Continuous inculsion of the dual of continuous included Banach spaces

Question

If $B$ and $C$ are Banach spaces and $B \subset C$ with the inclusion being continuous. If it true that the set of continuous linear functionals on $C$, $C'$, is continuous included in the set of continuous linear functionals on $B$, $B'$: $$ C' \subset B'$$ If it is true how do I prove it?

Answer 1

You have $\|x\|_C\le\alpha\|x\|_B$ for all $x\in B$ and some $\alpha > 0$. Define $\Phi : C'\to B'$ by $\Phi f := f|B$. Since for $x\in B$ we have $$ |(f|B)x| = |fx|\le\|f\|_{C'}\|x\|_C\le\alpha\|f\|_{C'}\|x\|_B, $$ $\Phi$ is well defined. Now, it remains for you to show that $\Phi$ is indeed bounded.