Orthonormal Set

Definition

The set $\{{\varphi }_i|i\in \mathbb{N}\}\subset C_p(a,b)$^[Piecewise Continuous] satisfying $({\varphi }_m,{\varphi }_n)=\{\begin{array}{ll}0& m\ne n\\ 1& m=n\end{array}$

Facts

Closed Orthonormal set

If there is no function $f$ in $C_p(a,b)=X$ or in subspace of $X$ which is orthogonal to the $m$th orthonormal set $\{{\varphi }_m\}$ an orthonormal set $\{{\varphi }_n\}$ is closed in $C_p(a,b)=X$ or in subspace of $X$,

Bessel’s Inequality for Orthonormal set

Since MSE $E$ is non-negative, $||f|{|}^2-\sum _{n=1}^N{c}_n^2\ge 0\Rightarrow \sum _{n=1}^N{c}_n^2\le ||f|{|}^2$

Let $s_N=\sum _{n=1}^N{c}_n^2$ then a sequence of functions $\{s_N(x)\}{,}_{N\in \mathbb{N}}$ is bounded increasing sequence
So $\{s_N(x)\}$ converge to $s\in \mathbb{R}\Rightarrow \underset{n\to \infty }{\lim }{c}_n^2=0$