Let $f$ be a linear functional on a normed vector space. It's open epigraph is convex, is it not? If so, then we can apply the Hahn-Banach separation theorem to conclude that there is a bounded linear functional $g$ separating this epigraph from the zero vector. Thus, there exists a bounded linear functional $g \leq f$.
But how could that possibly be true? Intuitively, it seems like if $g$ is less than $f$ in one direction, then it must be greater than $f$ in the opposite direction! What am I missing here?
Yes, if $f$ and $g$ are linear and $g\le f$ then $g=f$.
I take it that when you say "open epigraph" you mean the set $$E=\{(x,y):y>f(x)\}.$$The terminology "strict epigraph" might be better, because $E$ need not be open. (Hence I don't see what version of Hahn-Banach applies...)