Page 358 - Advanced Linear Algebra

P. 358

342 Advanced Linear Algebra

Here is the analog of Theorem 13.17.

Theorem 13.22 Let E ~¸" 2¹ be an arbitrary orthonormal family of

vectors in a Hilbert space . The two series
/
" and ((

2 2
converge or diverge together. If these series converge, then

i i ~ " ((
2 2
Proof. The first series converges if and only if for every , there exists a
finite set 0 2 such that

1q 0 ~ JÁ 1 finite ¬ " i
i

1
or equivalently,
1q 0 ~ JÁ 1 finite ¬ ((

1
and this is precisely what it means for the second series to converge. We leave
proof of the remaining statement to the reader.
The following is a useful characterization of arbitrary sums of nonnegative real
terms.

Theorem 13.23 Let ¸ 2¹ be a collection of nonnegative real numbers.

Then
~ sup (13.8 )
1 finite
2 12 1
provided that either of the preceding expressions is finite.
Proof. Suppose that

sup ~ 9 B

1 finite
12 1
Then, for any , there exists a finite set : 2 such that

9 9 c

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