Orthogonal and Orthonormal Sets Review

We will now review some of the recent material regarding orthogonal and orthonormal sets.

• On the Orthogonal and Orthonormal Sets in Inner Product Spaces page we said that if $H$ is an inner product space then a subset $S \subseteq H$ is said to be an Orthogonal Set if $\langle x, y \rangle = 0$ for all $x, y \in S$. If in addition $\| x \| = 1$ for every $x \in S$ then $S$ is said to be an Orthonormal Set.

• We then proved a bunch of important results regarding orthonormal sequences which are summarized in the table below.

Page	Theorem
The Pythagorean Identity for Inner Product Spaces	If $H$ is an inner product space and $\{ x_1, x_2, ..., x_n \}$ is an orthonormal set then for all $c_1, c_2, ..., c_n \in \mathbb{C}$, $\displaystyle{\biggr \| \sum_{k=1}^{n} c_kx_k \biggr \|^2 = \sum_{k=1}^{n} |c_k|^2}$.
Bessel's Inequality for Inner Product Spaces	If $H$ is an inner product space and $(x_n)_{n=1}^{\infty}$ is an orthonormal sequence in $H$ then for all $y \in H$, $\displaystyle{\sum_{n=1}^{\infty} |\langle y, x_n \rangle|^2 \leq \| y \|^2}$.
Convergence Criterion for Series in Hilbert Spaces	If $H$ is a Hilbert space and $(x_n)_{n=1}^{\infty}$ is an orthonormal sequence in $H$ then for every $y \in H$ the series $\displaystyle{\sum_{n=1}^{\infty} \langle y, x_n \rangle x_n}$ converges to some $z \in H$ such that $z - y \perp \{ x_1, x_2, ... \}$.
Hilbert Bases (Orthonormal Bases) for Hilbert Spaces	If $H$ is a Hilbert space then an orthonormal sequence of points $(x_n)_{n=1}^{\infty}$ is a Hilbert basis of $H$ if and only if $\{ x_1, x_2, ... \}^{\perp} = \{ 0 \}$.
Parseval's Identity for Inner Product Spaces	If $H$ is a Hilbert space and $(x_n)_{n=1}^{\infty}$ is a Hilbert basis of $H$ then for all $y \in H$, $\displaystyle{\sum_{n=1}^{\infty} |\langle y, x_n \rangle|^2 = \| y \|^2}$.
Separability Criterion for Hilbert Spaces	If $H$ is a Hilbert space then $H$ is separable if and only if $H$ has a countable Hilbert basis.