<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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.

Weak Convergence and Weak Convergence

Gifu National College of Technology Gifu, Japan

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

{"article-title":"Weak Convergence and Weak Convergence"}

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...

Weak Convergence and Weak Convergence

Online veröffentlicht: 30. Sept. 2015

Seitenbereich: 231 - 241

Eingereicht: 01. Juli 2015

DOI: https://doi.org/10.1515/forma-2015-0019

Schlüsselwörter
normed linear spaces, Banach spaces, duality and reflexivity, weak topologies, weak* topologies

© by Keiko Narita

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Weak Convergence and Weak Convergence

Online veröffentlicht: 30. Sept. 2015

Seitenbereich: 231 - 241

Eingereicht: 01. Juli 2015

DOI: https://doi.org/10.1515/forma-2015-0019

Schlüsselwörternormed linear spaces, Banach spaces, duality and reflexivity, weak topologies, weak* topologies

© by Keiko Narita

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Schlüsselwörter
normed linear spaces, Banach spaces, duality and reflexivity, weak topologies, weak* topologies