Hilbert Spaces Review

We will now review some of the recent material regarding Hilbert Spaces.

• On the Hilbert Spaces page we said that a Hilbert Space is an inner product space that is complete with respect to the norm induced by its inner product.

• On The Best Approximation Theorem for Hilbert Spaces page we proved that if $H$ is a Hilbert space and $K \subset H$ is a nonempty, closed, and convex subset of $H$ then if $h_0 \in H \setminus K$ there exists a unique $k_* \in K$ such that:

(1)

\begin{align} \quad \| k_* - h_0 \| = \inf_{k \in K} \| k - h_0 \| \end{align}

• On the Algebraic Complements of Closed Subspaces of Hilbert Spaces we proved that if $H$ is a Hilbert space and $M \subset H$ is a closed subspace of $H$ then:

(2)

\begin{align} \quad H = M \oplus M^{\perp} \end{align}

• That is, an algebraic complement for a closed subspace $M$ of $H$ is $M^{\perp}$.

• On the Density of the Span of Closed Subsets in Hilbert Spaces page we looked at two results. First we proved that if $H$ is a Hilbert space then:

(3)

\begin{align} \quad S^{\perp} = (\mathrm{span}(S))^{\perp} \end{align}

• We then proved that if $H$ is a Hilbert space and $S \subseteq H$ is a closed subset of $H$ then $\mathrm{span}(S)$ is dense in $H$ if and only if $S^{\perp} = \{ 0 \}$.

• On The Orthogonal Projection of a Hilbert Space onto a Closed Subspace page we defined a special operator. We said that if $H$ is a Hilbert space and $M \subseteq H$ is a closed subspace of $H$ so that $H = M \oplus M^{\perp}$ then the corresponding Orthogonal Projection operator is the bounded linear operator $P : H \to H$ defined for each $x = m + m'$ ($m \in M$, $m' \in M^{\perp}$) by:

(4)

\begin{align} \quad P(x) = m \end{align}

• We then proved some properties of the orthogonal projection which are outlined in the table below.

1	$\| x^2 \| = \| P(x) \|^2 + \|(I - P)(x) \|^2$ for all $x \in H$.
2	$\| P \| = 1$.
3	$\langle P(x), y \rangle = \langle x, P(y) \rangle$ for all $x, y \in H$.