Let $f \in L^2(0,T;H)$ where $H$ is a Hilbert space. Suppose that $$\langle f, u \rangle = 0\quad\text{for all $u \in C_c^\infty(0,T;H)$}$$ where the dual pairing is the one between $L^2(0,T;H)$ and its dual $L^2(0,T;H^*).$
Does it follow that $f\equiv 0$ the zero functional? I think yes, because picking $u = \psi(t)h$ where $\psi \in C_c^\infty(0,T)$ and $h \in H$, we get $$\int_0^T \psi(t)\langle f(t), h \rangle_{H^*,H}=0$$ for all $\psi$, hence $$\langle f(t), h \rangle = 0$$ almost every $t$. Since $h$ is arbitrary, $f(t)=0$ a.e. So $f=0.$ Agreed?