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, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Given any vector space V over a field F, the (algebraic) dual space V∗ (alternatively denoted by V ∨ {displaystyle V^{vee }} or V ′ {displaystyle V'} ) is defined as the set of all linear maps φ: V → F (linear functionals). Since linear maps are vector space homomorphisms, the dual space is also sometimes denoted by Hom(V, F). The dual space V∗ itself becomes a vector space over F when equipped with an addition and scalar multiplication satisfying: for all φ and ψ ∈ V∗, x ∈ V, and a ∈ F. Elements of the algebraic dual space V∗ are sometimes called covectors or one-forms. The pairing of a functional φ in the dual space V∗ and an element x of V is sometimes denoted by a bracket: φ(x) = or φ(x) = ⟨φ,x⟩. This pairing defines a nondegenerate bilinear mapping ⟨·,·⟩ : V∗ × V → F called the natural pairing. If V is finite-dimensional, then V∗ has the same dimension as V. Given a basis {e1, ..., en} in V, it is possible to construct a specific basis in V∗, called the dual basis. This dual basis is a set {e1, ..., en} of linear functionals on V, defined by the relation for any choice of coefficients ci ∈ F. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations