Dilations of von Neumann-Bounded Neighborhood of Origin in Topological Vector Space form Local Basis for Origin
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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 $\delta_n \to 0$.
Let $V$ be a von Neumann-bounded open neighborhood of ${\mathbf 0}_X$.
Then:
- $\BB = \set {\delta_n V : n \in \N}$ is a local basis for ${\mathbf 0}_X$.
Proof
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 ${\mathbf 0}_X$.
$\blacksquare$
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.15$: Theorem