Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$

My reference is Daniel Bump’s book, Automorphic Forms and Representations, Chapter 3.7. Let $ k$ be a number field, $ G = \operatorname{GL}_2$ , $ B$ and $ T$ the usual Borel subgroup and maximal torus of $ G$ . For $ \chi$ an unramified character of $ T(\mathbb A)/T(k)$ , let $ V$ be the space of “smooth” functions $ f: G(\mathbb A) \rightarrow \mathbb C$ satisfying $ f(bg) = \chi(b) \delta_B(b)^{\frac{1}{2}}f(g)$ which are right $ K$ -finite. For $ f \in V$ and $ g \in G(\mathbb A)$ , define the Eisenstein series

$ $ E(g,f) = \sum\limits_{\gamma \in B(k) \backslash G(k)} f(\gamma g)$ $

For suitable $ \chi$ , the series converges absolutely for all $ g \in G(\mathbb A)$ . Now $ E(g,f)$ has a “Fourier expansion,” which as explained by Bump is gotten as follows: the function $ $ \Phi: \mathbb A/k \rightarrow \mathbb C$ $ $ $ \Phi(x) = E( \begin{pmatrix} 1 & x \ & 1 \end{pmatrix}g,f)$ $ is continuous, hence is in $ L^2(\mathbb A/k)$ , and therefore has a “Fourier expansion” over the characters of $ \mathbb A/k$ . If $ \psi$ is a fixed character of $ \mathbb A/k$ , then $ \psi_{\alpha}: x \mapsto \psi(\alpha a)$ comprise the rest of them, for $ \alpha \in k$ . Then

$ $ \Phi(x) = \sum\limits_{a \in k} c_{\alpha}(g,f) \psi_{\alpha}(x) \tag{1}$ $

$ $ c_{\alpha}(g,f) = \int\limits_{\mathbb A/k} E( \begin{pmatrix} 1 & y \ & 1 \end{pmatrix} g,f) \psi(-\alpha y) dy$ $

According to Bump, we may simply set $ x = 0$ , giving us the Fourier expansion for the Eisenstein series

$ $ E(g,f) = \sum\limits_{\alpha \in k} c_{\alpha}(g,f)$ $

My question: Why is this last step valid? The right hand side of equation (1) converges to $ \Phi(x)$ in the $ L^2$ -norm. As far as I know, this is not an equation of pointwise convergence. In general, the Fourier series of a continuous function need not converge pointwise to that function everywhere (in the classical case $ \mathbb R/\mathbb Z$ , the Fourier series of a continuous function converges pointwise to that function almost everywhere).