To begin my question I wish to first clarify the definition of weak convergence FOR a sequence of functions.
We say that given sequence of functions, $\{f_{n}\}_{n=1}^{\infty}$, such that each $f_n$ is continuous in $[a,b]$, CONVERGES WEAKLY to $f$ if:
$\underset{n \to \infty}{\lim} \int_{a}^{b}f_n(x)g(x)dx = \int_{a}^{b}f(x)g(x)dx$,
for all $g(x)\in C[a,b]$.
Now, my real question is the following:
Let $\{e_n\}_{n=1}^{\infty}$ be a sequence of functions such that $e_n(x)=e^{-inx}$. Considering Parsevals Identity, what can be said about the weak convergence of $\{e_n\}_{n=1}^{\infty}$?
The first thing that came to mind was finding what the sequence converges to. Unfortunately I am having trouble evaluating
$\lim_{n \to \infty}e^{-inx}=\lim_{n \to \infty}cos(nx)-i\lim_{n \to \infty}sin(nx)$.
Either way, I try to relate to Parseval's identity which states that
$\sum_{n=-\infty}^{\infty}|a_n|^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^2$ such that $a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$
I realise that $\int_{-\pi}^{\pi}f(x)e^{-inx}dx$ is familiar with my weak convergence definition, but I am unsure of how to progress from there. Since the limits from Parseval's Identity are rather fixed. I would assume that I am trying to show weak convergence for $e_n(x) \in C[-\pi,\pi]$.
Thank you in advanced for your help.
Since $\sum_n |a_n|^2$ converges, the terms must go to $0$, so $e^{inx}$ converges weakly to $0$ in $C[-\pi,\pi]$.