Pointwise Maximum of Finite Family of Seminorms is Seminorm

Theorem
Let $\struct {K, \norm {\,\cdot\,}_K}$ be a normed division ring.

Let $X$ be a vector space over $K$.

Let $\II$ be a set of seminorms on $X$.

Define:


 * $\ds q = \max_{p \mathop \in \II} p$

where $\max$ denotes the pointwise maximum over $\II$.

Proof of $(\text N 2)$
Let $\lambda \in K$ and $x \in X$.

Then:

proving $(\text N 2)$.

Proof of $(\text N 3)$
Let $x, y \in X$.

Then for each $p \in \II$ we have:

Taking pointwise maximums over $p \in \II$ we have:


 * $\map q {x + y} \le \map q x + \map q y$