Is there some general (elementary) theory for Fourier series for Banach space valued functions? In particular, is there a citable statement like this:
Let $B$ be a (complex) Banach space and $f: \mathbb{S}^1 \rightarrow B$ be a ($\mathcal{C}^1$?) function. Then, the partial Fourier series $\sum\limits_{j = -N}^N a_j e^{ijt}$ with $a_j = \int\limits_0^{2\pi}f(t)e^{-ijt}\mathrm{d}t$ converges uniformly to $f$ for $N \rightarrow \infty$.
I think this should not be that difficult to prove, as standard treatments from real or complex analysis should carry over to the Banach space case. However, I am not that well-versed with Banach space valued integrals, so I'd rather cite this from somewhere... But I can only find treatments for the function spaces $L^p(\mathbb{S}, B)$ using much more theory, so I don't really understand those. Am I missing something?