International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 328481, 6 pages
doi:10.1155/2008/328481
Research Article
Norm Attaining Multilinear Forms on L1(μ)
Mathematics Department, Hebron University, P.O. Box 40, Hebron, West Bank, Palestine
Received 5 November 2007; Revised 23 March 2008; Accepted 9 June 2008
Academic Editor: Manfred Moller Moller
Copyright © 2008 Yousef Saleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Given an arbitrary measure μ, this study shows that the set of norm attaining multilinear
forms is not dense in the space of all continuous multilinear forms on L1(μ). However, we have the density if and only if μ is purely atomic. Furthermore, the study presents an example of a
Banach space X in which the set of norm attaining operators from X into X∗ is dense in the
space of all bounded linear operators L(X,X∗). In contrast, the set of norm attaining bilinear
forms on X is not dense in the space of continuous bilinear forms on X.