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Real and Complex Inner Product Spaces 207
under the inner product
B
º² ³Á ²! ³» ~ !
~
Such sequences are called square summable . Of course, for this inner product
to make sense, the sum on the right must converge. To see this, note that if
² ³Á ²! ³ M , then
(
² c ! ³ ~ (( c ((( ! b ! (
((
(
(
(
and so
( ( ( b ! (
! ( (
M
which implies that ² ! ³ M . We leave it to the reader to verify that is an
inner product space.
The following simple result is quite useful.
Lemma 9.1 If is an inner product space and º"Á %» ~ º#Á %» for all % = ,
=
then "~# .
The next result points out one of the main differences between real and complex
inner product spaces and will play a key role in later work.
Theorem 9.2 Let be an inner product space and let B ² = . ³
=
1)
º #Á$» ~ for all #Á$ = ¬ ~
)
2 If is a complex inner product space, then
=
º#Á #» ~ for all # = ¬ ~
but this does not hold in general for real inner product spaces.
Proof. Part 1) follows directly from Lemma 9.1. As for part 2), let #~ % b & ,
for %Á & = and - . Then
~ º ² % b &³Á % b &»
((
~ º%Á %» b º&Á &» b º%Á &» b º&Á %»
~ º %Á &» b º &Á %»
Setting ~ gives
º%Á &» b º&Á %» ~
and setting ~ gives
