# Continuous inculsion of the dual of continuous included Banach spaces

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?

• have you tried to see if this is true when $B$ and $C$ are finite dimensional vector spaces Mar 6, 2016 at 13:01

## 1 Answer

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.

• What is meant by $f|B$? Mar 6, 2016 at 13:20
• This is the restriction of $f$ to $B$. Mar 6, 2016 at 13:26