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

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