I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i n^2 \tau } = 1 + 2 \sum_{n = 1}^{\infty}e^{2 \pi i n^2 \tau } }$ in the context of Hilbert modular forms for real quadratic fields. It is well known that this function is holomorphic on the upper half plane $\mathbb{H}$, and I made sure to understand well and wrote down a detailed proof of this before asking this question.
The context of my question is as follows
Let $F = \mathbb{Q}(\sqrt{d})$ be a real quadratic field, with $d > 0$ a positive square-free integer such that $d \equiv 2, 3 \pmod{4}$. Assume also that $F$ has class number $1$. Also its ring of integers is $\mathcal{O}_F = \mathbb{Z}[\sqrt{d}] = \mathbb{Z} \oplus\sqrt{d} \, \mathbb{Z} = \{ n + m\sqrt{d} \mid n, m \in \mathbb{Z} \}$. Then there are two real embeddings $\sigma_i : F \hookrightarrow \mathbb{R}$ given by $\sigma_1(a + b\sqrt{d}) = a + b\sqrt{d}$ and $\sigma_2(a + b\sqrt{d}) = a - b\sqrt{d}$, so that $\sigma_1$ is just the identity map, and $\sigma_2$ gives the "conjugate" of an element.
Now for $z = (z_1, z_2) \in \mathbb{H} \times \mathbb{H}$ and $\alpha \in F$, we define the "trace" as follows $\operatorname{Tr}(\alpha z) := \sigma_1(\alpha) z_1 + \sigma_2(\alpha) z_2 = \alpha z_1 + \alpha' z_2$, where we wrote $\alpha'$ for the conjugate of $\alpha$.
Finally we define the holomorphic Hilbert modular theta function $\vartheta: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{C}$ by
$$ \vartheta(z) := \vartheta(z_1, z_2) := \sum_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)} $$
I'm trying to prove the absolute convergence of this infinite series.
The argument I found online To prove the convergence of the series, it is compared to the Jacobi theta function defined above. To be precise, the author observes that
$$ |\vartheta(z)| = \left| \sum_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)} \right| = \left| \sum_{x \in \mathcal{O}_F} e^{\pi i x^2 z_1} e^{\pi i x'^2 z_2} \right| \leq \left| \sum_{x \in \mathcal{O}_F} e^{2 \pi i x^2 z_1} \right|^{1/2} \left| \sum_{x \in \mathcal{O}_F} e^{2 \pi i x'^2 z_2} \right|^{1/2} $$
where apparently the Cauchy-Schwarz inequality has been used in the last step, although I'm not convinced that it is true as stated. In any case, then it is noted that since the Jacobi theta function defined above $\displaystyle{\theta(\tau) = \sum_{n = -\infty}^{\infty} e^{2 \pi i n^2 \tau }}$ is holomorphic on $\mathbb{H}$, then each of the two series on the right hand side of the above inequality are absolutely convergent on compact sets in $\mathbb{H}$, so that the original series $\vartheta(z_1, z_2)$ converges absolutely on compact subests of $\mathbb{H} \times \mathbb{H}$.
Questions
Can someone please help me understand how does the convergence of each of the series on the right hand side of the above inequality follows from the corresponding fact for the Jacobi theta function? I mean, since each of the two sums on the RHS of the inequality are over all algebraic integers $x \in \mathcal{O}_F = \mathbb{Z} \oplus \sqrt{d} \, \mathbb{Z}$, then the sums are actually double sums, so it doesn't seem to follow trivially.
Is the argument where the Cauchy-Schwarz inequality is used correct? If I'm not mistaken, I think that the absolute value should be inside the infinite series, like this $$ \left | \sum x_n y_n \right| \leq \left( \sum |x_n|^2 \right ) ^{1/2} \left( \sum |y_n|^2 \right ) ^{1/2} $$
Thank you very much for any help.
The convergence of the sum on the right-hand side, essentially is
absolute convergence of $$ \sum_{m,n\in \mathbb{Z}} e^{-am^2-bmn-cn^2}$$ where $ax^2+bxy+cy^2$ is a positive-definite quadratic form.
Because of absolute convergence, there is no problem with using C-S.
Note that absolute convergence implies convergence.
The actual point where you used C-S, need to be replaced by absolute values.