Dual space Totally Explained

Everything about Dual Space totally explained

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. 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's a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

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, for example, 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: ť

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

(a phi ) (x ) = a phi (x ) ,

for all

phi, psi

in V*, a in F and x in V.
Elements of the algebraic dual space V* are sometimes called covectors or one-forms. 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.
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 ť

phi(x)=[phi,x],quad	ext. 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's 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.
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 isn't finite-dimensional but has a Hamel basis 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 won't form a basis.
   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. 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 doesn't have a countable basis.
   This observation generalizes to any 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. Spaces for which the map Ψ is a bijection are called reflexive.
   The continuous dual can be used to define a new topology on V, called the weak topology.
   If the dual of V is separable, then so is the space V itself. The converse isn't true; the space l¹ is separable, but its dual is l^∞, which isn't separable.
   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.

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