Dilations of von Neumann-Bounded Neighborhood of Origin in Topological Vector Space form Local Basis for Origin

Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\Bbb F$.

Let $\sequence {\delta_n}_{n \mathop \in \N}$ be a strictly decreasing real sequence with $\delta_n \to 0$.

Then:

$\BB = \set {\delta_n V : n \in \N}$ is a local basis for ${\mathbf 0}_X$.

Let $U$ be an open neighborhood of ${\mathbf 0}_X$.

We show that there exists $N \in \N$ such that:

$\delta_N V \subseteq U$

Since $V$ is von Neumann-bounded, there exists $s > 0$ such that:

$V \subseteq t U$ for $t > s$.

Since $\delta_n \to 0$ and $\sequence {\delta_n}_{n \mathop \in \N}$ is a strictly decreasing real sequence, there exists $N \in \N$ with:

$\ds 0 < \delta_N < \frac 1 s$

so that:

$\ds \frac 1 {\delta_N} > s$

giving:

$\ds V \subseteq \frac 1 {\delta_N} U$

so that:

$\delta_N V \subseteq U$

Since $U$ was arbitrary, we have that $\BB$ is a local basis for ${\mathbf 0}_X$.

$\blacksquare$