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I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the dual space of $F$ is given by

$$F^{\ast}=E^{\ast}/K(F)$$

where $/$ denotes the quotient relation and $K(F)$ denotes the subspace of $E^{\ast}$ consisting of elements $\phi$ s.t. $\phi(e)=0$ for every $e\in F$?

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This is correct. By Hahn-Banach every continuous linear functional $\Phi$ on $F$ extends to $E$, so it is the restriction of en element of $E^*$. Those elements of $E^*$ which restrict to 0 on $F$ are exactly the elements of $K(F)$.

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