Let $A$ be a closed subspace of a Banach space, $V$. I want to show there is a functional $f'$ such that $A \subset \ker(f')$ and $f'(x) \ne 0$ for $x \in V-A$. I'm wondering what control we have when extending functions using Hahn-Banach? I want to say we can define $f:A \to \mathbb{R}$ as the $0$ function, and then extend it to $A+\mathbb{R}x$ by defining $f(x)=\alpha$ such that $|\alpha|\le ||x||$.
As I'm familiar with it, Hahn-Banach simply says that since there is a sub-linear functional (the norm) for which $f$ is less than for all $a \in A$, we can extend $f$ to the entire space $V$. There's nothing explicit about the statement that says we can force $f$ to be positive outside of $A$, only that we can extend $f$ in some way.
I'm wondering if, because my stated extension, is also less than the sublinear functional used in the extension, the extension is valid and we thus get a functional $f':V \to \mathbb{R}$ such that $f'|_A=0$ and $f'(x)\ne 0$ for $x \not \in A$?