Sequential Characterization of von Neumann-Boundedness in Topological Vector Space
Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $E \subseteq X$.
Then $E$ is von Neumann-bounded if and only if:
- for each sequence $\sequence {x_n}_{n \in \N}$ in $E$ and each sequence $\sequence {\alpha_n}_{n \in \N}$ in $\GF$ such that $\alpha_n \to 0$ as $n \to \infty$, we have $\alpha_n x_n \to {\mathbf 0}_X$ as $n \to \infty$.
Proof
Necessary Condition
Let $E \subseteq X$ be von Neumann-bounded.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$.
From Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood, there exists a balanced open neighborhood $V$ of ${\mathbf 0}_X$ such that $V \subseteq X$.
Since $E$ is von Neumann-bounded, there exists $t > 0$ such that:
- $E \subseteq t V$
so that:
- $\ds \frac 1 t E \subseteq V$
Then we have:
- $\ds \frac {x_n} t \in V$
Since $\alpha_n \to 0$, there exists $N \in \N$ such that:
- $\ds \cmod {\alpha_n} < \frac 1 t$ for $n \ge N$.
That is:
- $\ds \cmod {\alpha_n} t = \cmod {\alpha_n t} < 1$ for $n \ge N$.
Since $V$ is balanced, we have:
- $\alpha_n t V \subseteq V$
so that:
- $\alpha_n x_n \in V$ for $n \ge N$
since $x_n/t \in V$.
So, we have:
- $\alpha_n x_n \in V$ for $n \ge N$.
So $\alpha_n x_n \to \mathbf 0_X$ as $n \to \infty$.
$\Box$
Sufficient Condition
Let $E \subseteq X$ be such that:
- for all sequences $\sequence {x_n}_{n \in \N}$ is a sequence in $E$ and $\sequence {\alpha_n}_{n \in \N}$ is a sequence in $\GF$ such that $\alpha_n \to 0$, we have $\alpha_n x_n \to {\mathbf 0}_X$.
Suppose that $E$ is not von Neumann-bounded.
Then for some open neighbourhood $V$ of ${\mathbf 0}_X$ and all $s > 0$, there exists $t > s$ such that $E \not \subseteq t V$.
That is, there exists a sequence $\sequence {r_n}_{n \in \N}$ of positive real numbers such that $r_n \to \infty$ and $E \not \subseteq r_n V$ for each $n \in \N$.
Then $r_n^{-1} \to 0$ as $n \to \infty$.
For each $n \in \N$, pick $x_n \in E \setminus r_n V$.
Then we have $r_n^{-1} x_n \not \in V$ for each $n \in \N$.
So $\sequence {r_n^{-1} x_n}_{n \in \N}$ cannot converge to $\mathbf 0_X$.
This is a contradiction, so $E$ is von Neumann-bounded.
$\blacksquare$
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.30$: Theorem