A Reverse Theorem on the parallel to.parallel to-omega* Continuity of the Dual Map

作者：de Kock Mienie; Javier Garcia Pacheco Francisco

来源：Journal of Function Spaces, 2015, 2015: 864173.

DOI：10.1155/2015/864173

Given a Banach space X, x is an element of S-X, and J(X)(x) = {x* is an element of S-X* : x*(x) = 1}, we define the set J(X)*(x) of all x* is an element of S-X* for which there exist two sequences (x(n))(n epsilon N) subset of S-X \ {x} and (x(n)*)(n is an element of N) subset of S-X* such that (x(n))(n is an element of N) converges to x, (x(n)*)(n is an element of N) has a subnet omega*-convergent to x*, and x(n)*(x(n))(n is an element of N) = 1 for all n is an element of N. We prove that if X is separable and reflexive and X* enjoys the Radon-Riesz property, then J(X)*(x) is contained in the boundary of J(X)(x) relative to S-X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of J(X)(x) relative to S-X* is contained in J(X)*(x).

• 出版日期2015