Mosco convergence and the Kadec property. (English) Zbl 0672.46007
Let X be a real Banach space and \(\{C_ n\}^ a \)sequence of sets in X. The sequence \(\{C_ n\}\) is said to converge Wijsman to a set C if \(\lim_{n\to \infty}d(x,c_ n)=d(x,c)\) for all x in X, and to converge Mosco to C if
(i) For each x in C and sufficiently large n there are \(x_ n\) in \(C_ n\) for which \(\{x_ n\}\) converges to x, and
(ii) If \(x_{ni}\in C_{ni}\) for \(i=1,2,..\). and \(\{x_{ni}\}\) converges weakly to some x, then \(x\in C.\)
In this paper it is shown that Wijsman and Mosco convergence are the same for sequences of convex sets in X if and only if X is reflexive and the norm on \(X^*\) has the Kadec-property (i.e. whenever \(\{x^*_ n\}\subset X^*\) is such that \(\| x^*_ n\| =1\) for all n and \(\{x^*_ n\}\) is \(w^*\)-convergent to some \(x^*\in X^*\) with \(\| x^*\| =1\), then \(\{x^*_ n\}\) converges to x in norm).
(i) For each x in C and sufficiently large n there are \(x_ n\) in \(C_ n\) for which \(\{x_ n\}\) converges to x, and
(ii) If \(x_{ni}\in C_{ni}\) for \(i=1,2,..\). and \(\{x_{ni}\}\) converges weakly to some x, then \(x\in C.\)
In this paper it is shown that Wijsman and Mosco convergence are the same for sequences of convex sets in X if and only if X is reflexive and the norm on \(X^*\) has the Kadec-property (i.e. whenever \(\{x^*_ n\}\subset X^*\) is such that \(\| x^*_ n\| =1\) for all n and \(\{x^*_ n\}\) is \(w^*\)-convergent to some \(x^*\in X^*\) with \(\| x^*\| =1\), then \(\{x^*_ n\}\) converges to x in norm).
Reviewer: J.R.Holub
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