Let $H$ be a Hilbert space and let $\{e_n, n\in \mathbb N\}$ be an orthonormal sequence in $H$. Determine whether these series converge in $H$:
- $\sum \frac{e_n}{n}$
- $\sum\frac{e_n}{\sqrt{n}}$
The Parseval's identity tells us that for every $x \in H$, $$ \displaystyle \sum _m|\langle x,e_m\rangle |^2=\|x\|^2. $$ Let $x=\displaystyle\sum_n \frac{e_n}{n}$, then $|\langle x,e_m\rangle|^2=\displaystyle\sum_n \frac{1}{n^2}\quad \forall m \in \mathbb N$. Which is a convergent Riemann series. In other hand we have $$ \|x\|^2= \displaystyle\sum_n |\langle \frac{e_n}{n}, \frac{e_n}{n}\rangle|^2=\displaystyle\sum_n \frac{1}{n^2} $$
and there I got confused.
Is there a merciful soul that can help me? Thank's
$\sum a_ne_n$ converges if and only if $\sum|a_n|^{2} <\infty$. Hence the first series is convergent and the second one is not.
Proof: Recall that Hilbert spaces are complete so a series in $H$ converges if and only if the partial sums form a Cauchy sequence. $\|\sum\limits_{k=n}^{m} a_ne_n\|^{2}=\sum\limits_{k=n}^{m}|a_n|^{2}$ and a series $\sum b_n$ of non-negative numbers converges if and only if $\sum\limits_{k=n}^{m} b_k \to 0$ as $ m>n \to \infty$.