Page 342 - Advanced Linear Algebra

326 Advanced Linear Algebra

Theorem 13.2
)
1 If is a real inner product space, then
=

º"Á #» ~ ² " b#) c " c# ³
)
)
)

)
2 If is a complex inner product space, then
=

)
)
º"Á #» ~ ² " b#) c " c# ³b ² "b #) c " c # ³
)
)
)
)

The inner product also induces a metric on defined by
=
²"Á #³ ~ " c # )
)
Thus, any inner product space is a metric space.
Definition Let and > be inner product spaces and let  ² = Á >B . ³
=
)
1 is an isometry if it preserves the inner product, that is, if
º"Á #» ~ º"Á #»

for all "Á #  = .
2 A bijective isometry is called an isometric isomorphism . When ¢= ¦ >
)
is an isometric isomorphism, we say that = and > are isometrically
isomorphic.
It is easy to see that an isometry is always injective but need not be surjective,
even if =~ > .
B

Theorem 13.3 A linear transformation ²= Á > ³ is an isometry if and only
if it preserves the norm, that is, if and only if
) ) #~# )
)
for all #= .
The following result points out one of the main differences between real and
complex inner product spaces.

=
Theorem 13.4 Let be an inner product space and let  B ² = . ³
)
#

1 If º#Á $» ~ for all $  = , then ~ .
Á

)
2 If = is a complex inner product space and 8 ²#³~º #Á #»~ for all

#= , then ~ .
)
)
3 Part 2 does not hold in general for real inner product spaces.
Hilbert Spaces
Since an inner product space is a metric space, all that we learned about metric
spaces applies to inner product spaces. In particular, if ²% ³ is a sequence of