Can the multiplicative group $\mathbb{R}^*$ of nonzero real numbers be given the structure of a real vector space? I know that the positive reals can, namely by defining scalar multiplication by $a$ to be exponentiation by $a$ (i.e., $a\cdot x = x^a$), but this obviously doesn't work when we allow negative reals.
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closed as too localized by Andres Caicedo, Simon Thomas, Felipe Voloch, Harry Gindi, Qiaochu Yuan Dec 25 2010 at 5:47 |
