Elementary Properties of Sublinear Functionals

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $V$ be a real vector space with zero vector $0_{V}$, and let $p$ be a sublinear functional on $V$.

(i) Value at zero. $p(0_{V})=0$.

(ii) Reflection. $-p(-x)\le p(x)$ for every $x\in V$.

(iii) Dominated linear functionals. If $M$ is a linear subspace of $V$ and $f$ is a linear functional on $M$ with $f(x)\le p(x)$ for every $x\in M$, then

$$-p(-x)\le f(x)\le p(x)\qquad\text{for every }x\in M.$$

(iv) The norm. If $\lVert\cdot\rVert$ is a norm on $V$, then $\lVert\cdot\rVert$ is a sublinear functional on $V$.

(v) Lipschitz bound. If $\lVert\cdot\rVert$ is a norm on $V$ and $C\ge0$ is a real number with $p(x)\le C\lVert x\rVert$ for every $x\in V$, then

$$|p(x)-p(y)|\le C\,\lVert x-y\rVert\qquad\text{for all }x,y\in V.$$

(vi) Bound for dominated functionals. If $\lVert\cdot\rVert$ and $C$ are as in (v) and $M$ and $f$ are as in (iii), then $|f(x)|\le C\lVert x\rVert$ for every $x\in M$.