Seminorm is Sublinear Functional

Theorem

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

Let $p : X \to \R$ be a seminorm on $X$.

Then:

$p$ is a sublinear functional.

Proof

Since $p$ is a seminorm, we have:

$\map p {x + y} \le \map p x + \map p y$ for each $x, y \in X$

We also have:

$\map p {\lambda x} = \cmod \lambda \map p x$ for each $\lambda \in \R$ and $x \in X$.

and in particular:

$\map p {\lambda x} = \lambda \map p x$ for each $\lambda \in \R_{\ge 0}$ and $x \in X$.

So:

$p$ is a sublinear functional.

$\blacksquare$