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Given a Banach space X, x∈X, and Xx=x*∈X*:x*x=1, we define the set X*x of all x*∈X* for which there exist two sequences xnn∈N⊆X∖{x} and xn*n∈N⊆X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and  X* enjoys the Radon-Riesz property, then X*x is contained in the boundary of Xx relative to 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 Xx relative to X* is contained in X*x

journal of function spaces

A Reverse Theorem on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mfenced open="‖" close="‖" separators="|"><mml:mrow><mml:mo>·</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>Continuity of the Dual Map

A Reverse Theorem on the·-w*Continuity of the Dual Map