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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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SCImago Journal Rank (SJR) 2015: 0.134
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ISSN
1898-9934
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Volume 23, Issue 3 (Sep 2015)

Issues

Weak Convergence and Weak Convergence

Keiko Narita
  • Hirosaki-city Aomori, Japan
/ Yasunari Shidama
  • Shinshu University Nagano, Japan
/ Noboru Endou
  • Gifu National College of Technology Gifu, Japan
Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/forma-2015-0019

Abstract

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.

Keywords: normed linear spaces; Banach spaces; duality and reflexivity; weak topologies; weak* topologies

MSC: 46E15; 46B10; 03B35

MML identifier:: DUALSP03

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About the article

Received: 2015-07-01

Published Online: 2015-09-30

Published in Print: 2015-09-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2015-0019. Export Citation

© by Keiko Narita. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

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