Page 221 - Advanced Linear Algebra
P. 221
Chapter 9
Real and Complex Inner Product Spaces
We now turn to a discussion of real and complex vector spaces that have an
additional function defined on them, called an inner product , as described in the
following definition. In this chapter, will denote either the real or complex
-
field. Also, the complex conjugate of d is denoted by .
Definition Let be a vector space over - ~ s or - ~ d . An inner product
=
on is a function ºÁ»¢= d = ¦ - with the following properties:
=
1 )(Positive definiteness ) For all #= ,
º#Á #» and º#Á #» ~ ¯ # ~
)
2 For -~ d: (Conjugate symmetry )
º"Á #» ~ º#Á "»
For -~ s:(Symmetry )
º"Á #» ~ º#Á "»
3 )(Linearity in the first coordinate ) For all "Á # = and Á -
º " b #Á $» ~ º"Á $» b º#Á $»
)
A real or complex vector space , together with an inner product, is called a
(
=
(
real or complex inner product space.
)
If ?Á @ = , then we let
º?Á@ » ~ ¸º%Á&» % ?Á& @ ¹
and
º#Á ?» ~ ¸º#Á %» % ?¹
Note that a vector subspace of an inner product space is also an inner
=
:
product space under the restriction of the inner product of to .
:
=
