the language of tensors, components of elements of V relative to a basis are sometimes called contravariant, and components of elements of V* relative to the dual basis are called covariant. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually ^[1]

The pairing of a functional φ in the dual space V* and an element x of V is sometimes denoted by a bracket, such as

$\phi(x)=[\phi,x],\quad\text{or}\,\,\, \phi(x)=\langle\phi, x\rangle.$

For the former notation, see (Halmos 1974). For the latter, see (Misner, Thorne & Wheeler 1973). A similar notation, widely used in quantum physics, is the bracket notation. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical The bracket defines a nondegenerate bilinear mapping,

$[\cdot,\cdot] : V^* \times V \to F.$

The finite dimensional case

If V is finite-dimensional, then V* has the same dimension as V. In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, the dimension of a Vector space V is the cardinality (i Given a basis of V, it is possible to give a basis of V*, called the dual basis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In linear algebra a dual basis is a set of vectors that forms a basis for the Dual space of a vector space In detail, if {e₁,. . . ,e_n} is a basis for V, then the associated dual basis of V* is an n-tuple {e¹,. . . ,eⁿ} of linear functionals on V defined by the relation

$\mathbf{e}^i(c_1 \mathbf{e}_1+\cdots+c_n\mathbf{e}_n) = c_i$

For any choice of coefficients c_i (and hence any vector in V, since the e_i are assumed to be a basis. ) In particular, taking c_j=1, and every other coefficient zero gives the relation

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

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. In Mathematics, the real numbers may be described informally in several different ways Such a row acts on Rⁿ as a linear functional by ordinary matrix multiplication. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix

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.

The infinite dimensional case

If V is not finite-dimensional but has a Hamel basis^[2] e_α indexed by an infinite set A, then the same construction as in the finite dimensional case yields linearly independent elements e^α (α∈A) of the dual space, but they will not form a basis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors

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 i∈N, e_i is the sequence which is zero apart from the ith term, which is one. In Mathematics, a sequence is an ordered list of objects (or events The dual space of R^∞ is R^N, the space of all sequences of real numbers: such a sequence (a_n) is applied to an element (x_n) of R^∞ to give the number ∑_na_nx_n, which is a finite sum because there are only finitely many nonzero x_n. The dimension of R^∞ is countably infinite, whereas R^N does not have a countable basis. In Mathematics, the dimension of a Vector space V is the cardinality (i

This observation generalizes to any^[2] infinite dimensional vector space V over any field F: a choice of basis {e_α:α∈A} identifies V with the space (F^A)₀ of functions f:A→F such that f_α=f(α) is nonzero for only finitely many α∈A, where such a function f is identified with the vector

$\sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha$

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

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

$T\biggl(\sum_{\alpha\in A} f_\alpha e_\alpha\biggr) = \sum_{\alpha\in A} \theta_\alpha f_\alpha.$

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

Note that (F^A)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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction e. , there are linear isomorphisms

$V\cong (\mathbb F^A)_0\cong\bigoplus_{\alpha\in A} \mathbb{F}.$

On the other hand F^A is (again by definition), the direct product of infinitely many copies of F indexed by A, and so the identification

$V^*\cong \biggl(\bigoplus_{\alpha\in A}\mathbb{F}\biggr)^*\cong \prod_{\alpha\in A}\mathbb{F}^*\cong\prod_{\alpha\in A}\mathbb{F}\cong \mathbb{F}^A$

is a special case of a general result relating direct sums (of modules) to direct products. In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

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

If V is finite-dimensional, then V is isomorphic to V*. But we don't have a natural isomorphism unless we choose a basis in V. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In fact, any isomorphism Φ from V to V* defines a unique non-degenerate bilinear form on V by

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

and conversely every such non-degenerate bilinear product on a finite-dimensional space gives rise to an isomorphism from V to V*. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where

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*. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This map $Ψ$ is always injective^[2]; 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. )

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

$f^* (\varphi) = \varphi \circ f \,$

where $φ$ is an element of W*. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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 Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Note that one can identify (f*)* with f using the natural injection into the double dual.

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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a 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.

Continuous dual space

When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field. 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 Intuitively this is a function 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.

The continuous dual V ′ of a normed vector space V (e. 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 g. , a Banach space or a Hilbert space) forms a normed vector space. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis This article assumes some familiarity with Analytic geometry and the concept of a limit. A norm ||φ|| of a continuous linear functional on V is defined by

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

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.

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, as shown by the example of discontinuous linear map. In Mathematics, Linear maps form an important class of "simple" functions which preserve the algebraic structure of Linear spaces and are often

Examples

Let 1 < p < ∞ be a real number and consider the Banach space ℓ^p of all sequences a = (a_n) for which

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

is finite. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, a sequence is an ordered list of objects (or events Define the number q by 1/p + 1/q = 1. Then the continuous dual of ℓ^p is naturally identified with ℓ^q: given an element φ ∈ (ℓ^p)′, the corresponding element of ℓ^q is the sequence (φ(e_n)) where e_n denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (a_n) ∈ ℓ^q, the corresponding continuous linear functional φ on ℓ^p is defined by φ(b) = ∑_n a_n b_n for all b = (b_n) ∈ ℓ^p (see Hölder's inequality). In Mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental Inequality between integrals and an indispensable tool

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 supremum norm) and c₀ (the sequences converging to zero) are both naturally identified with ℓ¹. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematical analysis, the uniform norm assigns to real- or complex -valued bounded functions f the nonnegative number

Further properties

In analogy with the case of the algebraic double dual, there is always a naturally defined injective^[3] continuous linear operator Ψ : V → V ′′ from V into its continuous double dual V ′′. In case V is normed, this map is in fact an isometry, meaning ||Ψ(x)|| = ||x|| for all x in V. For the Mechanical engineering and Architecture usage see Isometric projection. Spaces for which the map Ψ is a bijection are called reflexive. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to

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. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n The converse is not true; the space l¹ is separable, but its dual is l^∞, which is not separable.

If V is a Hilbert space, then its continuous dual is a Hilbert space which is anti-isomorphic to V. This article assumes some familiarity with Analytic geometry and the concept of a limit. 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. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

Notes

^ See, for instance, (Misner, Thorne & Wheeler 1973)
^ ^a ^b ^c Several assertions in this article require the axiom of choice for their justification. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that R^N 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.
^ Injectivity holds if and only if V is Hausdorff. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Otherwise, the kernel is the smallest closed subspace containing {0}. In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that

References

Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 0387900934
Misner, Charles W. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician ; Thorne, Kip. S. & Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0

Rudin, Walter (1991), Functional analysis, McGraw-Hill Science, ISBN 978-0070542365

Dictionary

-noun

(mathematics) The vector space which comprises the set of linear transformations of a given vector space into its scalar field

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