Let $E$ be a infinite dimensional normed vector space.
1 - How to define a "not continuous" linear functional $f$ in $E$ such that the set $\ker(f)=\{x\in E:\ f(x)=0\}$ is dense in $E$ but $\ker(f)\neq E$?
2 - If $f$ is the functional defined above, can we find $y\in E\setminus\ker(f)$ such that the set $\{\lambda y:\ \lambda\in\mathbb{R}\}$ is not contained in $\ker(f)$?