Page 351 - Advanced Linear Algebra
P. 351
Hilbert Spaces 335
)
and so ~ , implying that ' ~!; . Thus, : ; . Part 2 follows from part
)
1.
Let us denote the closure of the span of a set of vectors by cspan ² : . ³
:
Theorem 13.13 Let be a Hilbert space.
/
1 If is a subset of , then
)
/
(
cspan²(³ ~ (
)
2 If is a subspace of , then
/
:
cl²:³ ~ :
)
3 If is a closed subspace of , then
/
2
2~ 2
Proof. We leave it as an exercise to show that ´ ²cspan ( ³ µ ~ ( . Hence
/ ~ cspan ²(³ p ´cspan ²(³µ ~ cspan ²(³ p (
But since ( is closed, we also have
/~ ( p (
and so by Theorem 13.12, cspan²(³ ~ ( . The rest follows easily from part
)
1.
In the exercises, we provide an example of a closed subspace 2 of an inner
product space for which 2 £ 2 . Hence, we cannot drop the requirement
=
that be a Hilbert space in Theorem 13.13.
/
Corollary 13.14 If is a subset of a Hilbert space , then span ² ( ³ is dense in
(
/
/ if and only if ( ~ ¸ ¹.
Proof. As in the previous proof,
/~ cspan ²(³ p (
and so ( ~ ¸ ¹ if and only if / ~ cspan ²(³ .
Hilbert Bases
We recall the following definition from Chapter 9.
Definition A maximal orthonormal set in a Hilbert space is called a Hilbert
/
/
basis for .
Zorn's lemma can be used to show that any nontrivial Hilbert space has a Hilbert
basis. Again, we should mention that the concepts of Hilbert basis and Hamel
(
)
basis a maximal linearly independent set are quite different. We will show
