Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods/Corollary 2

Corollary

Let $K \in \set {\R, \C}$.

Let $X$ be a topological vector space over $K$.

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

Then there exists a balanced open neighborhood $U$ of ${\mathbf 0}_X$ such that:

$U + U \subseteq W$

Proof

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollary, there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:

$V + V \subseteq W$

From Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood, there exists a balanced open neighborhood $U$ of ${\mathbf 0}_X$ such that $U \subseteq V$.

Then $U + U \subseteq V + V \subseteq W$.

$\blacksquare$