Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies $\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$ for all $\phi \in C_c^\infty(0,T)$.
My question is how to interpret the RHS. Should I think of $v(t)\phi(t)$ as $v(t)(\phi(t)$) (as $v(t) \in H^{-1}$)?