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 II

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.

This paper is a continuation of our paper . We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of ) with respect to summable set by a functional $L$ in real Hilbert space. After presenting certain useful lemmas, we prove our main theorem that the summability for an orthonormal infinite set in real Hilbert space is equivalent to its summability with respect to the square of norm, say $H(x) = (x, x)$. Then we show that the square of norm $H$ commutes with infinite sum operation if the orthonormal set under our consideration is summable. Our main theorem is due to .

#### MML Identifier: BHSP_7

The terminology and notation used in this paper have been introduced in the following articles                     

#### Contents (PDF format)

1. Necessary and Sufficient Condition for Summable Set
2. Equivalence of Summability

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