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

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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.



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Proof

Proof that $B$ is convex

Let $t \in \closedint 0 1$ and $x, y \in B$.

Then:

\(\ds \map p {t x + \paren {1 - t} y}\) \(\le\) \(\ds t \map p x + \paren {1 - t} \map p y\) Seminorm Axiom $\text N 2$: Positive Homogeneity, Seminorm Axiom $\text N 3$: Triangle Inequality
\(\ds \) \(<\) \(\ds t + \paren {1 - t}\)
\(\ds \) \(=\) \(\ds 1\)

so:

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

$\Box$


Proof that $B$ is balanced

Let $s \in \GF$ such that $\cmod s \le 1$.

Let $x \in B$.

Then, we have by Seminorm Axiom $\text N 2$: Positive Homogeneity 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.

$\Box$


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 Seminorm Axiom $\text N 2$: Positive Homogeneity, we have:

$\map p {n x} < n$ if and only if $\map p x < 1$

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 Seminorm Axiom $\text N 2$: Positive Homogeneity, 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:

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

$\blacksquare$


Sources

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