Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

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

- This paper is a continuation of our paper [22]. We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of [22]) 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 [8].

- Necessary and Sufficient Condition for Summable Set
- Equivalence of Summability

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*Journal of Formalized Mathematics*, 15, 2003. - [22]
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*Journal of Formalized Mathematics*, 15, 2003.

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