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 [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
The terminology and notation used in this paper have been
introduced in the following articles
[16]
[19]
[6]
[1]
[17]
[9]
[4]
[5]
[20]
[18]
[12]
[13]
[14]
[3]
[7]
[10]
[15]
[11]
[2]
[21]
[22]
-
Necessary and Sufficient Condition for Summable Set
-
Equivalence of Summability
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Received April 17, 2003
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