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I would like to know if the following property that is true for Banach spaces, it holds for Fréchet space as well.

The question is:

Let $X$ be a Banach reflexive space and $Y$ be a Fréchet space. Suppose that $ X ⊂ Y$ dense and the injection of $X$ in $Y$ continuous. Then: $$L^{\infty} (0,T;X)\cap C_w([0,T] ;Y)= C_w([0,T] ;X), $$ Where $C_w([0, T]; Y)$, we represent the space of weakly continuous functions form $[0, T]$ into $Y$. This means that if $u \in C_w([0,T] ;Y)$, the mapping $$t\mapsto \langle u(t), y'\rangle$$ is continuous on $[0, T]$ for all $y' \in Y′$ dual of $ Y.$?

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