# How many dimensions can be embedded within a small number of dimensions

I was wondering how many dimensions can be embedded within a 2D space, or more generally within N-dimensions. Is there a formal demonstration?

This question came to me when I read about the Holographic principle, it says is possible to encode a volume inside a black hole into the area that surrounds it. So there is a one to one correspondence between 3D and 2D.

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I am not sure the tag vector-space is appropriate for this question. It might be better to pose this question on physics SE, too. – Fabian Dec 30 '12 at 15:39
@Fabian The Holographic principle is just an exemplification. – rraallvv Dec 30 '12 at 15:43
You should have clarified what you mean with embedding. The holographic principle conserves some notion of locality. – Fabian Dec 30 '12 at 15:50
I've edited the question for clarification – rraallvv Dec 30 '12 at 16:07

The Abelian group $(\mathbb R,+)$ is isomorphic to that of $(\mathbb R^k,+)$, for any finite $k$, and in fact even more.