Orthonormal Bases

# Orthonormal Bases

 Definition: Let $H$ be a Hilbert space. An Orthonormal Basis for $H$ is an orthonormal sequence $(e_n)$ of $H$ such that every $h \in H$ can be written uniquely as $\displaystyle{h = \sum_{n=1}^{\infty} \langle e_n, h \rangle e_n}$.

Before we characterize orthonormal bases, we need the following definition:

 Definition: Let $H$ be a Hilbert space. An orthonormal sequence $(e_n)$ of $H$ is said to be Complete if $h \perp (e_n)$ implies that $h = 0$. That is, the only vector in $H$ that is orthogonal to every $e_n$ is $h = 0$.

The following theorem tells us exactly when an orthonormal sequence is an orthonormal basis.

 Theorem 1: Let $H$ be a Hilbert space and let $(e_n)$ be an orthonormal sequence in $H$. Then $(e_n)$ is an orthonormal basis of $H$ if and only if $(e_n)$ is complete.
• Proof: $\Rightarrow$ Suppose that $(e_n)$ is an orthonormal basis of $H$. Suppose that $h$ is orthogonal to every $e_n$. Write $\displaystyle{h = \sum_{n=1}^{\infty} \langle e_n, h \rangle e_n}$. Then $\langle e_n, h \rangle = 0$ for each $n \in \mathbb{N}$ and thus $h = 0$. So $(e_n)$ is complete.
• $\Leftarrow$ Suppose that $(e_n)$ is complete. Recall that for each $h \in H$ we have that:
(1)
\begin{align} \quad h - \sum_{n=1}^{\infty} \langle e_n, h \rangle e_n \end{align}
• is orthogonal to every $e_n$. So for every $h \in H$ we have that:
(2)
\begin{align} \quad h = \sum_{n=1}^{\infty} \langle e_n, h \rangle e_n \end{align}
• So $(e_n)$ is an orthonormal basis of $H$. $\blacksquare$