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B-Completeness and the Open-Mapping Theorem

In: The Open Mapping and Closed Graph Theorems in Topological Vector Spaces

Author

Listed:
  • Taqdir Husain

    (McMaster University)

Abstract

In Chapter 3 we tried to find pairs E and F of 1.c. spaces for which statements (A) and (B) (Chapter 3, § 1) are true. To this end we proved several theorems. Continuing our search for such pairs, an apparent generalization of Theorems 3, 7, and 9 (Chapter 3) is suggested to the reader as follows: If E is a complete 1.c. space and F a t-space then statements (A) and (B) (Chapter 3, § 1) are true. Unfortunately this generalization is not true as will be shown in the sequel. One needs a stronger notion than that of completeness in order to have the open mapping and closed graph theorems. This is exactly where B-completeness comes in. The notion of B-completeness is due to V. Pták [28]. The results of this chapter are based on [28], [31] and [32]. Because of its repeated use we first prove the following:

Suggested Citation

  • Taqdir Husain, 1965. "B-Completeness and the Open-Mapping Theorem," Springer Books, in: The Open Mapping and Closed Graph Theorems in Topological Vector Spaces, chapter 4, pages 45-58, Springer.
    • Handle: RePEc:spr:sprchp:978-3-322-96210-2_4
    DOI: 10.1007/978-3-322-96210-2_4
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