I'm supposed to prove that if $f$ is a discontinuous linear functional $H\rightarrow \mathbb C$, each of its fibers $f^{-1} \left\{ \alpha \right\} $ is dense. I already know the kernel, i.e $f^{-1} \left\{ 0 \right\} $ is dense, and I would like to say something like $f^{-1} \left\{ \alpha \right\} =v+ f^{-1} \left\{ 0 \right\}$ for some $v\in f^{-1} \left\{ \alpha \right\}$ by some sort of homogeneity to deduce all fibers are dense. However, I'm not sure how to prove this equality.
Lastly, in my exercise, $H$ was taken to be a Hilbert space, but think we just need $H$ to be normed (not even complete). Is this correct?