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

L , ⋃ n∈N Z n ∈ L . This however, follows easily from <strong>the</strong>definition <strong>of</strong> <strong>the</strong> family L .❦ ✂✁Let X be a normed linear space. Then we say that an elementx ∗ ∈ B X ∗ is weak ∗ exposed if <strong>the</strong>re exists an element x ∈ Xsuch that y ∗ (x) < x ∗ (x) for all y ∗∈ B X ∗ \ {x ∗ }. It isnot difficult to show that if Exp(B X ∗) denotes <strong>the</strong> set <strong>of</strong> allweak ∗ exposed points <strong>of</strong> B X ∗ <strong>the</strong>n Exp(B X ∗) ⊆ Ext(B X ∗).However, if X is a separable L 1 -predual <strong>the</strong>n <strong>the</strong> relationshipbetween Exp(B X ∗) and Ext(B X ∗) is much closer.Lemma 2 If X is a separable L 1 -predual, <strong>the</strong>n Exp(B X ∗) =Ext(B X ∗).Let us also pause for a moment to recall that if B is anyboundary <strong>of</strong> a Banach space X <strong>the</strong>nExp(B X ∗) ⊆ B ∩ Ext(B X ∗) ⊆ Ext(B X ∗) ⊆ B weak∗ .