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

∗ ∈ B such that b ∗ (x) = ‖x‖ ≠ 0. Then for any y ∗ ∈ B Y ∗we havey ∗ (x) ≤ |y ∗ (x)| ≤ ‖y ∗ ‖‖x‖ ≤ ‖x‖ = b ∗ (x) = (b ∗ | Y )(x).In particular, e ∗ (x) ≤ b ∗ | Y (x). Since b ∗ | Y ∈ B Y ∗ andy ∗ (x) < e ∗ (x) for all y ∗ ∈ B Y ∗ \ {e ∗ }, it must be <strong>the</strong> casethat e ∗ = b ∗ | Y .❦ ✂✁The Main ResultsTheorem 3 Let B be any boundary for a Banach space Xthat is an L 1 -predual and suppose that {x n : n ∈ N} ⊆ X,<strong>the</strong>n {x n : n ∈ N} σ(X,B) ⊆ {x n : n ∈ N} σ(X,Ext(B X ∗)) .Pro<strong>of</strong>: In order to obtain a contradiction let us suppose that{x n : n ∈ N} σ(X,B) ⊈ {x n : n ∈ N} σ(X,Ext(B X ∗)) .Choose x ∈ {x n : n ∈ N} σ(X,B) \ {x n : n ∈ N} σ(X,Ext(B X ∗)) .Then <strong>the</strong>re exists a finite set {e ∗ 1, e ∗ 2, . . .,e ∗ m} ⊆ Ext(B X ∗)