# Intuition Behind Balanced Sets

*Solution:*

[Added in response to the comments below:] Despite what is written below,
*balanced* needn't imply convex; it simply doesn't follow from the definition. So one should take the following discussion as rather an explanation about *convex* balanced sets, for whatever that is worth.

[Original answer follows:]

*Balanced* means that the set is symmetric around the origin (i.e. invariant under $v \mapsto -v$) and is convex. These are nice properties, generalizing some useful features of balls around the origin in $\mathbb R^n$. (Incidentally, if you didn't have the condition $|\alpha| = 1$, then any balanced subset with non-empty interior (in $\mathbb R^n$ say) would be all of $\mathbb R^n$, so that wouldn't be such an interesting condition.)

Also, any vector space will have plenty of such sets; you can form them just by taking the convex hull of a collection of points invariant under $v \mapsto -v$. What is more relevant in functional analysis is whether the origin of a topological vector space contains a *basis* of balaced neighbourhoods. The topological vector space is then called *locally convex* and this has important consequences for the space (especially for
its duality theory).

Note that balls around the origin in a normed space are automatically balanced, so normed spaces are locally convex; so again, in the topological setting, one sees that this is a way of capturing some the geometric aspects of normed vector spaces, without insisting on the existence of a norm. (So, while normed spaces are not closed under certain operations, such as forming infinite direct sums, locally convex spaces are --- any direct sum (just to take one example) of a collection of locally convex spaces has a natural locally convex topology of its own.)