17.5. WEAK AND WEAK ∗ TOPOLOGIES 463

Corollary 17.5.6 If X is weakly separable and K ⊆ X ′ is compact in the weak ∗ topology,then K is sequentially compact. That is, if { fn}∞

n=1 ⊆ K, then there exists a subsequencefnk and f ∈ K such that for all x ∈ X,

limk→∞

fnk (x) = f (x).

Proof: By Theorem 17.5.5, K is a metric space for the metric described there and it iscompact. Therefore by the characterization of compact metric spaces, Proposition 7.6.5 onPage 144, K is sequentially compact. This proves the corollary.

17.5.3 Eberlein Smulian TheoremNext consider the weak topology. The most interesting results have to do with a reflexiveBanach space. The following lemma ties together the weak and weak ∗ topologies in thecase of a reflexive Banach space.

Lemma 17.5.7 Let J : X → X ′′ be the James map

Jx( f )≡ f (x)

and let X be reflexive so that J is onto. Then J is a homeomorphism of (X , weak topology)and (X ′′, weak ∗ topology).This means J is one to one, onto, and both J and J−1 arecontinuous.

Proof: Let f ∈ X ′ and let

B f (x,r)≡ {y : | f (x)− f (y)|< r}.

Thus B f (x,r) is a subbasic set for the weak topology on X . I claim that

JB f (x,r) = B f (Jx,r)

where B f (Jx,r) is a subbasic set for the weak ∗ topology. If y∈ B f (x,r) , then ∥Jy− Jx∥=∥x− y∥< r and so JB f (x,r)⊆ B f (Jx,r) . Now if x∗∗ ∈ B f (Jx,r) , then since J is reflexive,there exists y ∈ X such that Jy = x∗∗ and so

∥y− x∥= ∥Jy− Jx∥< r

showing that JB f (x,r) = B f (Jx,r) . A typical subbasic set in the weak ∗ topology is of theform B f (Jx,r) . Thus J maps the subbasic sets of the weak topology to the subbasic sets ofthe weak ∗ topology. Therefore, J is a homeomorphism as claimed.

The following is an easy corollary.

Corollary 17.5.8 If X is a reflexive Banach space, then the closed unit ball is weakly com-pact.

Proof: Let B be the closed unit ball. Then B = J−1 (B∗∗) where B∗∗ is the unit ball inX ′′ which is compact in the weak ∗ topology. Therefore B is weakly compact because J−1

is continuous.