bidualization

Bidualization refers to forming the bidual X'' of a normed space X. If X' denotes the continuous dual (the space of all bounded linear functionals on X), then X'' is the dual space of X'. The canonical embedding J_X: X -> X'' is defined by J_X(x)(f) = f(x) for every f in X'. This map is linear and isometric, so X can be regarded as a subspace of X'' through J_X.

If X is complete, i.e., a Banach space, then X'' is also a Banach space. A space X is called reflexive when J_X is surjective, meaning X is isometrically isomorphic to its bidual. Finite-dimensional spaces are reflexive; many classical Banach spaces are reflexive for 1 < p < ∞ (for example L^p with 1 < p < ∞). Spaces such as c0 and l^1 are not reflexive, and in those cases X'' is strictly larger than X (for instance, c0'' ≅ l∞).

The bidual construction extends to maps. For a bounded linear operator T: X -> Y, the adjoint T': Y' -> X' is defined by T'(g) = g ∘ T, and the bidual T'': X'' -> Y'' is defined by T''(Φ) = Φ ∘ T'. One has T'' ∘ J_X = J_Y ∘ T, so the bidualization is functorial: it maps spaces and bounded linear maps to their biduals and corresponding bidual maps.

Conceptually, X'' is naturally a dual space (the dual of X'), and by Goldstine’s theorem the canonical image J_X(X) is weak-* dense in X''. The bidual framework is central to studying reflexivity and the geometric structure of Banach spaces; non-surjectivity of J_X measures the failure of reflexivity. For example, c0 embeds into l∞ as its bidual.