biduals

The bidual of a vector space is the dual space of its dual. In functional analysis, given a normed vector space X over the real or complex numbers, its dual X consists of all continuous linear functionals on X. The bidual, denoted X or (X), is the space of continuous linear functionals on X. The elements of X act on X by evaluation, and each element of X can be naturally identified with an element of X through the canonical embedding. This embedding sends a vector x∈X to the functional x̂ defined by x̂(f)=f(x) for all f∈X. It is linear and preserves the norm, making it an isometric embedding.

A normed space X is reflexive if the canonical map from X into X is surjective, meaning every element of the bidual corresponds to some vector in X. Reflexive spaces possess many attractive properties: every bounded sequence has a weakly convergent subsequence, and the unit ball is weakly compact. Classical examples of reflexive spaces include the Lp spaces for 1<p<∞ and the sequence spaces ℓp for the same range of p. In contrast, spaces such as L^1 and ℓ∞ are not reflexive because their biduals strictly contain the original space.

The bidual concept extends beyond Banach spaces to locally convex spaces and topological vector spaces, where continuous duals and reflexivity are defined with respect to weaker topologies. In category theory, the dual of an object in an abstract category can be described, and iterated duals (the bidual) may yield interesting adjoint relationships. Nonetheless, in most analyses the term "bidual" specifically refers to the second dual of a normed linear space.