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Very smooth points of spaces of operators

T. S. S. R. K. Rao
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Abstract

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is an $M$-ideal in the space of bounded operators, a very smooth operator $T$ attains its norm at a unique vector $x$ (up to a constant multiple) and $T(x)$ is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon--Nikodým property. We give an example of a smooth Banach space without any very smooth points.

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T. S. S. R. K. Rao. 2003-12-05. Very smooth points of spaces of operators. https://arxiv.org/abs/math/0312116

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