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

Proposition 3 Let B be any boundary for a Banach space Xthat is an L 1 -predual and suppose that A is a separable Bairespace. If X is ℵ 0 -monolithic in <strong>the</strong> σ(X, Ext(B X ∗)) <strong>topology</strong><strong>the</strong>n for each continuous mapping f : A → (X, σ(X, B))<strong>the</strong>re exists a dense subset D <strong>of</strong> A such that f is continuouswith respect to <strong>the</strong> norm <strong>topology</strong> on X at each point <strong>of</strong> D.Pro<strong>of</strong>: Fix ε > 0 and consider <strong>the</strong> open set:O ε := ⋃ {U ⊆ A : U is open and ‖ · ‖ − diam[f(U)] ≤ 2ε}.We shall show that O ε is dense in A. To this end, let Wbe a nonempty open subset <strong>of</strong> A and let {a n : n ∈ N} be acountable dense subset <strong>of</strong> W. Then by continuityf(W) ⊆ {f(a n ) : n ∈ N} σ(X,B) ;which is norm separable by Corollary 2. Therefore <strong>the</strong>re existsa countable set {x n : n ∈ N} in X such that f(W) ⊆