Let $X,Y,Z$ be some Banach spaces and assume that $X \hookrightarrow Y$, i.e $X$ is embedded continuously into $Y$. Can we claim then that $ Z \cap X \hookrightarrow Z \cap Y$?
By a set theory approach, this seems correct but I'm not sure about the injections here. Maybe my question is quite elementary but since I haven't taken many courses in functional analysis, I would really appreciate if somebody could confirm or reject this via a counterexample.
EDIT: My question is motivated by the following spaces:
$X \equiv W^{1,2}(0,T;L^2(\Omega))$, $Y \equiv W^{1,2}(0,T;W^{-1,2}(\Omega))$ and $Z \equiv L^2(0;T;W^{1,2}(\Omega))$
Thanks in advance!