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

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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∗ .
The fact that Ext(B X ∗) ⊆ B weak∗ follows from Milman’s<strong>the</strong>orem and <strong>the</strong> fact that B X ∗ = co weak∗ (B); which in turnfollows from a separation argument.Let us also take this opportunity to observe that if B X denotes<strong>the</strong> closed unit ball in X <strong>the</strong>n B X is closed in <strong>the</strong> σ(X, B)<strong>topology</strong> for any boundary B <strong>of</strong> X.Finally, let us end this part <strong>of</strong> <strong>the</strong> talk with one more simpleobservation that will turn out to be useful in our laterendeavours.Proposition 2 Suppose that Y is a linear subspace <strong>of</strong> a Banachspace (X, ‖ · ‖) and B is any boundary for X. Then foreach e ∗ ∈ Exp(B Y ∗) <strong>the</strong>re exists b ∗ ∈ B such that e ∗ = b ∗ | Y .Pro<strong>of</strong>: Suppose that e ∗ ∈ Exp(B Y ∗) <strong>the</strong>n <strong>the</strong>re exists anx ∈ Y such that y ∗ (x) < e ∗ (x) for each y ∗ ∈ B Y ∗ \ {e ∗ }.By <strong>the</strong> fact that B is a boundary <strong>of</strong> (X, ‖ · ‖) <strong>the</strong>re exists a
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