# Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Hideki Sakurai; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

- Volume: 20, Issue: 4, page 271-274
- ISSN: 1426-2630

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topHideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. "Banach’s Continuous Inverse Theorem and Closed Graph Theorem." Formalized Mathematics 20.4 (2012): 271-274. <http://eudml.org/doc/267765>.

@article{HidekiSakurai2012,

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.},

author = {Hideki Sakurai, Hiroyuki Okazaki, Yasunari Shidama},

journal = {Formalized Mathematics},

language = {eng},

number = {4},

pages = {271-274},

title = {Banach’s Continuous Inverse Theorem and Closed Graph Theorem},

url = {http://eudml.org/doc/267765},

volume = {20},

year = {2012},

}

TY - JOUR

AU - Hideki Sakurai

AU - Hiroyuki Okazaki

AU - Yasunari Shidama

TI - Banach’s Continuous Inverse Theorem and Closed Graph Theorem

JO - Formalized Mathematics

PY - 2012

VL - 20

IS - 4

SP - 271

EP - 274

AB - 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.

LA - eng

UR - http://eudml.org/doc/267765

ER -

## References

top- [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [2] Czesław Bylinski. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.
- [3] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [4] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [5] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [7] Czesław Bylinski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.
- [8] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249-253, 2003.
- [9] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
- [10] Isao Miyadera. Functional Analysis. Riko-Gaku-Sya, 1972.
- [11] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.
- [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.[Crossref] Zbl1276.46015
- [13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
- [14] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
- [15] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
- [16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. FormalizedMathematics, 1(2):297-301, 1990.
- [17] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
- [18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

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