Page 226 - Advanced Linear Algebra

210 Advanced Linear Algebra

Theorem 9.5
)
1 ²"Á #³ ‚ and ²"Á #³ ~ if and only if " ~ #
2)(Symmetry )
²"Á #³ ~ ²#Á "³
3)(The triangle inequality )
²"Á #³  ²"Á $³ b ²$Á #³

=
Any nonempty set , together with a function ¢ = d = ¦ s that satisfies the
properties of Theorem 9.5, is called a metric space and the function is called

a metric on . Thus, any inner product space is a metric space under the metric
=
(9.2 ).
Before continuing, we should make a few remarks about our goals in this and
the next chapter. The presence of an inner product, and hence a metric, permits
the definition of a topology on = , and in particular, convergence of infinite
sequences. A sequence ²# ³ of vectors in converges to #  = if
=

lim ) )#c # ~
¦B
Some of the more important concepts related to convergence are closedness and
closures, completeness and the continuity of linear operators and linear
functionals.

In the finite-dimensional case, the situation is very straightforward: All
subspaces are closed, all inner product spaces are complete and all linear
operators and functionals are continuous. However, in the infinite-dimensional
case, things are not as simple.

Our goals in this chapter and the next are to describe some of the basic
properties of inner product spaces—both finite and infinite-dimensional—and
then discuss certain special types of operators (normal, unitary and self-adjoint)
in the finite-dimensional case only. To achieve the latter goal as rapidly as
possible, we will postpone a discussion of convergence-related properties until
Chapter 12. This means that we must state some results only for the finite-
dimensional case in this chapter.
Isometries
An isomorphism of vector spaces preserves the vector space operations. The
corresponding concept for inner product spaces is the isometry .

Definition Let and > be inner product spaces and let  ² = Á >B . ³
=