Volume 16, 2004

University of Bialystok

Copyright (c) 2004 Association of Mizar Users

**Yasumasa Suzuki**- 2-14-18 Take, Yokosuka City, Kanagawa Pref., Japan

- We introduce the arithmetic addition and multiplication in the set of bounded real sequences and introduce the norm also. This set has the structure of the Banach space.

- The Banach Space of Bounded Real Sequences
- The Banach Space of Bounded Functions

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Binary operations.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [8]
Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama.
Real linear space of real sequences.
*Journal of Formalized Mathematics*, 15, 2003. - [9]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Jaroslaw Kotowicz.
Properties of real functions.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Jan Popiolek.
Real normed space.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Yasunari Shidama.
Banach space of bounded linear operators.
*Journal of Formalized Mathematics*, 15, 2003. - [19]
Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama.
Banach space of absolute summable real sequences.
*Journal of Formalized Mathematics*, 15, 2003. - [20]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [22]
Andrzej Trybulec.
Function domains and Fr\aenkel operator.
*Journal of Formalized Mathematics*, 2, 1990. - [23]
Andrzej Trybulec.
On the sets inhabited by numbers.
*Journal of Formalized Mathematics*, 15, 2003. - [24]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [25]
Wojciech A. Trybulec.
Subspaces and cosets of subspaces in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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