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Let $(V,+)$ be an abelian group. Can we have two multiplication map $\phi_1$ and $\phi_2$ from $\mathbb R\times V\to V$ such that $(V, +, \phi_1)$ and $(V, +, \phi_2)$ are two different vector space $( \phi_1\neq \phi_2)$.

zapkm
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  • Sorry for bad editing. First time posting from mobile device, will figure out soon.. – zapkm Jan 20 '16 at 15:03
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    The sign you want is $ – Pedro Jan 20 '16 at 15:03
  • In principle it could be possible that the same group $V$ has two distinct structures as a vector space. The problem here is that the base field $\Bbb{R}$ is "too complicated". – Crostul Jan 20 '16 at 15:10

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