Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Hideki Sakurai; Hiroyuki Okazaki; Yasunari Shidama

doi:10.2478/v10037-012-0032-y

.blurhash-client-img { display: none !important; }

Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Formalized Mathematics

Volume 20 (2012): Issue 4 (December 2012)

By: Hideki Sakurai, Hiroyuki Okazaki and Yasunari Shidama

Open Access

|Feb 2013

Abstract

In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[2] Czesław Bylinski. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.
Search in Google Scholar
[3] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[4] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[5] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
[7] Czesław Bylinski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.
Search in Google Scholar
[8] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249-253, 2003.
Search in Google Scholar
[9] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
Search in Google Scholar
[10] Isao Miyadera. Functional Analysis. Riko-Gaku-Sya, 1972.
Search in Google Scholar
[11] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.
Search in Google Scholar
[12] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51-59, 2011, doi: 10.2478/v10037-011-0009-2.10.2478/v10037-011-0009-2
Search in Google Scholar
[13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
Search in Google Scholar
[14] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
Search in Google Scholar
[15] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
Search in Google Scholar
[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. FormalizedMathematics, 1(2):297-301, 1990.
Search in Google Scholar
[17] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
Search in Google Scholar
[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar
[20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10037-012-0032-y | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

Journal RSS Feed

Language: English

Page range: 271 - 274

Published on: Feb 2, 2013

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2013 Hideki Sakurai, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 20 (2012): Issue 4 (December 2012)