Positively Homogeneous and Subadditive Functions

# Positively Homogeneous and Subadditive Functions

Definition: Let $X$ be a linear space. A function $p : X \to [0, \infty)$ is said to be Positively Homogeneous if for all $\lambda > 0$ and for all $x \in X$ we have that $p(\lambda x) = \lambda p(x)$. |

Definition: Let $X$ be a linear space. A function $p : X \to [0, \infty)$ is said to be Subadditive if for all $x, y \in X$ we have that $p(x + y) \leq p(x) + p (y)$. |

Let $X$ be a normed linear space. One of the simplest example of a positive homogeneous and subadditive function is:

(1)\begin{align} \quad p(x) = \| x \| \end{align}

Then clearly for all $x, y \in X$ and $\lambda > 0$ we have that:

(2)\begin{align} \quad p(\lambda x) = \| \lambda x \| = |\lambda| \| x \| = \lambda \| x \| = \lambda p(x) \end{align}

(3)
\begin{align} \quad p(x + y) = \| x + y \| \leq \| x \| + \| y \| = p(x) + p(y) \end{align}