I am have been thinking lately of the sequence of functions $$ f_n = \sin nx $$
and its limit as n tends to infinity. I am quite comfortable with the fact that viewing this sequence in $\mathcal{C}([a,b],\mathbb{R})$ this has no limit.
However I have recently finished a course at on Hilbert Spaces, where we were teased with function spaces like $L^2([-\pi,\pi])$ and $L^1([a,b])$. Now we were not given any rigorous definitions of these spaces but I am wondering if this sequence does have a limit in a more abstract space.
I am mainly wondering about its limit in $L^2([-\pi,\pi])$. The reason I am wondering about this space in particular is because to me it seems to the limit of this function sequence, if it exists, will be 1 and -1 at an infinite number of points in the interval $[-\pi,\pi]$, however the set of these points seems countable and therefore the set of such points have a measure of zero.
If this seems a little like a waffle session, I apologise, I do not have a lot of experience in measure and integration theory just yet. (I will be doing that next year)
$\int_{-\pi}^{\pi} |f_n-f_m|^{2} =2\pi$ for $n \neq m$ which means that the distance between $f_n$ and $f_m$ in the space $L^{2} [-\pi,\pi]$ is $\sqrt {2\pi}$ for $n \neq m$. It follows that the sequence $\{f_n\}$ is not Cauchy, hence not convergent in $L^{2} [-\pi,\pi]$.