Let a Sobolev norm that involves time, i.e., $$ \|u\|_{L_2(0,T;L_2(\Omega))}^2 = \int_0^T\|u(t)\|_{L_2(\Omega)}^2\,dt, $$ where $\Omega$ is bounded domain of $\mathbb{R}^2$ and $T>0.$
I was wondering if the following mapping is norm,
$$ a(v) = \int_0^T\|v(t)\|_{L_2(\Omega)}^2\,dt + \|v_t(T)\|_{L_2(\Omega)}^2, $$ for a sufficiently smooth function $v.$ If yes, in what space is it norm?