If $v \in W^{1,\infty}\left([0,T];X\right)$ and $\|u\|_V\leq \|v\|_X$, do we have $u\in W^{1,\infty}\left([0,T];V\right)$?

Question

Let $V, X$ be two Banach spaces, and $v:[0,T]\rightarrow X ,\ u:[0,T]\rightarrow V .$

if $v \in W^{1,\infty}\left([0,T];X\right)$ and we have this relation $$\|u\|_V\leq \|v\|_X,$$

Is there a result which says that$$u\in W^{1,\infty}\left([0,T];V\right)?$$