When does convergence in $L^2$ imply convergence in $C[0,1]$ Suppose $f_n: [0,1] \mapsto \mathbb{R} $ are continuous functions. I'm interested in knowing under what conditions does $f_n \stackrel{L^2[0,1]}{\to}f$ imply $\sup_{x \in [0,1]} |f_n(x)-f(x)| \to 0$. Clearly one example that implies this is if each $f_n$ is uniformly bounded, and has uniformly bounded first and second derivatives (the result follows from Arzela-Ascoli in this case, Does Lp-convergence and uniform boundedness in $C^2$, imply $C^{1}$ convergence?). Is this basically necessary and sufficient? Or are there weaker conditions under which convergence in mean-square implies convergence uniformly?
 A: The theorem of Ascoli-Arzelá is necessary and sufficient. By this I mean that, if the sequence of continuous functions $f_n$ converges uniformly to some continuous function, then $f_n$ is uniformly bounded and equicontinuous. This is not hard to prove.
However, the a priori knowledge that $f_n\to f$ in $L^2(0,1)$ does give an important piece of information: if $f_n$ is uniformly bounded and equicontinuous, then $f_n\to f$ uniformly. Indeed, by the theorem of Ascoli-Arzelá, $f_n$ and each of its subsequences have a subsequence that converges uniformly to some continuous function. Such function must be $f$, because uniform convergence implies $L^2$ convergence on bounded intervals. And so we can conclude that $f_n\to f$ uniformly, as claimed.
The point is that Ascoli-Arzelá gives uniform convergence of a subsequence to some continuous function, which is completely unknown a priori. The $L^2$ convergence allows us to conclude that the whole sequence $f_n$, not just a subsequence, converges to $f$, not just to some function.
A: Yes there are weaker conditions. It will work as soon as you have an interpolation inequality of this kind for some fixed constant $t\in (0,1]$
$$
\|u\|_{L^\infty} \leq \|u\|_{L^2}^t \|u\|_{X}^{1-t}
$$
and that you have uniform bounds in the $X$ norm, since then replacing $u$ by $f_n-f$ and using the fact that $\|f_n-f\|_{X} \leq \|f_n\|_{X} + \|f\|_{X} \leq C$, you obtain
$$
\|f_n-f\|_{L^\infty} \leq C^{1-t} \|f_n-f\|_{L^2}^t
$$
Now, there are a lot of interpolation inequalities of this kind, due to the theory of interpolation of spaces and of Sobolev type embeddings. Examples of spaces $X$ are the Hölder spaces $C^s$ for any $s>0$ (in particular $C^1$, but also $C^{0,1/2}$ for example), the Sobolev spaces $W^{s,p}$ with $s > 1/p$ (see e.g. Gagliardo-Nirenberg Sobolev's inequalities here) so for example $H^1 = W^{1,2}$ ...
