Bases, Orthogonal, Orthonormal, and Complete Sets

We begin by defining orthogonal sets and orthonormal sets. So, let $\{g_n\}$ be a subset of the space $H$. $\{g_n\}$ is said to be orthogonal if $<g_n,g_m>=0$, if $n \neq m$, and it is said to be orthonormal is $<g_n,g_m>=\delta_{nm}$, where

$$\delta_{nm}=\left\{ \begin{array}{ll} 1 & \mbox{if $n=m$} \\ 0 & \mbox{if $n \neq m$} \end{array}\right.$$

The set $\{g_n\}$ is said to be complete if its span is dense in $H$. One can show that $\{g_n\}$ is complete if and only if the only element orthogonal to $\{g_n\}$ is zero.

The set $\{e_n\}$ is said to be a basis for the Hilbert space $H$ if for every $f \in H$, there exists a unique sequence of scalars $\{c_n\}$ such that $f=\sum_n c_n e_n$. A basis $\{e_n\}$ is said to be orthonormal if it is orthogonal and $\Vert e_n\Vert=1$, for all $n$. It is standard that if $\{e_n\}$ is an orthonormal basis for $H$, then for all $f \in H$, we have

$f=\sum_n <f,e_n>e_n$

and

$\Vert f\Vert^2=\sum_n \vert<f,e_n>\vert^2$

Next, we define unconditional bases and Riesz bases.

Definition: A basis $\{e_n\}$ for the Hilbert space $H$ is called unconditional if whenever $\sum_n c_n e_n \in H$, $\sum_n \vert c_n\vert e_n \in H$.

Definition: A subset $\{g_n\}$ of the Hilbert space $H$ is called a Riesz basis if there exists an orthonormal basis $\{e_n\}$ for $H$ and an isomorphism $A$, such that $Ag_n=e_n$, for all $n$.

Definition: Let $H$ be a Hilbert space and let $\{g_n\}$ and $\{h_n\}$ be subsets of $H$. If $\sum_n \Vert g_n-h_n\Vert^2 < \infty$, then $\{g_n\}$ and $\{h_n\}$ are said to be quadratically close.