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338 Advanced Linear Algebra

)
(
converges absolutely. But if 13.5 converges unconditionally, then so does
³
(13.4 ). The last part of the theorem follows from the continuity of the norm.
Now let E ~ ¸"Á "Á Ã ¹ be a countably infinite orthonormal set in / . The

Fourier expansion of a vector %/ is defined to be the sum
B
% ~ º%Á " »" (13.6 )
V
~
(
)
To see that this sum converges, observe that for any € , 13.3 gives

( (º%Á " »  % )
)
~
and so
B

( (º%Á " »  % )
)
~
which shows that the series on the left converges. Hence, according to Theorem
(
13.17, the Fourier expansion 13.6 converges unconditionally.
)
Moreover, since the inner product is continuous,
º% c %Á" » ~ º%Á" » c º%Á" » ~
V
V

ž
and so %c%  ´span ² ³µ ~ ´cspan ² ³µ ž . Hence, is the best approximation
E
E
V
% V
to in cspan E ³ . Finally, since c % % V ž % V , we again have
²
%
%
%
)) ~ V % )) c ) c % % )  V ))
and so
))
))%%
V
with equality if and only if %~% V , which happens if and only if %  cspan ²E . ³
Thus, the following analog of Theorem 13.16 holds.
Theorem 13.18 Let E ~ ¸"Á "Á Ã ¹ be a countably infinite orthonormal set in

/
a Hilbert space . For any  % / , the Fourier expansion
B
% ~ º%Á " »"
V
~
%
²
%
of converges unconditionally and is the best approximation to in cspan E . ³
We also have Bessel's inequality
V
))
))%%