Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## On Some Properties of Real Hilbert Space. Part I

Hiroshi Yamazaki
Shinshu University, Nagano
Yasumasa Suzuki
Take, Yokosuka-shi, Japan
Takao Inoue
The Iida Technical High School, Nagano
Yasunari Shidama
Shinshu University, Nagano

### Summary.

In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a necessary and sufficient condition for summability of an infinite set of vectors of real Hilbert space as our main theorem. The last theorem is due to [8].

#### MML Identifier: BHSP_6

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

#### Contents (PDF format)

1. Preliminaries
2. Summability
3. Necessary and Sufficient Condition for Summability

#### Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. 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] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[8] P. R. Halmos. \em Introduction to Hilbert Space. American Mathematical Society, 1987.
[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[10] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[11] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Journal of Formalized Mathematics, 5, 1993.
[12] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.
[13] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.
[14] Jan Popiolek. Introduction to Banach and Hilbert spaces --- part I. Journal of Formalized Mathematics, 3, 1991.
[15] Jan Popiolek. Introduction to Banach and Hilbert spaces --- part III. Journal of Formalized Mathematics, 3, 1991.
[16] Andrzej Trybulec. Semilattice operations on finite subsets. Journal of Formalized Mathematics, 1, 1989.
[17] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[18] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[19] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[20] Wojciech A. Trybulec. Binary operations on finite sequences. Journal of Formalized Mathematics, 2, 1990.
[21] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[23] Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel's inequality. Journal of Formalized Mathematics, 15, 2003.