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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?

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

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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.

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

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