In a lecture on model order reduction, the following expression is said to be equivalent to the $L^2$ norm of $h(t): \mathbb{R} \rightarrow \mathbb{R}^{r \times m}$, if $h(t) \rightarrow 0$ for $t \rightarrow \infty$.
$$ \left( \int_0^\infty ||h(t - \tau)||^2_F d\tau \right)^{1/2}, $$
where $||f||_F := \sqrt{\text{trace}(f^Tf)}$ is the Frobenius norm and the $L^2$ norm of $h(t)$ is elsewhere defined as
$$ ||f(t)||_{L^2} = \left( \int_0^\infty||f(t)||^2_F dt \right)^{1/2}. $$
Is this correct? And if so, why?