Definition:Sublinear Functional

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Let $V$ be a vector space over $\R$.

Let $p : V \to \R$ be a function.

We say that $p$ is a sublinear functional if and only if:

$(1): \quad$ $\map p {x + y} \le \map p x + \map p y$ for each $x, y \in V$
$(2): \quad$ $\map p {\lambda x} = \lambda \map p x$ for each $x \in V$ and $\lambda \in \R_{\ge 0}$.