Page 353 - Advanced Linear Algebra

Hilbert Spaces 337

Theorem 13.16 Let E ~¸" Á Ã Á " ¹ be a finite orthonormal set in a Hilbert

space / . For any %  / , the Fourier expansion % V of % is the best
%
²
approximation to in span E ³ . We also have Bessel's inequality
))%%
))
V
or equivalently,

( (º%Á " »  % ) (13.3 )
)
~
with equality if and only if % span ²E . ³
The Countably Infinite-Dimensional Case
In the countably infinite case, we will be dealing with infinite sums and so
questions of convergence will arise. Thus, we begin with the following.

Theorem 13.17 Let E ~ ¸"Á "Á Ã ¹ be a countably infinite orthonormal set in

a Hilbert space . The series
/
B
" (13.4 )

~
converges in if and only if the series
/
B

(( (13.5 )
~
converges in . If these series converge, then they converge unconditionally
s
(that is, any series formed by rearranging the order of the terms also
(
)
converges . Finally, if the series 13.4 converges, then
)

B B
~ " ((

i i
~ ~
and the partial sums of
Proof. Denote the partial sums of the first series by
the second series by . Then for 

) ) c ~ i i ( ~ ( " ~ ( ( c
~ b ~ b
Hence ² ³ is a Cauchy sequence in if and only if ² ³ is a Cauchy sequence
/

in . Since both / s and are complete, ² s ³ converges if and only if ² ³
converges.
(
If the series 13.5 converges, then it converges absolutely and hence
)
(
unconditionally. A real series converges unconditionally if and only if it