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.