I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real valued functions on $[0,1]$ and the sequence of partial sums $S_m(t)=\sum_{n=1}^m f_n(t)$ converges in $\mathrm{L}^2([0,1])$ and is equicontinuous. Does this imply the uniform convergence of $(S_m)_{m\in\mathbb{N}}$ in $[0,1]$?
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Since $S_n(t)$ converges in $L^2$ to a function $S(t)$, it is bounded at least at a point. Since it is equicontinuous, every subsequence, by Ascoli-Arzelà, has a sub-subsequence that converges uniformly. The limit is the same function $S(t)$, hence $S_n$ itself converges uniformly.
answered May 17, 2014 at 20:23
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$\begingroup$ (recall that in any topological space a sequence $s_n$ converges to a point $s$ if and only if every subsequence has a sub-subsequence converging to $s$) $\endgroup$ Commented May 17, 2014 at 20:35
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$\begingroup$ We may also say: for an equicontinuous sequence $(s_n)$ in $C([0,1])$, either a subsequence converges uniformly, or a subsequence diverges uniformly (i.e. $\min_{0\le t\le1} |s_n(t)|\to\infty $. However the latter is not the case here, since $(s_n)$ is bounded in $L^2$ $\endgroup$ Commented May 17, 2014 at 21:35