Hilbert Spaces

We call an inner product space complete if its underlying vector space is complete with respect to the induced norm, that is, a Banach space.

Definition (Hilbert Space)

A Hilbert space is a complete inner product space.

Example   ${\mathbb{C}}^n$ ${\ell }^2$ $L^2$ are Hilbert spaces, $C([0,1])$ is not

Every nonempty closed convex subset of a Hilbert space has the nearest point property.

In other words, if $\mathcal{H}$ is a Hilbert space, $S\subset \mathcal{H}$ is nonempty, closed and convex, and $x$ is a point in $\mathcal{H}$, then there exists a unique point $y\in S$ such that

$$\Vert x-y\Vert =\inf \left\{\Vert x-s\Vert :s\in S\right\}.$$

Proof   TODO $\square$

Recall that a linear mapping $P:X\to X$ on a vecor space is called a projection if $P^2=P$. It is called an orhogonal projection if $(x-Px)\perp Px$ for all $x\in X$ (or, equivalently, $P^{\ast }=P$).

The range of an orthogonal projection is a closed linear subspace.

Orthogonal Projection Theorem

Suppose $U$ is a closed linear subspace of an inner product space $X$. The mapping $P_U:X\to X$, which assigns each $x\in X$ to its nearest point in $U$, is an orhogonal projection with range $U$, called the orhogonal projection onto $U$ If $U=\{0\}$, then $P_U=0$. Otherwise $\Vert P_U\Vert =1$.

The mapping $U\mapsto P_U$ describes a 1-1 correspondence between the closed linear subspaces of $X$ and the orhogonal projections in $X$.

• theorem about complements