# 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?

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.

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 $$H^1 = W^{1,2}$$ ...

• Thanks so much! Very useful information Commented Sep 28, 2020 at 23:02
• I found Agmon's inequality, which is a version of this bounding the esssup norm by the produce of $L^2$ norm and a Sobolev norm, en.wikipedia.org/wiki/Agmon%27s_inequality , but do you have a reference for this inequality for the Holder space case? Commented Sep 29, 2020 at 15:17
• It should follows from interpolation of spaces and Sobolev embeddings. The first allows to control the $W^{s,p}$ norm by the product of $C^{,\alpha}$ with $L^2$ norm (look at interpolation of Besov and Triebel Lizorkin spaces). I will look for references if I have time. Commented Sep 29, 2020 at 16:35