If I have two (seperable) Banach spaces $\hat{B}, B$, s.t. $\hat{B}$ is continuously embedded into $B$, i.e. $i: \hat{B} \to B$ is injective and continuous. Now I endow $B$ and $\hat{B}$ with their respective Borel-$\sigma$-algebras.
Can I already conclude that $i(\hat{B})$ is a measurable subspace of $B$?
Can I conclude that if I have a measurable, linear bounded map $\hat{l}$ on $\hat{B}$, that $$l: B \to \mathbb{R}$$ $$l = \hat{l}\text{ on }i(\hat{B}), ~ l=0 \text{ on }B\setminus i(\hat{B})$$ is measurable?
$i(\hat{B})$ is even a closed subspace of $B$ since the image of a Banach space under a continuous, linear map is again a Banach space.
As a result, $l = \hat{l} \mathbb{1}_{i(\hat{B})}$ is measurable as a product of measurable functions.