Let $f: X\to \mathbb R$ be an unbounded linear functional, where $X$ is a Banach space. Then $\ker f$ is dense in $X$.
I tried to prove contrapositive of the statement.
Suppose that $\ker f$ is not dense in $X$. Then there exists a $p\in X$ not in closure of $\ker f, f(p)$ is non zero. It follows that range $f= \mathbb R.p$.
But I got stuck while trying to prove that $f$ should be bounded:
$|f(x)|=\|r_x.p\|$ for some $r\in \mathbb R$ for all $x: \|x\|=1$. From here, $|f(x)|\le |r_x|\|p\|$. Not sure, how to bound $r_x$ from here.
