Assume that $X,Y$ are Banach spaces, such that $X$ is compactly embedded into $Y$. Further, let $f_n: [0,T]\rightarrow X$ be a sequence of functions such that
\begin{equation} \sup_{t \in [0,T]}|\langle f_n(t) - f(t),g \rangle_{X \times X'}| \rightarrow 0 \ \ \ \forall g \in X'. \end{equation}
Now I want to show that
\begin{equation} \sup_{t \in [0,T]} \| f_n(t) - f(t) \|_Y \rightarrow 0, \end{equation}
but i don't know how. By the above given weak convergence for fixed $t$ and the compact embedding we know of course that
\begin{equation} \| f_n(t) - f(t) \|_Y \rightarrow 0 \ \ \ \forall t \in [0,T] \end{equation}
but I don't see why this convergence also holds true in the supremum. I also searched for embedding theorems which might give us immediately that C([0,T],X) is compactly embedded in C([0,T],Y), but I can't find anything. The continuity of this embedding is clear, but I can't see that it is really compact.
Thank you for any hints.