I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things here. I'm studying spectral theory by myself as part of my research activity and the following question arose.

Let $H$ be a Hilbert space and $Q: \mathcal{H}\to \mathbb{C}$ a function such that:

(1) There exists $C>0$ such that $|Q(x)| \le C||x||^{2}$ for every $x\in H$

(2) $Q(x+y)+Q(x-y) = 2Q(x) + 2Q(y)$ for every $x,y \in H$ and

(3) $Q(\lambda x) = |\lambda|^{2}Q(x)$ for every $x \in H$ and $\lambda \in \mathbb{C}$.

**Question:** Is there some bounded linear operator $A \in H$ such that $Q(x) = \langle Ax, x\rangle$?

The answer to this question seems to be affirmative and a sketch of a possible approach is given here (page 7, Lemma 12.2.7). The idea is to define: \begin{eqnarray} \Psi(x,y) = \frac{1}{4}[Q(x+y)-Q(x-y)+iQ(x+iy)-iQ(x-iy)] \tag{1}\label{1} \end{eqnarray} where $\{e_{\alpha}\}_{\alpha \in I}$ is an orthonormal basis of $H$ and then define $A$ by means of the rule: \begin{eqnarray} Ax = \sum_{\alpha \in I}\Psi(x,e_{\alpha})e_{\alpha}\tag{2}\label{2} \end{eqnarray} But I'm puzzled with this approach since I was not able to prove that $\sum_{\alpha \in I}\Psi(x,e_{\alpha})e_{\alpha}$ converges in the first place. All I could prove was $|\Psi(x,y)| \le K(||x||^{2}+||y||^{2})$ for some $K > 0$. As you can see in my previous post on math stack, it seems that the convergence problem is a bit tricky indeed.

In summary: I don't know how to prove that (\ref{2}) converges and, thus, I don't quite understand the proof of the result. However, I believe it's possible to find a more direct proof, maybe using Riesz Representation Theorem ideas (although $Q$ here is not linear) or something like that. I'd appreciate any help on either way.