Page 359 - Advanced Linear Algebra
P. 359
Hilbert Spaces 343
Hence, if ; 2 is a finite set for which ; : , then since ,
9 9 c
; :
and so
i 9c i
;
(
which shows that converges to . Finally, if the sum on the left of 13.8)
9
(
)
converges, then the supremum on the right is finite and so 13.8 holds.
The reader may have noticed that we have two definitions of convergence for
countably infinite series: the net version and the traditional version involving
the limit of partial sums. Let us write
B
% % and
o b ~
for the net version and the partial sum version, respectively. Here is the
relationship between these two definitions.
Theorem 13.24 Let / be a Hilbert space. If % / , then the following are
equivalent:
)
1 % converges net version to % ( )
o b
B
2 % ) converges unconditionally to %
~
Proof. Assume that 1 holds. Suppose that is any permutation of o b . Given
)
any , there is a finite set : o b for which
; :Á ; finite ¬ % c %
i
i
;
Let us denote the set of integers ¸ Á Ã Á ¹ by and choose a positive integer
0
such that ²0 ³ : . Then for we have
²0 ³ ²0 ³ : ¬ i % ² ³ c % ~
i %
c %
~
²0 ³
)
and so 2 holds.
