Weak Convergence and Weak Convergence

Keiko Narita; Yasunari Shidama; Noboru Endou

Formalized Mathematics (2015)

  • Volume: 23, Issue: 3, page 231-241
  • ISSN: 1426-2630

Abstract

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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.

How to cite

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Keiko Narita, Yasunari Shidama, and Noboru Endou. "Weak Convergence and Weak Convergence." Formalized Mathematics 23.3 (2015): 231-241. <http://eudml.org/doc/276419>.

@article{KeikoNarita2015,
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.},
author = {Keiko Narita, Yasunari Shidama, Noboru Endou},
journal = {Formalized Mathematics},
keywords = {normed linear spaces; Banach spaces; duality and reflexivity; weak topologies; weak* topologies; weak topologies},
language = {eng},
number = {3},
pages = {231-241},
title = {Weak Convergence and Weak Convergence},
url = {http://eudml.org/doc/276419},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Keiko Narita
AU - Yasunari Shidama
AU - Noboru Endou
TI - Weak Convergence and Weak Convergence
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 3
SP - 231
EP - 241
AB - 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.
LA - eng
KW - normed linear spaces; Banach spaces; duality and reflexivity; weak topologies; weak* topologies; weak topologies
UR - http://eudml.org/doc/276419
ER -

References

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