Question

If a sub additive functional defined on a normed space X is nonnegative outside a sphere {x Illxll = r}, show that it is nonnegative for all x E X

Answer 1

Suppose that $f$ is a sub additive functional on $X$. Then, by definition, we have

$\forall x,y \in X \quad f(x+y)\leq f(x)+f(y).$ For any $x\in X$ outside a sphere $f(x)\geq0$ . Suppose that $\tilde{x}$ lies either inside a sphere or on its boundary. This means that $||\tilde{x}||=\alpha\leq r$. We consider two cases:

1. $\alpha>0$ . We set $n=[\frac{r}{\alpha}]+2$ , where $[]$ denotes the truncation. Then $n\alpha>r$ . It means, that $||n\tilde{x}||>r$ and $n\tilde{x}$ lies outside the sphere. Using sub additivity, we get $0\leq f(n\tilde{x})=f((n-1)\tilde{x}+\tilde{x})\leq f(\tilde{x})+f((n-2)\tilde{x}+\tilde{x})\leq...\leq n\,f(\tilde{x})$

Therefore, we have $f(\tilde{x})\geq0$ .

2. $\alpha=0$ . Using the definition of the norm, we have $\tilde{x}=0$ . Assume that $x\in X$ is outside the sphere. Then, $f(x+0)\leq f(0)+f(x).$ From the latter we get $f(0)\geq0$ .

Thus, $f(x)\geq0$ for all $x\in X.$