Interior of Balanced Set containing Origin in Topological Vector Space is Balanced

Theorem

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

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

Let $B \subseteq X$ be a balanced set such that:

${\mathbf 0}_X \in B^\circ$

where $B^\circ$ is the interior of $B$.

Then $B^\circ$ is balanced.

Proof

Let $\lambda \in \Bbb F$ have $0 < \cmod \lambda \le 1$.

Then, we have:

$\lambda B \subseteq B$

From Interior of Subset, we have:

$\paren {\lambda B}^\circ \subseteq B^\circ$

Then from Dilation of Interior of Set in Topological Vector Space is Interior of Dilation we have:

$\lambda B^\circ \subseteq B^\circ$

for $\lambda \in \Bbb F$ with $0 < \cmod \lambda \le 1$.

By hypothesis, we have:

${\mathbf 0}_X \in B^\circ$

and so:

$\set { {\mathbf 0}_X} \subseteq B^\circ$

So we have:

$\lambda B^\circ \subseteq B^\circ$

if $\lambda = 0$.

So:

$\lambda B^\circ \subseteq B^\circ$

for all $\lambda \in \Bbb F$ with $\cmod \lambda \le 1$, so $B^\circ$ is balanced.

$\blacksquare$

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.13$: Theorem