I need to prove that: If a nonzero linear functional $f$ on a Banach Space $X$ is discontinuous then the nullspace $N_f$ is dense in $X$.
To prove that $N_f$ is dense, it suffices to show that $\overline N_f = X$ which is equivalent to $(X \setminus N_f)^o=\emptyset$. (the interior of complement of $N_f$ is null set.) Since $f$ is a linear functional and is discontinuous, it has to be unbounded. I don't know exactly how to utilize these observations.
Also on a related topic, I'm a little confused about how to exploit the a Linear Functional $f:X \to R$ or a Linear Operator $T:X \to Y$ being unbounded. Can I say that if a linear operator is unbounded then exists a sequence $<x_n>$ in $X$ s.t. $||Tx_n|| > n^2||x_n||$ for each $n$ or $||Tx_n|| > n||x_n||$ ?