bidual

Bidual refers to the second dual of a normed space in functional analysis. If X is a normed space, its continuous dual X' consists of all bounded linear functionals on X. The bidual X'' is the dual space of X', i.e., the set of all continuous linear functionals on X'. There is a canonical embedding J from X into X'' defined by J(x)(f) = f(x) for every f in X'. This embedding is linear and isometric, so X can be identified with a subspace of X''.

X is said to be reflexive if J is surjective, meaning X is isometrically isomorphic to its bidual X''. Finite-dimensional spaces are reflexive, and many classical spaces, such as Lp spaces for 1 < p < ∞, are reflexive. In contrast, spaces like l1 and c0 are not reflexive; for example, X' ≅ l∞ and X'' is strictly larger than X'.

The bidual X'' is always a Banach space, since it is the dual of X'. The Banach–Alaoglu theorem implies that the closed unit ball of X'' is compact in the weak*-topology (the topology sigma(X'', X')). The canonical embedding shows X as a subspace of X'', and X is reflexive exactly when this subspace equals X''.

In addition, for Banach algebras A, A'' carries natural Arens products extending the original multiplication, and A is called Arens regular if the two Arens products coincide. In operator algebra contexts, the bidual of a C*-algebra can be identified with a von Neumann algebra under suitable representations.