560 CHAPTER 21. BANACH SPACES

This diagram follows from Theorem 21.2.10 on Page 544, the theorem on adjoints. Nowlet y∗∗ ∈ Y ′′. Is y∗∗ = JY y for some y ∈ Y ? Since X is reflexive, i∗∗y∗∗ = JX (y) for some y.I want to show that y ∈ Y . If it is not in Y then since Y is closed, there exists x∗ ∈ X ′ suchthat x∗ (y) ̸= 0 but x∗ (Y ) = 0. Then i∗x∗ = 0. Hence

0 = y∗∗ (i∗x∗) = i∗∗y∗∗ (x∗) = J (y)(x∗) = x∗ (y) ̸= 0,

a contradiction. Hence y ∈ Y . Letting JY denote the James map from Y to Y ′′ and x∗ ∈ X ′,

y∗∗ (i∗x∗) = i∗∗y∗∗ (x∗) = JX (y)(x∗)

= x∗ (y) = x∗ (iy) = i∗x∗ (y) = JY (y)(i∗x∗)

Since i∗ is onto, this shows y∗∗ = JY (y) . ■

Theorem 21.5.14 (Eberlein Smulian) The closed unit ball in a reflexive Banachspace X, is weakly sequentially compact. By this is meant that if {xn} is contained in theclosed unit ball, there exists a subsequence,

{xnk

}and x ∈ X such that for all x∗ ∈ X ′, it

follows that x∗(xnk

)→ x∗ (x) .

Proof: Let {xn} ⊆ B ≡ B(0,1). Let Y be the closure of the linear span of {xn}. ThusY is a separable. It is reflexive because it is a closed subspace of a reflexive space so theabove lemma applies. By the Banach Alaoglu theorem, the closed unit ball B∗Y in Y ′ isweak ∗ compact. Also by Theorem 21.5.5, B∗Y is a metric space with a suitable metric. Thefollowing diagram illustrates the rest of the argument.

B∗∗ Y ′′ i∗∗ 1-1→ X ′′

B∗Y Y ′ weak∗ separable i∗ onto← X ′

BY Y separable i→ X

Thus B∗Y is complete and totally bounded with respect to this metric and it follows thatB∗Y with the weak ∗ topology is separable. This implies Y ′ is also separable in the weak ∗topology. To see this, let {y∗n} ≡ D be a weak ∗ dense set in B∗Y and let y∗ ∈ Y ′. Let p bea large enough positive rational number that y∗/p ∈ B∗. Then if A is any finite set from Y,there exists y∗n ∈ D such that ρA (y

∗/p− y∗n) <ε

p . It follows py∗n ∈ BA (y∗,ε) showing thatrational multiples of D are weak ∗ dense in Y ′. Letting BY = B∩Y, this BY is the closedunit ball in Y and Y ′ is weak ∗ separable. Therefore, by Corollary 21.5.10, BY is weaklysequentially compact. Thus there exists

{xnk

}such that xnk → x ∈ BY weakly in Y. Letting

x∗ ∈ X∗, i∗x∗ ∈ Y ′ and so

x∗(xnk

)= i∗x∗

(xnk

)→ i∗x∗ (x) = x∗ (x)

and so in fact, xnk → x weakly in X . ■The following is the form of the Eberlein Smulian theorem which is often used.

Corollary 21.5.15 Let {xn} be any bounded sequence in a reflexive Banach space X .Then there exists x ∈ X and a subsequence,

{xnk

}such that for all x∗ ∈ X ′, it follows that

limk→∞ x∗(xnk

)= x∗ (x) .