# Application of the Hahn- Banach theorem

Let $E$ be a normed space and $F$ be a subspace of $E$. Show that $F$ is dense in $E$ if and only if all the linear and continuous functional on $E$ satisfying $f\vert _F=0$ are identically zero ($f = 0$).

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i think it related to Hahn - Banach extension on $l_p$ to $c_0$ because $l_p$ is dense in $c_0$, so all extensions are unique and itself. –  daisy Dec 18 '12 at 11:42
If $F$ is dense then every continuous function (linear or not) which is zero on $F$ must be zero on all of $E$.
If $F$ is not dense then the closure of $F$ is a proper linear subspace. Take a point $x$ outside $\overline{F}$ and use Hahn-Banach to find a linear functional which is zero on $F$ and non-zero in $x$.