Let $\mathcal{H}$ be a Hilbert space, let $\mathcal{D} \subset \mathcal{H}$ be a dense subspace and let $(\eta_x)_{x \in X} \subset \mathcal{H}$ be indexed by a $\sigma$-finite measure space $(X, \Sigma_X, \mu_X)$. Suppose there exists a $K > 0$ such that, for all $f \in \mathcal{D}$, $$ \int_X | \langle f, \eta_x \rangle |^2 \; d\mu_X (x) \leq K \| f \|^2_{\mathcal{H}} \quad \quad \quad \quad (*)$$ Then the inequality $(*)$ holds for all $f \in \mathcal{H}$.
I am aware of a proof of the above stated result that is quite ingenious and heavily depends on the fact that the involved measure space is $\sigma$-finite. However, I am curious if this is really necessary.
I am approaching the above stated result by considering the linear operator $$C : \mathcal{H} \to L^2 (X), \; f \mapsto \{ \langle f, \eta_x \rangle \}_{x \in X},$$ which is bounded on $\mathcal{D}$. However, boundedness of a dense subspace is, in general, not sufficient for a linear operator to be bounded on its entire domain. Thus the result is not trivial, and is a special property of $C$.
Any help or comment is highly appreciated.
Let $f_0 \in H $. By density, there is a sequence $(f_n)_n $ in $D $ such that $f_n \to f $.
Now, for $F_n (x) := |\langle f_n, \eta_x\rangle |^2$, we have $F_n \to F_0$ pointwise. Hence, by Fatou's Lemma, $$ \int F_0 \leq \liminf_n \int F_n \leq \liminf K \|f_n \|^2 = K \|f\|^2. $$
Thus, no fancy functional analysis is neede; only Fatou's Lemma.