2003/12/05 by T. S. S. R. K. Rao
Mathematics · #math.FA
published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 1, February 2003, pp. 53-64 · 12 pages, no figures, no tables
arxiv created 2003/12/05 · arxiv updated 2009/12/01
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.