Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Yasumasa Suzuki**- 2-14-18 Take, Yokosuka-shi, Kanagawa, Japan
**Noboru Endou**- Gifu National College of Technology
**Yasunari Shidama**- Shinshu University, Nagano

- A continuation of [5]. As the example of real norm spaces, we introduce the arithmetic addition and multiplication in the set of absolute summable real sequences and introduce the norm also. This set has the structure of the Banach space.

- l1\_Space:The Space of Absolute Summable Real Sequences
- l1\_Space is Banach

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama.
Hilbert space of real sequences.
*Journal of Formalized Mathematics*, 15, 2003. - [6]
Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama.
Real linear space of real sequences.
*Journal of Formalized Mathematics*, 15, 2003. - [7]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jan Popiolek.
Real normed space.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Konrad Raczkowski and Andrzej Nedzusiak.
Series.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [14]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [15]
Wojciech A. Trybulec.
Subspaces and cosets of subspaces in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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