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 [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].

MML Identifier: BHSP_7

Contents (PDF format)

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

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