A question about the symbol $n$ in the additive group $\mathbb{R}^n$

Question

As we know the symbol $n$ determines the dimension of $\mathbb{R}^n$ in the category of vector spaces. Also, in the category of manifolds, $n$ determines the dimension of $n$-manifold $\mathbb{R}^n$.

Now, my question is that:

Is there any algebraic property related to the additive group $(\mathbb{R}^n ,+)$ which is determined by $n$?

score 3 · Accepted Answer · answered Aug 29 '18 at 07:52

3

This answer proves that as additive groups $\mathbb{R}^n \cong \mathbb{R}^m$ for all $n,m$.

answered Aug 29 '18 at 07:52

Ivan Di Liberti

4,881

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As $K$-vector spaces even, for any countable subfield $K$ of $\mathbb{R}$ – Maxime Ramzi Aug 29 '18 at 08:57
@IvanDiLiberti Thank you very much. – M.Ramana Aug 29 '18 at 11:16
@Max This is a good point for me. Thank you so much. – M.Ramana Aug 29 '18 at 11:19

A question about the symbol $n$ in the additive group $\mathbb{R}^n$

1 Answers1