Fourier series with $l^2(\mathbb{R})$ coefficient converge almost everywhere in $[0,1)$

Question

Let $(a_n)$ be a sequence in $l^2(\mathbb{R})$. I want to prove for almost every $x\in[0,1)$, that $$\lim_{n\to\infty} \sum_{k=0}^n a_k e^{2\pi i k x }$$ exists. The relevant text says the Carleson's Theorem would be an overkill, so I'm wondering if there is an elementary way. Any help is appreciated.

Answer 1

The spaces $\ell^2(\mathbb{R})$ and $L^2([0,1],\mathbb{C})$ are isometrically isomorphic. The sequence $\alpha_n = (a_1,...,a_n,0,0,...)$ satisfies $\alpha_n \stackrel{\ell^2}{\to} \alpha = (a_1,a_2,...)$. Similarly $f_n(x) = \sum_{j=1}^n a_j e^{2\pi i j x}$ satisfies $f_n \stackrel{L^2}{\to} f$, where $f(x) = \sum_{j=1}^\infty a_j e^{2\pi i j x}$, and this limit must exist in $L^2([0,1],\mathbb{C})$, since the sequence $f_n$ is Cauchy. Therefore the limit in question must exist almost everywhere and coincide with $f$.