I know the theorem that assures this is called "Riesz-Fischer Theorem" but doing a bit of research I found out that Riesz and Fischer proved simultaneously that $L^2$ is complete without mentioning any concept of "complete" nor "$L^2$".
So my question is: Who did the proof that $L^p$ is complete for $p\neq 2$ and when did these spaces start being relevant? Any hep or reference would help!
According to Wikipedia, $L^p$ spaces for general $p$ were introduced in 1910 by Riesz in the paper
which is freely available online at http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002263491. My German is not very good, but it appears to me that this paper considers $L^p$ spaces for all $1<p<\infty$, and on page 468, it states and proves the completeness of $L^p$ for all such $p$ and calls this result a generalization of a theorem of Fischer (citing Fischer's paper which proved the completeness of $L^2$).