The following is from my homework. PLEASE don't reveal all the solution, but leave at least something for my imagination.
Let $X$ be a normed space. Let $\phi,\psi : X → \mathbb C$ be linear functionals such that $\ker \phi ⊂ \kerψ$. Show that there exists $λ \in \mathbb C$ such that $ψ = \lambda\phi$.
The reason that I'm asking for help is that this proposition don't even sound right to me. The result is VERY strong, while $\ker \phi \subset \ker\psi$ isn't (at least the way I see it).
What I've tried so far was assuming that $\not \exists \lambda \in C$, therefore there are $x_1,x_2\in X$ with $\psi(x_1)=\lambda_1\phi(x_1)$, $\psi(x_2)=\lambda_1\phi(x_2)$, and $\lambda_1 \neq \lambda_2$. I tried to find $z\in X$ that would be in $\ker\phi - \ker\psi$, using the linearity.
My main problem was that I don't feel I have enough "tools" - no strong theorems and nothing much to work with. This is the kind of help I would like the most - if you could give such lead.
Thanks so much!