Page 236 - Advanced Linear Algebra
P. 236
220 Advanced Linear Algebra
)
3 Bessel's inequality holds for all #= , that is
))##
) )
V
Proof. For part 1), since
º# c #Á" » ~ º#Á" » c º#Á" » ~
V
V
V
V
it follows that # c # : . Also, if # c : for : , then c #: and
c # ~ ²# c #³ c ²# c ³ :
V
V
V
and so ~# V . For part 2), if : , then # c #: implies that
²# c #³ ²# c ³ and so
V
V
)
) )#c ~ #c#b#c )V ~ ) )#c# V b V ) )#c
V
Hence, ) )#c is smallest if and only if ~ # V and the smallest value is
) )#c# V . We leave proof of Bessel's inequality as an exercise.
(
Theorem 9.15 The projection theorem ) If is a finite-dimensional subspace
:
of an inner product space , then
=
:~ : p :
In particular, if #= , then
#~# b ²# c #³: p :
V
V
It follows that
dim²= ³ ~ dim²:³ b dim²: ³
V
Proof. We have seen that # c # : and so = ~ : b : . But : q : ~ ¸ ¹
and so =~ : p : .
The following example shows that the projection theorem may fail if is not
:
finite-dimensional. Indeed, in the infinite-dimensional case, : must be a
complete subspace, but we postpone a discussion of this case until Chapter 13.
Example 9.6 As in Example 9.5, let =~ M and let be the subspace of all
:
sequences with finite support, that is, is spanned by the vectors
:
~ ² ÁÃÁ Á Á Áó
where has a in the th coordinate and 's elsewhere. If ~ % ² % ³ : , then
% ~ º%Á » ~ for all and so % ~ . Therefore, : ~ ¸ ¹. However,
:p : ~ : £ M
The projection theorem has a variety of uses.
