# Functional Space C (ω), C 0 (ω)

Katuhiko Kanazashi; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

- Volume: 20, Issue: 1, page 15-22
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topKatuhiko Kanazashi, Hiroyuki Okazaki, and Yasunari Shidama. " Functional Space C (ω), C 0 (ω) ." Formalized Mathematics 20.1 (2012): 15-22. <http://eudml.org/doc/268270>.

@article{KatuhikoKanazashi2012,

abstract = {In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.},

author = {Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama},

journal = {Formalized Mathematics},

language = {eng},

number = {1},

pages = {15-22},

title = { Functional Space C (ω), C 0 (ω) },

url = {http://eudml.org/doc/268270},

volume = {20},

year = {2012},

}

TY - JOUR

AU - Katuhiko Kanazashi

AU - Hiroyuki Okazaki

AU - Yasunari Shidama

TI - Functional Space C (ω), C 0 (ω)

JO - Formalized Mathematics

PY - 2012

VL - 20

IS - 1

SP - 15

EP - 22

AB - In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

LA - eng

UR - http://eudml.org/doc/268270

ER -

## References

top- Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
- Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
- Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.
- Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237-242, 2004.
- Noboru Endou. Banach space of absolute summable complex sequences. Formalized Mathematics, 12(2):191-194, 2004.
- Noboru Endou. Complex Banach space of bounded linear operators. Formalized Mathematics, 12(2):201-209, 2004.
- Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93-102, 2004.
- Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109-117, 2004.
- Noboru Endou. Complex valued functions space. Formalized Mathematics, 12(3):231-235, 2004.
- Noboru Endou. Continuous functions on real and complex normed linear spaces. Formalized Mathematics, 12(3):403-419, 2004.
- Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
- Katuhiko Kanazashi, Hiroyuki Okazaki, and Yasunari Shidama. Banach algebra of bounded complex-valued functionals. Formalized Mathematics, 19(2):121-126, 2011, doi: 10.2478/v10037-011-0019-0.[Crossref] Zbl1276.47091
- Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
- Chanapat Pacharapokin, Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Complex function differentiability. Formalized Mathematics, 17(2):67-72, 2009, doi: 10.2478/v10037-009-0007-9.[Crossref]
- Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
- Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.[Crossref]
- Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
- Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
- Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
- Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.