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 $e\in F$?