Page 360 - Advanced Linear Algebra
P. 360
344 Advanced Linear Algebra
)
)
Next, assume that 2 holds, but that the series in 1 does not converge. Then
there exists an such that for any finite subset 0 o b , there exists a finite
subset with q 1 0 ~ J for which
1
i % i
1
of
From this, we deduce the existence of a countably infinite sequence 1
mutually disjoint finite subsets of o b with the property that
max²1 ³ ~ 4 b ~ min²1 b ³
and
i % i
1
Now we choose any permutation o ¦ ¢ b o b with the following properties
1) ²´ Á 4 µ³ ´ Á 4 µ
) ¹ , then
2 if 1 ~ ¸ Á ÁÃÁ Á "
Á Á ² b Á Á Ã Á ² b" c ³ ~ Á "
2
³ ~
² ³ ~
The intention in property 2 is that for each , takes a set of consecutive
)
.
integers to the integers in 1
For any such permutation , we have
b"c
i ² ³ ~ i i % % i
~ 1
which shows that the sequence of partial sums of the series
B
² ³
%
~
is not Cauchy and so this series does not converge. This contradicts 2 and
)
)
)
)
shows that 2 implies at least that 1 converges. But if 1 converges to & / ,
then since 1 implies 2 and since unconditional limits are unique, we have
)
)
)
)
&~ %. Hence, 2 implies 1 .
Now we can return to the discussion of Fourier expansions. Let
E ~¸" 2¹ be an arbitrary orthonormal set in a Hilbert space /. Given
any %/ , we may apply Theorem 13.16 to all finite subsets of , to deduce
E
