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Here is a problem so beautiful that I had to share it. I found it in Paul Halmos's autobiography. Everyone knows that $\mathbb{C}$ is a vector space over $\mathbb{R}$, but what about the other way around?

Problem: Prove or disprove: $\mathbb{R}$ can be written as vector space over $\mathbb{C}$

Of course, we would like for $\mathbb{R}$ to retain its structure as an additive group.

Potato
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    Depends on what structures on $\mathbb{R}$ you want to preserve. I don't think this is a well-specified question. – Qiaochu Yuan Jun 06 '12 at 23:08
  • All you want to preserve is the structure as an additive group. – Potato Jun 06 '12 at 23:12
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    @QiaochuYuan: What about the question don't you understand? – Thomas Jun 06 '12 at 23:25
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    @Thomas: there's nothing I don't understand. The question as it is written just doesn't specify what structure on $\mathbb{R}$ is supposed to be retained. For example, by restriction of scalars, every complex vector space is also a real vector space. Is the induced real vector space structure on $\mathbb{R}$ supposed to have any relationship to its usual real vector space structure? Halmos doesn't specify this. – Qiaochu Yuan Jun 06 '12 at 23:36
  • The structure was specified when I first posted the problem. It's in the last line. – Potato Jun 06 '12 at 23:38

2 Answers2

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If you want the additive vector space structure to be that of $\mathbb{R}$, and you want the scalar multiplication, when restricted to $\mathbb{R}$, to agree with multiplication of real numbers, then you cannot.

That is, suppose you take $\mathbb{R}$ as an abelian group, and you want to specify a "scalar multiplication" on $\mathbb{C}\times\mathbb{R}\to\mathbb{R}$ that makes it into a vector space, and in such a way that if $\alpha\in\mathbb{R}$ is viewed as an element of $\mathbb{C}$, then $\alpha\cdot v = \alpha v$, where the left hand side is the scalar product we are defining, and the right hand side is the usual multiplication of real numbers.

If such a thing existed, then the vector space structure would be completely determined by the value of $i\cdot 1$: because for every nonzero real number $\alpha$ and every complex number $a+bi$, we would have $$(a+bi)\cdot\alpha = a\cdot \alpha +b\cdot(i\cdot \alpha) = a\alpha + b(i\cdot(\alpha\cdot 1)) = a\alpha + b\alpha(i\cdot 1).$$ But say $i\cdot 1 = r$. Then $(r-i)\cdot 1 = 0$, which is contradicts the properties of a vector space, since $r-i\neq 0$ and $1\neq \mathbf{0}$. So there is no such vector space structure.

But if you are willing to make the scalar multiplication when restricted to $\mathbb{R}\times\mathbb{R}$ to have nothing to do with the usual multiplication of real numbers, then you can indeed do it by transport of structure, as indicated by Chris Eagle.

davidlowryduda
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Arturo Magidin
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As an additive group, $\mathbb{R}$ is isomorphic to $\mathbb{C}$ (they're both continuum-dimensional rational vector spaces). Clearly $\mathbb{C}$ can be given the structure of a $\mathbb{C}$-vector space, thus $\mathbb{R}$ can too.

Chris Eagle
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  • The expansion is in the parenthesis: They are both continuum-dimensional rational vector spaces. What more is there to say? Things that are isomorphic as vector spaces are ipso facto also isomorphic as additive groups. – hmakholm left over Monica Jun 06 '12 at 23:16
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    The cardinality of the vector space equals the cardinality of the basis times the cardinality of the base field. If $|\Bbb Q|=\aleph_0$ and $|V|=\mathcal{c}$ then... – anon Jun 06 '12 at 23:25
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    Wouldn't the existence of a countable basis for $\Bbb R$ over $\Bbb Q$ immediately imply that $\Bbb R$ was a countable union of countable sets and therefore countable? – MJD Jun 06 '12 at 23:27
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    @MarkDominus: Without assuming the continuum hypothesis, you don't get that uncountable => at least continuum-sized. – Noah Stein Jun 06 '12 at 23:37
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    So what? I don't need CH to know that $|{\Bbb R}| > \aleph_0$, and that's all I need here. – MJD Jun 06 '12 at 23:46
  • I think you've only shown that the dimension is uncountable, not that it is continuum-dimensional. – Potato Jun 06 '12 at 23:53