|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'585
Articles rated: 2609
25 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Weak* sequential closures in Banach space theory and their applications | | M. I. Ostrovskii
; | | Date: |
14 Mar 2002 | | Journal: | in: "General Topology in Banach Spaces", ed. by T. Banakh and A. Plichko, Nova Science, New York, 2001, pp. 21-34 | | Subject: | Functional Analysis; General Topology MSC-class: 46B10, 46B03, 54A20, 47G10 | math.FA math.GN | | Abstract: | Let X be a Banach space. Given a subset A of the dual space X* denote by $A_{(1)}$ the weak* sequential closure of A, i.e., the set of all limits of weak*-convergent sequences in A. The study of weak* sequential closures of linear subspaces of the duals of separable Banach spaces was initiated by S.Banach. The first results of this study were presented in the appendix to his book "Theorie des operations lineaires" (1932). It is natural to suppose that the reason for studying weak* sequential closures by S. Banach and S. Mazurkiewicz was the lack of acquaintance of S. Banach and his school with concepts of general topology. Although the name "General topology" was introduced later, the subject did already existed. F. Hausdorff introduced topological spaces in his book published in 1914, Alexandroff-Urysohn (1924) studied compactness, and A.Tychonoff published his theorem on compactness of products in 1929. Also J.von Neumann introduced the notion of a weak topology in his paper published in 1929. Using the notions of a topological space and the Tychonoff theorem, more elegant treatment of weak and weak* topologies, and the duality of Banach spaces was developed by L.Alaoglu, N.Bourbaki and S.Kakutani (1938-1940). Nevertheless, an "old-fashioned" treatment of S.Banach still attracts attention. It happens because the "sequential" approach is very useful in several contexts. The main purpose of the paper is to describe the history of this direction of research and to give an up-to-date (1999) survey of the results on weak* sequential closures and their applications. | | Source: | arXiv, math.FA/0203139 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER