Given a sublinear functional $p$ in a real vector space $X$, show that there exists a linear functional $f$ in $X$ such that $-p(-x)\leq f(x)\leq p(x)$.
I am trying to use the Hahn-Banach theorem taking the vector subspace $G_{x_0} = \{ tx_0 : t\in \mathbb{R}\}$ for any $x_0\in X$ and properties of sublinear functional ($p(\lambda x) = \lambda p(x)$ for $\lambda \geq 0$ and $p(x+y)\leq p(x) + p(y)$ for every $x,y\in X$) to find a relation between $g: G_{x_0} \to \mathbb{R}$ and $p$.
Any hint?
Best Answer
Notice that you can apply Hahn-Banach to a functional defined on the trivial subspace $Y =\{0\}$ of $X$. There is exactly one such functional given by $f(0) = 0$. We must have $p(0) = 0$ by positive homogeneity so $f(x) \leq p(x)$ for all $x \in Y$.
Then Hahn-Banach yields an extension of $f$ to a functional on all of $X$ satisfying $f(x) \leq p(x)$ for all $x \in X$. Hence $f(-x) \leq p(-x)$ for all $x$, which implies $f(x) \geq -p(-x)$ for all $x$.