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Let $X$ be a linear space over the complex field. Let $X_{\Bbb R}$ be the space obtained from $X$ by restricting the scalars to the real field. I proved that $X_{\Bbb R}$ is a real linear space. But is it true that every real linear space is of the form $X_{\Bbb R}$ for some complex vector space?

paf
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Marios Gretsas
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    It seems false since the restriction of scalars of a $\Bbb C$-vector space of dim. $n$ is a $\Bbb R$-vector space of dim. $2n$. – paf Aug 20 '16 at 01:17
  • @paf Why is this true? A one-dimensional $\mathbb{C}$-vector space is just the field of scalars $\mathbb{C}$, and restricting to real scalars just gives $\mathbb{R}$, which is a one-dimensional real vector space. – angryavian Aug 20 '16 at 01:21
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    So what do you call "restriction of scalars"? Because the usual definition (https://en.wikipedia.org/wiki/Change_of_rings) gives $\Bbb R^2$ for restriction of scalars of $\Bbb C$. Maybe do you think of "real form", i.e. vector space $Y$ s.t. $Y\otimes_{\Bbb R}\Bbb C=X$. In this case, see http://math.stackexchange.com/questions/23470/realification-and-complexification-of-vector-spaces – paf Aug 20 '16 at 01:27
  • @paf Thanks for the references, I did not know about this. – angryavian Aug 20 '16 at 01:30
  • @angryavian Doesn't that give $\mathbb{C}$ again (I am taking the "restriction" as simply the restriction of the scalar: you have still the set $\mathbb{C}$, and the scalars are in $\mathbb{R}$; as such, it is a two-dimensional real vector space) – Clement C. Aug 20 '16 at 01:30
  • @ClementC. I thought "restricting the scalars to the real field" just meant throwing away all non-real scalars, and didn't realize it meant something different. – angryavian Aug 20 '16 at 01:32

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No; if $V$ is a $\Bbb{C}-$vector space of finite dimension $n$, then $V_\Bbb{R}$ is of dimension $2n$, so an odd-dimensional real vector space cannot be realized as a restricted version of a complex vector space. However, every $2n$-dimensional real vector space is isomorphic to $\Bbb{R}^{2n}$, which in turn can be viewed as $\Bbb{C}^n$, restricting its scalars to the reals.

florence
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