In our Lecture we had the Hahn Banach theorem:
Let $X$ be real linear space and $Y$ is its subspace. Also let $p$ be a sublinear functional defined on $X$ and a linear funtional $f$ defined on $Y$ with $f(x) \leq p(x)$ for all $x \in Y$ . Then exists a linear functional $F$ defined on $X$, with $F(x) \leq p(x)$ for all $x \in X$ and $F(x) = f(x)$ for $x\in Y$.
Our proof begins with define $P$ on $X$ with $P(x):=inf(p(x-y)+f(y))$ for $x \in X$. We have: $p(x-y)+f(y) \geq p(-y)-p(-x)-f(-y)\geq -p(-x)> - \infty $ . Especially $P(x)> -\infty$
But i can not see where the first inequality comes from. I think the second one is because of:
$p(-y)-p(-x)-f(-y)\geq -p(-x)$
$p(-y)-f(-y)\geq -p(-x)+p(-x)$
$p(-y)-f(-y)\geq 0$
and this is true because $f(x) \leq p(x)$ for all $x \in Y$ or am I wrong? Sorry for the bad english. Thanks for any help, I really try to understand the proof!
Since $f$ is linear, we have that $f(y)=-f(-y)$. Furthermore, $p(-y)=p(x-y-x) \leq p(x-y)+p(-x)$, by subadditivity. I think you can use these observations to see why the first inequality is true.