Dilation of Closure of Set in Topological Vector Space is Closure of Dilation

Theorem
Let $F$ be a topological field.

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

Let $A \subseteq X$.

Let $\lambda \in F \setminus \set {0_F}$.

Then we have:


 * $\lambda A^- = \paren {\lambda A}^-$

where $A^-$ denotes the closure of $A$.