Page 357 - Advanced Linear Algebra
P. 357
Hilbert Spaces 341
Then ²& ³ is a Cauchy sequence, since
) )& c & i ~ %c % i i ~ %c % i
0 0 0 c 0 0 c 0
i i b i % % i b ¦
0c0 0 c0
Since is assumed complete, we have & ² ³ ¦ . &
=
5
Now, given , there exists an such that
5 ¬ & c & ~ % c & 2
)
i
)
i
0
Setting ~ max ¸5Á ° ¹ gives for ; 0 Á finite,
;
%c & ~ % %c & b
i i i i
; 0 ; c 0
i & i b i % % c i b
0 ; c 0
and so 2 % converges to . &
The following theorem tells us that convergence of an arbitrary sum implies that
only countably many terms can be nonzero so, in some sense, there is no such
thing as a nontrivial uncountable sum.
Theorem 13.21 Let A ~¸% 2¹ be an arbitrary family of vectors in an
inner product space . If the sum
=
%
2
can be nonzero.
converges, then at most a countable number of terms %
Proof. According to Theorem 13.20, for each , we can let 0 2 0 ,
finite, be such that
1q 0 ~ JÁ 1 finite ¬ % i
i
1
Let 0~ 0 . Then is countable and
0
¤ 0 ¬ ¸ ¹ q 0 ~ J for all ¬ % ) for all ¬ % ~
)
