Page 347 - Advanced Linear Algebra
P. 347
Hilbert Spaces 331
with the property that
Á 5 ¬ % c % ) )
Clearly, we can choose 5 5 Ä , in which case
) c 5 % )%
5 b
and so
B B
% c % ) 5 ) B
5 b
~ ~
Thus, according to hypothesis, the series
B
²% 5 c % ³
5 b
~
converges. But this is a telescoping series, whose th partial sum is
% c 5 % 5 b
and so the subsequence ²% ³ converges. Since any Cauchy sequence that has a
5
convergent subsequence must itself converge, the sequence ²% ³ converges and
=
so is complete.
An Approximation Problem
Suppose that is an inner product space and that is a subset of . It is of
=
:
=
:
considerable interest to be able to find, for any %= , a vector in that is
closest to in the metric induced by the inner product, should such a vector
%
exist. This is the approximation problem for .
=
Suppose that %= and let
~ % inf c ) )
:
Then there is a sequence for which
) ~% c ) ¦
as shown in Figure 13.1.
