My professor claimed that
if $f\in L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$, then $$f\left(t\right)=\sum_{k\in\mathbb{Z}}c\left(k\right)\exp\left(2\pi ikt\right).$$
I think that this is false.
Should it be that $f\in L^2\left(\left[0,1\right)\right)$ instead?
On
Yes, if he means convergence for every $x$ then this is false. Of course it's true for convergence in the $L62$ norm, and (although this is a very difficult theorem) it's true for almost everywhere convergence.
It's well known that there exists a continuous (hence integrable) function on the circle with a Fourier series that diverges at at least one point. This is trivial from the Uniform Boundedness Principle.
By $[0,1)$, they mean the Torus, which is compact. So $f \in C(\mathbb{T})$ implies it's in $L^\infty(\mathbb{T})$ and, in particular, in $L^1$ and $L^2$.