Inner Product Space

is a vector space on , An inner product on is , such that

Def. A complete inner product space is called Hilbert space.

In this section, we will assume a Hilbert space.

Proposition.

Thm. If is a NVS and satisfies the parallelogram law, then there exists compatible with .

Proof. We can define .

Definition. Orthogonality complement
Suppose , then we the orthogonality complement of by

Proposition. Orthogonality complement

Theorem. (distance to a convex closed subset is attained. )
There exists a unique such that

Proof. Suppose . If , there exists such that . Since is closed, we conclude .

Suppose , let such that . Then is a cauchy sequence. since

Thus . Suppose there is another , then

Corollay.
A direct result of the above theorem is
For all , there exists unique in such that

Proof. We consider the translation set of by

Easy to verify is a closed convex set. Then there is such that

Lemma. Assume a closed convex set. Given , then is the optimal approximation of iff

Proof. Define

If is the optimal estimation, then

This implies

and thus

Theorem. (Projection Theorem)

Suppose is a closed subspace of Hilbert space , then for all , there exists unique such that

i.e .

Proof. Since is a closed convex set, then there exists optimal estimation . There remains to show .

Given , define

Since is the best estimation, we have

In addition, we have

This implies

Thus .

Corollary. Suppose is a closed subspace. Then

For all , we have

Thus . Conversely, suppose , then Projection theorem implies

where .

This implies .