Page 352 - Advanced Linear Algebra
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336 Advanced Linear Algebra
later in this chapter that any two Hilbert bases for a Hilbert space have the same
cardinality.
Since an orthonormal set E is maximal if and only if E ~ ¸ ¹ , Corollary
13.14 gives the following characterization of Hilbert bases.
E
Theorem 13.15 Let be an orthonormal subset of a Hilbert space / . The
following are equivalent:
)
1 E is a Hilbert basis
2) E ~ ¸ ¹
3 E ) is a total subset of , that is, cspan E / ² ³ ~ / .
)
Part 3 of this theorem says that a subset of a Hilbert space is a Hilbert basis if
and only if it is a total orthonormal set.
Fourier Expansions
We now want to take a closer look at best approximations. Our goal is to find an
explicit expression for the best approximation to any vector from within a
%
closed subspace of a Hilbert space . We will find it convenient to consider
:
/
three cases, depending on whether : has finite, countably infinite, or
uncountable dimension.
The Finite-Dimensional Case
Suppose that E ~¸" Á Ã Á " ¹ is an orthonormal set in a Hilbert space / .
Recall that the Fourier expansion of any %/ , with respect to , is given by
E
% ~ º%Á " »"
V
~
where º%Á " » is the Fourier coefficient of with respect to " % . Observe that
V
V
º% c %Á" » ~ º%Á" » c º%Á" » ~
and so %c% span² ³ . Thus, according to Theorem 13.9, the Fourier
V
E
²
%
% V
expansion is the best approximation to in span E ³ . Moreover, since
V
%c% % V , we have
%
%
)) ~ V % )) c ) c % % ) V ))
and so
V
))%%
))
with equality if and only if %~% V , which happens if and only if % span ²E . ³
Let us summarize.
