Page 237 - Advanced Linear Algebra
P. 237
Real and Complex Inner Product Spaces 221
Theorem 9.16 Let = be an inner product space and let : be a finite-
dimensional subspace of .
=
1) : ~ :
)
2 If ? = and dim ²º?»³ B , then
? ~ º ? »
Proof. For part 1), it is clear that : : . On the other hand, if # : , then
Z
the projection theorem implies that #~ b Z where : and : . Then
Z
#
Z is orthogonal to both and and so is orthogonal to itself. Hence, Z ~
and #~ : and so : ~ : . We leave the proof of part 2) as an exercise.
Characterizing Orthonormal Bases
We can characterize orthonormal bases using Fourier expansions.
Theorem 9.17 Let E ~¸" Á Ã Á " ¹ be an orthonormal subset of an inner
product space and let ~ : º E » . The following are equivalent:
=
)
1 E is an orthonormal basis for .
=
E
2) º » ~ ¸ ¹
)
3 Every vector is equal to its Fourier expansion, that is, for all #= ,
#~#
V
)
4 Bessel's identity holds for all #= , that is,
) )
V
))#~#
5 Parseval's identity holds for all #Á $ = , that is,
)
V
º#Á $» ~ ´#µ h ´$µ V E E
where
´#µ h ´$µ ~ º#Á " »º$Á " » b Ä b º#Á " »º$Á " »
V
V E
E
is the standard dot product in - .
E
))
Proof. To see that 1) implies 2), if #º » is nonzero, then E r ¸#° # ¹ is
orthonormal and so is not maximal. Conversely, if is not maximal, there is
E
E
an orthonormal set for which E F . Then any nonzero # F ± E is in
F
º» . Hence, 2) implies 1). We leave the rest of the proof as an exercise.
E
The Riesz Representation Theorem
We have been dealing with linear maps for some time. We now have a need for
conjugate linear maps.
Definition A function ¢= ¦ > on complex vector spaces is conjugate linear
if it is additive,
²# b # ³ ~ # b #
