Open Ball with respect to Seminorm is Convex, Balanced and Absorbing

Theorem
Let $\GF \in \set {\R, \C}$.

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

Let $p$ be a seminorm on $X$.

Let $d_p$ be the pseudometric induced by $p$.

Let $B$ be the open unit ball in $\struct {X, d_p}$.

That is:


 * $B = \set {x \in X : \map p x < 1}$

Then $B$ is convex, balanced and absorbing.

Proof that $B$ is convex
Let $t \in \closedint 0 1$ and $x, y \in B$.

Then:

so:


 * $t x + \paren {1 - t} y \in B$.

Proof that $B$ is balanced
Let $s \in \GF$ such that $\cmod s \le 1$.

Let $x \in B$.

Then, we have by we have:


 * $\map p {s x} = \cmod s \map p x \le \map p x < 1$

so $s x \in B$.

So, we have:


 * $s B \subseteq B$

for all $s \in \GF$ with $\cmod s \le 1$.

So $B$ is balanced.

Proof that $B$ is absorbing
From Characterization of Convex Absorbing Set in Vector Space, it is enough to show that:
 * $\ds X = \bigcup_{n \mathop = 1}^\infty n B$

By, we have:


 * $\map p {n x} < n$ $\map p x < 1$

for each $n \in \N$.

So, we have:


 * $n B = \set {x \in X : \map p x < n}$

Clearly we have:


 * $\ds \bigcup_{n \mathop = 1}^n n B \subseteq X$

Now let $x \in X$, then we have:


 * $\ds \map p {\frac x {2 \map p x} } = \frac 1 2 < 1$

from, so:


 * $\ds \frac x {2 \map p x} \in B$

Then we have:


 * $x \in \paren {2 \map p x} B$

Taking $N \in \N$ with $N \ge \map p x$, we have $x \in N B$, and so:


 * $\ds x \in \bigcup_{n \mathop = 1}^\infty n B$

So:


 * $\ds X = \bigcup_{n \mathop = 1}^\infty n B$