# A subset that can be scaled to be the whole space, or that can contain a scaled version of the whole space

The following is from Mariano's comments on my earlier question

1. In a topological vector space, why is the following true:

if a neighborhood U of zero contains a scaled copy of the whole space, then it is in fact the whole space.

Is "a neighborhood U of zero contains a scaled copy of the whole space" the same as "a scaled copy of a neighborhood U of zero is the whole space"?

Sorry, I should have said: $\alpha V=V$, for $\alpha\ne0$. One inclusion is obvious. For the other inclusion, note if $v\in V$, then $v=\alpha(\alpha^{-1}v)\in\alpha V$. – David Mitra Feb 24 '12 at 16:40