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I think $\mathbb R^3 \not\subseteq \mathbb R^4$ since $\mathbb R^4$ contains no ordered triples. So, $\mathbb R^3$ is not a suspace of $\mathbb R^4$. Does that make sense?

user306944
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  • It is isomorphic to a subspace of $\mathbb{R}^4$, namely the subspace with one of the coordinates fixed at 0. This is usually what people mean when they say it is a subspace. – siegehalver Jan 22 '16 at 00:36
  • See number 14 http://math.stackexchange.com/a/1024302/137487 – ThePortakal Jan 22 '16 at 00:37
  • To elaborate on other comments, you are correct when you say $\mathbb{R}^3 \nsubseteq \mathbb{R}^4$ as sets, since $\mathbb{R}^4$ does not contain any triples. However, there is a subspace of $\mathbb{R}^4$ that is isomorphic to $\mathbb{R}^3$ (see @ChristopherHalverson's response). – Kevin Sheng Jan 22 '16 at 00:40

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Yes, this is correct. However, $\mathbf{R}^4$ contains subspaces which are isomorphic to $\mathbf{R}^3$.

fkraiem
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