Very Smooth Points of Spaces of Operators

[摘要] 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.

[学科分类] 数学（综合）

[关键词] Very smooth points;spaces of operators;𝑀-ideals.