Convergence of Series in Hilbert Spaces

Question

Let $\{e_j\}_{j\in\mathbb N}$ be some subset of a Hilbert space $H$. If

$$\|f\|^2=\sum|\langle f,e_j\rangle|^2$$

for every $f\in H$, how may I show that

$$\sum\langle f,e_j\rangle e_j$$

converges for every $f\in H$?

Answer 1

Define $S:H\to H$ by $f\mapsto\sum\langle f,e_j\rangle e_j$. $S$ is well defined since, for every $f\in H$,

$$\begin{align} \|\sum\langle f,e_j\rangle e_j\|^2&=\sup_{\|g\|=1}|\langle\sum\langle f,e_j\rangle e_j,g\rangle|^2\\ &=\sup_{\|g\|=1}|\sum\langle f,e_j\rangle\langle e_j,g\rangle|^2\\ &\leq\sup_{\|g\|=1}(\sum|\langle f,e_j\rangle|^2)(\sum|\langle e_j,g\rangle|^2)\\ &=\sup_{\|g\|=1}\|f\|^2\|g\|^2\\ &=\|f\|^2. \end{align}$$

Therefore, if $f\in H$,

$$\begin{align} \langle Sf,f\rangle&=\langle\sum\langle f,e_j\rangle e_j,f\rangle\\ &=\sum\langle f,e_j\rangle\langle e_j,f\rangle\\ &=\sum\langle f,e_j\rangle\overline{\langle f,e_j\rangle}\\ &=\sum|\langle f,e_j\rangle|^2\\ &=\|f\|^2\\ &=\langle f,f\rangle. \end{align}$$

This implies that $Sf=f$ for every $f\in H$.