I need an help with the following exercise.
Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I don't know if these hypothesis are enough to conclude, but my exercise states like that.
I know that if $f\in E'$, since $f$ is continuous and $\{\alpha\}$ is closed in $\Bbb R$, then $H$ is close in $E$. But I don't know how to prove the other implication. Any suggestion?