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The distance $dist(B;X)$ when $B$ is a boundary of $ball(X^{**})$

Abstract : Let $X$ be a real Banach space and $\bdy$ a boundary of the unit ball $B(X^{**})$ of the bidual $X^{**}$ (which means that for each $x^*\in X^*$ there is $b\in \bdy$ such that $\la b,x^*\ra=\|x^*\|$). We show that $dist(\bdy,X)=dist(B(X^{**}),X)$ where $dist(A,X)$ denotes the sup of all $dist(a, X)$ with $a\in A$. Since $\co^{w^*}(\bdy)=B(X^{**})$ this is in contrast with the fact that in general strict inequality can occur between $dist (K,X)$ and \linebreak $dist(\co^{w^*}(K),X)$ even for a $w^*$-compact $K\subset X^{**}$.
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Submitted on : Friday, July 30, 2010 - 3:51:10 PM
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A. S. Granero, J. M. Hernández, H. Pfitzner. The distance $dist(B;X)$ when $B$ is a boundary of $ball(X^{**})$. Proceedings of the American Mathematical Society, American Mathematical Society, 2011, 139 (3), pp.1095-1098. ⟨hal-00507504⟩

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