Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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"?

    I have thought about this for a while but don't know why.

  2. In a vector space, is it true that if a subset U of zero contains a scaled copy of the whole space, then it is in fact the whole space? I think it is not true when the base field of the vector space is a finite set?

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

Thanks and regards!

share|improve this question
1  
well, a scaled copy of the whole space is the whole space! –  t.b. Feb 24 '12 at 16:28
    
@t.b.: Really, in both TVS and vector space? Then Mariano introduced some unnecessary complication to that quote. –  Tim Feb 24 '12 at 16:30
    
@DavidMitra: How is the first equality true? –  Tim Feb 24 '12 at 16:32
1  
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
    
Thanks, @DavidMitra and t.b.! –  Tim Feb 24 '12 at 18:40

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.