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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!

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    well, a scaled copy of the whole space *is* the whole space!2012-02-24
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    @t.b.: Really, in both TVS and vector space? Then Mariano introduced some unnecessary complication to that quote.2012-02-24
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    @DavidMitra: How is the first equality true?2012-02-24
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    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 $v=\alpha(\alpha^{-1}v)\in\alpha V$.2012-02-24
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    Thanks, @DavidMitra and t.b.!2012-02-24

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