I just started a self-study of functional Analysis from "Functional analysis, Sobolev spaces and partial differential equations" by Haim Brezis. I struggle to understand the underlying assumptions for the proof of Hahn-Banach theorem on the extension of function. In brief, the theorem is written as follows
Let $p : E \rightarrow \mathbb{R} $ be a Minkowski function.
Let $G \subset E$, $g : G \rightarrow \mathbb{R} $ such that $g \leq p $ is a linear function.
Then there exists a linear function $f$ such that $f=g, \forall x \in G $ and $f \leq p, \forall x \in E$.
The definition of a Minkowski is
$1. p(\lambda x) = \lambda p(x) \forall \lambda > 0$,
$2. p(x+y) \leq p(x)+p(y)$.
My question is why we need the first strict condition for this theorem. Why any convex function does not work with the condition $p(\lambda x) \leq \lambda p(x), \forall \lambda \geq 1$.
First, for the case that $p(0)=0$ holds, one can show that your conditions on $p$ imply that $p$ is a Minkowski function.
But (perhaps) more important is that Hahn-Banach only compares the function $p$ to a linear function ($f$ or $g$). For comparing a function $p$ with a linear function in this context, it suffices to only consider the ``worst'' values of $p$ along each ray $\{t x : t>0\}$.
To formalize this, suppose $p$ is convex. We define $q$ via $q(x):=\inf \{t p(x/t) : t>0 \}$. Then one can show that $q$ is a Minkowski function and that $$ g \leq p \iff g \leq q $$ holds for all linear functions $g$ which are defined on a linear subspace (I will not give the detailed proof here for these claims).
From this one can conclude that the Hahn-Banach theorem also holds for convex functions $p$, and not just Minkowski functions.