Timeline for Is there an Isomorphism between R and C under addition? [duplicate]
Current License: CC BY-SA 2.5
12 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 11, 2010 at 20:06 | vote | accept | Daniel Miller | ||
| Jul 10, 2010 at 23:36 | history | edited | CommunityBot |
insert duplicate link
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| Jul 10, 2010 at 23:36 | history | closed |
Gerald Edgar Qiaochu Yuan Allen Knutson Kevin H. Lin Jonas Meyer |
exact duplicate | |
| Jul 10, 2010 at 21:04 | comment | added | Pietro Majer | Btw, talking about $\mathbb{R}$ as $\mathbb{Q}$ vector space, I'd like to mention that one can produce a Vitali set as a direct summand $V$ of the subspace $\mathbb{Q}$. This way $V$ can't have null Lebesgue measure since $\mathbb{R}=\cup_{q\in\mathbb{Q}} (V+q)$, and it can't have positive measure since $V−V\subset V$ is not a neighborhood of 0, as it should be were it a measurable set with positive measure. | |
| Jul 10, 2010 at 18:26 | comment | added | Pete L. Clark | This is an old saw, probably in part due to it being a favorite question (even on exams) of Paul Halmos. He liked to state it in the form "Can $\mathbb{R}$ be given the structure of a $\mathbb{C}$-vector space?" He describes this in his "automathography" <b>I Want to Be a Mathematician</b>. | |
| Jul 10, 2010 at 17:10 | comment | added | Arturo Magidin | @Martin: that works if $|V|>|K|$; otherwise, note that $K$ is a vector space over itself, but the dimension is not $|K|=\aleph_0$, it's $1$. | |
| Jul 10, 2010 at 17:07 | comment | added | Martin Brandenburg | If $K$ is a countable field and $V$ is a vector space, then its dimension equals its cardinality. In particular, such spaces are isomorphic as soon as they are equipotent. | |
| Jul 10, 2010 at 12:59 | comment | added | Gerry Myerson | Let the record show that Mariano beat me by two minutes. | |
| Jul 10, 2010 at 12:45 | answer | added | Gerry Myerson | timeline score: 5 | |
| Jul 10, 2010 at 12:43 | comment | added | Mariano Suárez-Álvarez | If you are asking whether there exists an isomorphism $\mathbb C\to\mathbb R$ as abelian groups, then the answer is yes: both are in fact $\mathbb Q$-vector spaces of the same dimension, so they are isomorphic as such. | |
| Jul 10, 2010 at 12:39 | history | asked | Daniel Miller | CC BY-SA 2.5 |