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

Proof of $(\text N 2)$

Let $\lambda \in K$ and $x \in X$.

Then:

\(\ds \map q {\lambda x}\)

\(=\)

\(\ds \max_{p \mathop \in \II} \map p {\lambda x}\)

\(\ds \)

\(=\)

\(\ds \max_{p \mathop \in \II} \norm \lambda_K \map p x\)

$(\text N 2)$ in the definition of the seminorm

\(\ds \)

\(=\)

\(\ds \norm \lambda_K \max_{p \mathop \in \II} \map p x\)

\(\ds \)

\(=\)

\(\ds \norm \lambda_K \max_{p \mathop \in \II} \map p x\)

proving $(\text N 2)$.

$\Box$

Proof of $(\text N 3)$

Let $x, y \in X$.

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

\(\ds \map p {x + y}\)

\(\le\)

\(\ds \map p x + \map p y\)

$(\text N 3)$ in the definition of the seminorm

\(\ds \)

\(\le\)

\(\ds \max_{p \mathop \in \II} \map p x + \max_{p \mathop \in \II} \map p y\)

Definition of Pointwise Maximum of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \map q x + \map q y\)

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

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

$\blacksquare$