Let be $L_2(\mathbb{T})=\{f:\mathbb{T}\rightarrow\mathbb{R} \ | \int_{\mathbb{T}}|f|^2d\mu <+\infty\}$, $\pi:\mathbb{R}\rightarrow \mathbb{T}$ the function that for every $x$ associates his equivalence class in $\mathbb{T}$, so $x\in \mathbb{R} \ : \pi(x)=[x]=\{y\in R\ | \exists k\in \mathbb{Z}:y=x+2\pi k\}\in \mathbb{T}$, and where $\mu:\mathbb{R}\rightarrow [0,+\infty]$ is a positive measure defined as $$\mu(E)=\frac1{2\pi}|\pi^{-1}(E)\cap [0, 2\pi]|$$ where $|*|:S\rightarrow [0,+\infty]$ is the Lebesgue measure. Proove that: $$M=\{f \in L_2(\mathbb{T}): \sum_{n=-\infty}^{\infty}\hat{f}(n) \ is \ convergent\}$$ is Baire first-category set in $L_2(\mathbb{T})$, where $\hat{f}(n)$ are the terms of the Fourier sequence of $f$ defined as: $$\hat{f}(n)=\frac1{2\pi}\int_{0}^{2\pi}f(t)e^{-int}dt$$
I've already proved the density, but I'm not able to prove that it is a first-category Baire set. I think that I have to use some kind of properties of the Hilbert spaces. I know for sure from the properties of the Hilbert spaces that the Fourier series: $$\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{inx}$$ converges in $L_2$ to $f$, but I don't know how to move forward, any help?
Define for $k \ge 1$ $$V_k:=\{ f \in L_2(\mathbb{T}) \, : \, \forall n \in [2^k, 2^{k+1}), \; \hat{f}(n)>2^{-k} \} \,, $$ and for $\ell \ge 1$, let $$ W_\ell:=\cup_{\ell=k}^\infty V_k \,. $$ Then each $V_k$ is open (by continuity of $f \mapsto \hat{f}(n)$) so $W_\ell$ is open for every $\ell$. Next, observe that $$h_k(t):=2^{-k}\sum_{n=2^k}^{2^{k+1}-1} e^{int} $$ satisfies $$\|h_k\|_2^2=2^{-k} \quad (*)$$ for every $k \ge 1$. Now for every $k>\ell$ and $f \in$ span$\{e^{i nt } : n \in [0, 2^{k})\}$, the function $f+2h_k$ is in $W_\ell$, so it follows from $(*)$ that $W_\ell$ is dense in $L_2(\mathbb{T})$. Thus by Baire's category theorem, $$W:=\cap_{\ell=1}^\infty W_\ell$$ is a dense $G_\delta$ set in $L_2(\mathbb{T})$. Moreover, $W$ is disjoint from
$$M=\{f \in L_2(\mathbb{T}): \sum_{n=-\infty}^{\infty}\hat{f}(n) \ is \ convergent\} \,,$$ and therefore $M \subset W^c$ is a Baire first-category set in $L_2(\mathbb{T})$.