Sublinear Functionals

Sublinear Functionals

 Definition: Let $X$ be a linear space. A function $p : X \to \mathbb{R}$ is said to be Subadditive if for all $x, y \in X$ we have that $p(x + y) \leq p(x) + p (y)$.
 Definition: Let $X$ be a linear space. A function $p : X \to \mathbb{R}$ is said to be Nonnegatively Homogeneous if for all $\lambda \geq 0$ and for all $x \in X$ we have that $p(\lambda x) = \lambda p(x)$.
 Definition: Let $X$ be a linear space. A Sublinear Functional is a function $p : X \to \mathbb{R}$ that is subadditive and nonnegatively homogeneous.

For example, if $p$ is a seminorm on $X$ then $p$ is a sublinear functional.

Furthermore, if $X$ be a normed linear space then the norm itself, i.e., $p(x) = \| x \|$ is a sublinear functional.
Clearly for all $x, y \in X$ and $\lambda > 0$ we have that:

(1)
\begin{align} \quad p(\lambda x) = \| \lambda x \| = |\lambda| \| x \| = \lambda \| x \| = \lambda p(x) \end{align}
(2)
\begin{align} \quad p(x + y) = \| x + y \| \leq \| x \| + \| y \| = p(x) + p(y) \end{align}