Page 341 - Advanced Linear Algebra

Chapter 13

Hilbert Spaces

Now that we have the necessary background on the topological properties of
metric spaces, we can resume our study of inner product spaces without
qualification as to dimension. As in Chapter 9, we restrict attention to real and
complex inner product spaces. Hence will denote either or .
-
s
d
A Brief Review
Let us begin by reviewing some of the results from Chapter 9. Recall that an
inner product space = over - is a vector space = , together with an inner
product ºÁ »¢ = d = ¦ - . If - ~ s , then the inner product is bilinear and if
-~ d, the inner product is sesquilinear.

An inner product induces a norm on , defined by
=
))#~ j º#Á #»

We recall in particular the following properties of the norm.

Theorem 13.1
1 )(The Cauchy-Schwarz inequality ) For all "Á #  = ,
( (º"Á #»  ) ) ) # )"

with equality if and only if "~ # for some - .
2 )(The triangle inequality ) For all "Á #  = ,

) )"b#  ) ) )" b # )
with equality if and only if "~ # for some - .
3)(The parallelogram law )

) )"b# b ) )"c# ) ~ " ) ) b # )
We have seen that the inner product can be recovered from the norm, as follows.