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

for all j ∈ N and all 1 ≤ k ≤ m. Thus,⋂{y ∈ X : |b ∗ k(x) − b ∗ k(y)| < ε} ∩ {x n : n ∈ N} = ∅.1≤k≤mThis contradicts <strong>the</strong> fact that x ∈ {x n : n ∈ N} σ(X,B) ; whichcompletes <strong>the</strong> pro<strong>of</strong>.❦ ✂✁Corollary 1 Let B be any boundary for a Banach spaceX that is an L 1 -predual. Then every relatively countablyσ(X, B)-compact subset is relatively countably σ(X, Ext(B X ∗))-compact. In particular, every norm bounded, relatively countablyσ(X, B)-compact subset is relatively weakly compact.Pro<strong>of</strong>: Suppose that a nonempty set C ⊆ X is relativelycountably σ(X, B)-compact. Let {c n : n ∈ N} be any sequencein C <strong>the</strong>n by Theorem 3∅ ≠ ⋂ {c k : k ≥ n} σ(X,B) ⊆ ⋂ {c k : k ≥ n} σ(X,Ext(B X ∗))n∈Nn∈N