Journal of Formalized Mathematics
Volume 16, 2004
University of Bialystok
Copyright (c) 2004 Association of Mizar Users

## Banach Space of Bounded Real Sequences

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

### Summary.

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 terminology and notation used in this paper have been introduced in the following articles [21] [6] [27] [29] [28] [15] [20] [3] [1] [2] [24] [23] [9] [4] [5] [7] [26] [22] [16] [17] [13] [11] [12] [10] [25] [14] [8] [19] [18]

#### Contents (PDF format)

1. The Banach Space of Bounded Real Sequences
2. The Banach Space of Bounded Functions

#### Bibliography

[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.