Reflexive Banach space

A Banach space $X$ is reflexive when its canonical embedding $J_X:X\to X^{\ast \ast }$ is surjective. Equivalently, its closed unit ball is weakly compact.

If $Z$ is a Banach space and $X$ is a norm-closed reflexive linear subspace of $Z^{\ast }$, evaluation gives the displayed surjection. Restriction $R:Z^{\ast \ast }\to X^{\ast }$ is weak-star continuous and maps the closed unit ball onto the target ball by the Hahn-Banach theorem. The Goldstine theorem then makes $T(B_Z)$ weak-star dense in $B_{X^{\ast }}$, where $T=R{J}_Z$. Reflexivity of $X$ makes this target weak-star topology equal to its weak topology. Convexity and the Hahn-Banach separation theorem convert weak density of this ball image to norm density.

To pass from density to surjectivity, approximate a desired $f$ by $T{z}_1$ with $\Vert z_1\Vert \le \Vert f\Vert$ and residual at most $\Vert f\Vert /2$. Repeat on each residual. The chosen vectors have summable norms, with sum at most $2\Vert f\Vert$, and the residuals tend to zero geometrically. Completeness of $Z$ produces $z={\sum }_n{z}_n$ with $Tz=f$. Using any smaller residual ratio proves the infimum lifting norm equals $\Vert f\Vert$, although that infimum need not be attained. This argument proves onto-ness, rather than merely density of the evaluation functionals.

A nonempty norm-closed convex subset $C$ of a reflexive Banach space attains its distance to zero. For $d={\inf }_C\Vert x\Vert$, the sets $C\cap (d+1/n)B_X$ are nonempty weakly closed subsets of one weakly compact ball, and form a decreasing family. Compactness gives a point in their intersection, of norm $d$. Mazur theorem supplies weak closedness of $C$ and the balls; no sequential-compactness theorem is required.

A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.

Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. This is the sequential form of weak compactness supplied by the Eberlein-Šmulian theorem.