Page 222 - Advanced Linear Algebra

206 Advanced Linear Algebra

We will study bilinear forms (also called inner products ) on vector spaces over
d
s
fields other than or in Chapter 11. Note that property 1) implies that º#Á #»
is always real, even if is a complex vector space.
=
If -~ s , then properties 2) and 3) imply that the inner product is linear in both
coordinates, that is, the inner product is bilinear . However, if -~ d , then
º$Á " b #» ~ º " b #Á$» ~ º"Á$» b º#Á$» ~ º$Á"» b º$Á#»

This is referred to as conjugate linearity in the second coordinate. Specifically,
a function ¢ = ¦ > between complex vector spaces is conjugate linear if
²" b #³ ~ ²"³ b ²#³

and
² "³ ~ ²"³

for all "Á #  = and  d . Thus, a complex inner product is linear in its first
coordinate and conjugate linear in its second coordinate. This is often described
by saying that a complex inner product is sesquilinear . (Sesqui means “one and
a half times.”)

Example 9.1
1) The vector space s is an inner product space under the standard inner
product, or dot product, defined by

º² Á à Á ³Á ² Á à Á ³» ~ b Ä b

The inner product space s is often called -dimensional Euclidean
space.
2) The vector space d is an inner product space under the standard inner
product defined by

º² Á à Á ³Á ² Á à Á ³» ~ b Ä b

This inner product space is often called -dimensional unitary space .
3) The vector space *´ Á µ of all continuous complex-valued functions on the
closed interval ´ Á µ is a complex inner product space under the inner
product

º Á » ~ ²%³ ²%³ %

Example 9.2 One of the most important inner product spaces is the vector space
³
M ² of all real (or complex) sequences with the property that

((  B