Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$.
Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$?
*$F$ is positive-definite if for every $a_n \in \mathbb{C}$ and $f_n \in L_2(\mathbb{R})$, one has $$\sum_{n,m} a_n a_m^* F(f_n - f_m) \geq 0$$ where $z^*$ is the complex conjugate of $z$.
In general, the continuity of a positive definite function at zero implies continuity everywhere. See for example Section 2 of Roger Horn's paper ``QUADRATIC FORMS IN HARMONIC ANALYSIS AND THE BOCHNER-EBERLEIN THEOREM'', Proc. AMS, 1975.
I believe your result is a special case.