Let $E$ be a normed vector space and $T$ be a non-zero linear functional. I want to show that $E\setminus\ker T$ is dense in $E$ and that, if $T$ is not continuous then $E\setminus\ker T$ is connected.
For the first part, I tried to build a sequence $(x_n)\subset E\setminus\ker T$ that converges to an arbitrary $x\in E$. I tried to utilise the fact that $E=\operatorname{span}(v)\oplus\ker T$, if $v\in E\setminus\ker T$ to construct such sequence but without much sucess. (I can write $nx$ as $\lambda_n v+h_n$, where $h_n\in \ker T$. Then I wanted to show that the sequence $x_n:=x-h_n/n\in E\setminus\ker T$ converges to $x$.)
For the second part, I had the following strategy: If $E\setminus\ker T$ is not connected, then there is a set $A\subset E\setminus\ker T$ which is both open and closed. I wanted to use this to prove that $T$ is continuous (at least at $0$). But without much sucess.