I'm wondering about the following:
Let$\ X $ be a topological vector space. Then one could pick balanced neighborhoods$\ W $ and$\ U $ of$\ 0 $ such that
$\ \overline{U} + \overline{U} \subset W $, where $\ U+U:=\{u_1 + u_2 | u_1,u_2 \in U \} $
I was faced this question while reading Rudin's "Functional Analysis". I'm able to prove this, but I think there should be a more elegant and easier way to do it.
Listed are the properties I used:
- using continuity of$\ + $ one can easily show that there exist $\ U, W $ as above such that $\ U+U \subset W $
- Use that every topological vector space has a balanced local base (local means here at$\ 0 $.)
- If $\ \mathcal{B} $ is a local base (in the above sense) for a topological vector space$\ X $ then every member of$\ \mathcal{B} $ contains the closure of some member of$\ \mathcal{B} $.
- and the last property I used was:$\ \overline{U_1} + \overline{U_2} \subset \overline{U_1+U_2} $ where$\ U_1,U_2 \subset X $
As you can see I need a lot of theory / basic properties about topological vector spaces and I'm just wondering if there's not an easier way. Thx for suggestions.
cheers
math