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