@TedShifrin: I was mistaken. The post on meta does refer to the main questions page. The meta post now references your math.meta post, so I have cleaned up comments on both posts (none of yours were removed).
@Vrouvrou Just look at the sequence of $x_n=n/2$. For even $n$ it limits to $1$, for odd $n$ it diverges to $\infty$
If you view the Fourier series as writing an element of an (infinite dimensional) Hilbert space as a linear combination of $sin$'s and $cos$'s, what is the condition on the functions that are within this space?
I was just wondering, given the idea that the Fourier series can be thought of as the basis expansion of an element of a vector space given the basis $cos(x)$ and $sin(x)$, what is the requirement of $f(x)$ to be an element of this space since presumably it can't be any old function
