A linear operator $A$ is defined from $L^2(0,T)$ to $C([0,T);\mathbb{H})$ where $\mathbb{H}$ is a Hilbert space. At some point, dual of this operator is needed; a detour to calculate the dual, and avoid dealing with dual space of continuous functions is to have $A$ as an operator from $L^2(0,T)$ to $L^2([0,T);\mathbb{H})$. Now consider $f\in L^2(0,T)$ and $w\in L^2([0,T);\mathbb{H})$ for which
$$\langle Af,w\rangle_{L^2([0,T);\mathbb{H})}=\langle f,A^*w\rangle_{L^2(0,T)}$$
coincides with
$$\langle f,Bw\rangle_{L^2(0,T)}$$
for a $B:L^2([0,T);\mathbb{H})\to L^2(0,T)$. Is this true to expect that $B=A^*:C([0,T);\mathbb{H})\to L^2(0,T)$?