Page 361 - Advanced Linear Algebra

Hilbert Spaces 345

that

sup ( (º%Á " »  % )
)
1 finite
1‹2 1
and so Theorem 13.23 tells us that the sum
( (º%Á " »
2
converges. Hence, according to Theorem 13.22, the Fourier expansion
% ~ º%Á " »"
V
2
of also converges and
%

%
)) ~ V ( % º Á " ( »
2
Note that, according to Theorem 13.21, is a countably infinite sum of terms of
% V
and so is in cspan ²E . ³
the form º%Á " »"
(
The continuity of infinite sums with respect to the inner product Theorem
)
13.19 implies that
º% c %Á" » ~ º%Á" » c º%Á" » ~
V
V

ž
and so %c%  ´ span² ³µ ~ ´ cspan² ³µ ž . Hence, Theorem 3.9 tells us that % V
E
E
V
is the best approximation to in cspan E ³ . Finally, since c % % V ž % V , we again
²
%
have
)) ~ V % )) c ) c % % )  V ))
%
%
and so
V
))%%
))
with equality if and only if %~% V , which happens if and only if %  cspan ²E . ³
Thus, we arrive at the most general form of a key theorem about Hilbert spaces.
Theorem 13.25 Let E ~¸" “  2¹ be an orthonormal family of vectors in

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