Let $H$ be a Hilbert space and let $f_n \in H$ be a sequence of orthogonal elements i.e $<f_n,f_m>=0 $ if $n\ne m$. Define the element $F_N= f_1 + f_2 +...+ f_N$ for each $ N\in \mathbb N$. Prove that the following are equivalent:
$i)$ The limit of $F_N$ exist when $N\to \infty$
$ii)$ The weak limit of $F_N$ exist when $N\to \infty$
I need help with $ii) \Rightarrow i)$.
If $F_N$ converges wekly then it is a bounded sequence in $H$ (this is true in all locally convex spaces). Moreover, by Pythagoras, $\|F_N-F_M\|^2 = \sum\limits_{n=M+1}^N \|f_n\|^2 \to 0$ (for $N>M\to\infty$). Therefore, $F_N$ is a Cauchy sequence and hence convergent.