Banach space
Bidual

If X {\displaystyle X} is a normed space, the (continuous) dual X ″ {\displaystyle X''} of the dual X ′ {\displaystyle X'} is called bidual, or second dual of X. {\displaystyle X.} For every normed space X, {\displaystyle X,} there is a natural map, { F X: X → X ″ F X (x) (f) = f (x) for all x ∈ X, and for all f ∈ X ′ {\displaystyle {\begin{cases}F_{X}\colon X\to X''\\F_{X}(x)(f)=f(x)&{\text{ for all }}x\in X,{\text{ and for all }}f\in X'\end{cases}}}

This defines F X (x) {\displaystyle F_{X}(x)} as a continuous linear functional on X ′, {\displaystyle X^{\prime },} that is, an element of X ′ ′. {\displaystyle X^{\prime \prime }.} The map F X: x → F X (x) {\displaystyle F_{X}\colon x\to F_{X}(x)} is a linear map from X {\displaystyle X} to X ′ ′. {\displaystyle X^{\prime \prime }.} As a consequence of the existence of a norming functional f {\displaystyle f} for every x ∈ X, {\displaystyle x\in X,} this map F X {\displaystyle F_{X}} is isometric, thus injective.

For example, the dual of X = c 0 {\displaystyle X=c_{0}} is identified with ℓ 1, {\displaystyle \ell ^{1},} and the dual of ℓ 1 {\displaystyle \ell ^{1}} is identified with ℓ ∞, {\displaystyle \ell ^{\infty },} the space of bounded scalar sequences. Under these identifications, F X {\displaystyle F_{X}} is the inclusion map from c 0 {\displaystyle c_{0}} to ℓ ∞. {\displaystyle \ell ^{\infty }.} It is indeed isometric, but not onto.

If F X {\displaystyle F_{X}} is surjective, then the normed space X {\displaystyle X} is called reflexive (see below). Being the dual of a normed space, the bidual X ″ {\displaystyle X''} is complete, therefore, every reflexive normed space is a Banach space.

Using the isometric embedding F X, {\displaystyle F_{X},} it is customary to consider a normed space X {\displaystyle X} as a subset of its bidual. When X {\displaystyle X} is a Banach space, it is viewed as a closed linear subspace of X ′ ′. {\displaystyle X^{\prime \prime }.} If X {\displaystyle X} is not reflexive, the unit ball of X {\displaystyle X} is a proper subset of the unit ball of X ′ ′. {\displaystyle X^{\prime \prime }.} The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. In other words, for every x ″ {\displaystyle x''} in the bidual, there exists a net (x i) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} in X {\displaystyle X} so that sup i ∈ I ‖ x i ‖ ≤ ‖ x ″ ‖, x ″ (f) = lim i f (x i), f ∈ X ′. {\displaystyle \sup _{i\in I}\left\|x_{i}\right\|\leq \|x''\|,\ \ x''(f)=\lim _{i}f\left(x_{i}\right),\quad f\in X'.}

The net may be replaced by a weakly*-convergent sequence when the dual X ′ {\displaystyle X'} is separable. On the other hand, elements of the bidual of ℓ 1 {\displaystyle \ell ^{1}} that are not in ℓ 1 {\displaystyle \ell ^{1}} cannot be weak*-limit of sequences in ℓ 1, {\displaystyle \ell ^{1},} since ℓ 1 {\displaystyle \ell ^{1}} is weakly sequentially complete.
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