On the topology of pointwise convergence on the ... - CARMA

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Lemma 1 Let Y be a closed separable linear subspace <strong>of</strong> aBanach space X and suppose that L ⊆ Ext(B X ∗) is weak ∗Lindelöf. Then <strong>the</strong>re exists a closed separable linear subspaceZ <strong>of</strong> X, containing Y , such that for any l ∗ ∈ L and any x ∗ ,y ∗ ∈ B Z ∗ if l ∗ | Z = 1 2 (x∗ + y ∗ ) <strong>the</strong>n x ∗ | Y = y ∗ | Y .Using this Lemma we can obtain <strong>the</strong> following <strong>the</strong>orem.Theorem 2 Let X be a Banach space and let L ⊆ Ext(B X ∗)be a weak ∗ Lindelöf subset. Then <strong>the</strong> set <strong>of</strong> all Z in S X suchthat {l ∗ | Z : l ∗ ∈ L} ⊆ Ext(B Z ∗) forms a rich family.Pro<strong>of</strong>: Let L denote <strong>the</strong> family <strong>of</strong> all closed separable linearsubspaces Z <strong>of</strong> X such that {l ∗ | Z : l ∗ ∈ L} ⊆ Ext(B Z ∗). Weshall verify that L is a rich family <strong>of</strong> closed separable linearsubspaces <strong>of</strong> X. So first let us consider an arbitrary closedseparable linear subspace Y <strong>of</strong> X, with <strong>the</strong> aim <strong>of</strong> showingthat <strong>the</strong>re exists a subspace Z ∈ L such that Y ⊆ Z. We
egin by inductively applying Lemma 1 to obtain an increasingsequence (Z n : n ∈ N) <strong>of</strong> closed separable linear subspaces<strong>of</strong> X such that: Y ⊆ Z 1 and for any l ∗ ∈ L and any x ∗ , y ∗ ∈B Z∗n+1if l ∗ | Zn+1 = 1 2 (x∗ + y ∗ ) <strong>the</strong>n x ∗ | Zn = y ∗ | Zn .We now claim that if Z := ⋃ n∈N Z n <strong>the</strong>n l ∗ | Z ∈ Ext(B Z ∗)for each l ∗ ∈ L. To this end, suppose that l ∗ ∈ L andl ∗ | Z = 1 2 (x∗ + y ∗ ) for some x ∗ , y ∗ ∈ B Z ∗.Then for each n ∈ N,l ∗ | Zn+1 = (l ∗ | Z )| Zn+1 = 1 2 (x∗ +y ∗ )| Zn+1 = 1 2 (x∗ | Zn+1 +y ∗ | Zn+1 )and x ∗ | Zn+1 , y ∗ | Zn+1 ∈ B Z∗n+1Therefore, by constructionx ∗ | Zn = y ∗ | Zn . Now since ⋃ n∈N Z n is dense in Z and both x ∗and y ∗ are continuous we may deduce that x ∗ = y ∗ ; which inturn implies that l ∗ | Z ∈ Ext(B Z ∗). This shows that Y ⊆ Zand Z ∈ L .To complete this pro<strong>of</strong> we must verify that for each increasingsequence <strong>of</strong> closed separable subspaces (Z n : n ∈ N) in
Page 1 and 2: On the</st
Page 3 and 4: oundary of X. In t
Page 5: The raison d’être for rich famil
Page 9 and 10: The fact that Ext(B X ∗) ⊆ B we
Page 12 and 13: for all j ∈ N and all 1 ≤ k ≤
Page 14 and 15: Proposition 3 Let B be any boundary
Page 16 and 17: Proof: In order to
Page 18: Corollary 3 Suppose that A is a top

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