An extension of James's compactness theorem

TL;DR

This paper extends James's compactness theorem to certain Banach space operators, showing that under specific conditions, norm attainment implies a weaker form of continuity.

Contribution

It introduces a new extension of James's theorem, linking norm attainment to (w^*,w) continuity for operators between Banach spaces.

Findings

T is (w^*,w) continuous under the given conditions

Norm attainment at F implies weaker continuity of T

Extension of classical James's theorem to new operator classes

Abstract

Let X and Y be Banach spaces and F a subset of B_{Y^*}. Endow Y with the topology \tau_F of pointwise convergence on F. Let T: X^* \to Y be a bounded linear operator which is (w^*, \tau_F) continuous. Assume that every vector in the range of T attains its norm at an element of F. Then it is proved that T is (w^*,w) continuous.