Comparing two linear functions

Question

Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a $\lambda\in \Bbb{C}$ such that for any $x\in X$ we have $h(x)=\lambda f(x)$.

Answer 1

If $f=0$ then choose $\lambda=0$. If $f\neq0$ then let $x_0\in X$ such that $f(x_0)\neq0$ or we may suppose that $f(x_0)=1$. For any $x\in X$ we have $f(x-f(x)x_0)=0$ and so $h(x-f(x)x_0)=0$. This means $h(x)=f(x)h(x_0)$. Thus it's enough to set $\lambda=h(x_0)$.