Related to this question (I sometimes read through the book to refresh my background in functional analysis).

The theorem starts with picking balanced neighbours $U$, $W$ such that

$$

bar{U} + bar{U} subset W

$$

But the conclusion of the theorem is reached through the chains of inclusions

$$

Lambda(V) subset Lambda x – Lambda (E) subset bar{U} – bar{U} subset W

$$

I’m comparing $bar{U} + bar{U} subset W$ against $bar{U} – bar{U} subset W$. In chapter 1 the definition of balanced is:

A set $B subset X$ (topological vector space) is said to be balanced if $alpha B subset B$ for every $alpha in Phi$ with $|alpha| leq 1$.

Therefore I suppose we have the inclusion $- bar{U} subset bar{U}$ (since $|-1|leq 1$) and therefore

$$

bar{U} – bar{U} subset bar{U} + bar{U}

$$

However my question is whether or not is $- bar{U} subset bar{U}$ or $- bar{U} = bar{U}$, I think the latter is the definition of symmetric set.