Page 346 - Advanced Linear Algebra
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330 Advanced Linear Algebra
subspace of a complete inner product space : Z and as such it is closed.
:
:
However, is dense in and so ~ : Z : : Z , which shows that is complete.
Infinite Series
Since an inner product space allows both addition of vectors and convergence of
sequences, we can define the concept of infinite sums, or infinite series.
Definition Let = be an inner product space. The th partial sum of the
=
sequence ²% ³ in is
~ % b Ä b %
If the sequence ² ³ of partial sums converges to a vector = , that is, if
) ) c ¦ as ¦ B
then we say that the series % converges to and write
B
%~
~
We can also define absolute convergence.
is said to be absolutely convergent if the series
Definition A series %
B
) ) %
~
converges.
The key relationship between convergence and absolute convergence is given in
the next theorem. Note that completeness is required to guarantee that absolute
convergence implies convergence.
Theorem 13.8 Let be an inner product space. Then is complete if and only
=
=
if absolute convergence of a series implies convergence.
Proof. Suppose that is complete and that ) % B . Then the sequence )
=
of partial sums is a Cauchy sequence, for if , we have
) ) c ~ i i ) % ) % ¦
~ b ~ b
Hence, the sequence ² ³ converges, that is, the series % converges.
Conversely, suppose that absolute convergence implies convergence and let
²% ³ be a Cauchy sequence in = . We wish to show that this sequence
converges. Since ²% ³ is a Cauchy sequence, for each , there exists an 5
