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I'm reading an analysis book and sometimes the author uses several denotations: $\mathbb R^{m+n}$, $\mathbb R^m\times\mathbb R^n$ or $\mathbb R^m\oplus\mathbb R^n$(for me they are essentially the same thing).

The first two spaces are homeomorphic to each other, I think when he uses $\mathbb R^m\times\mathbb R^n$ he want to emphasize some open subsets $U\subset \mathbb R^m$ and $V\subset \mathbb R^n$. However there is another problem, these spaces are diffeomorphic? if don't, we can't say they are essentially the same in the context of real analysis.

If my reasoning is right so fat, What I really don't understand is why he uses the direct sum $\mathbb R^m\oplus\mathbb R^n$. Does he want to emphasize the underline operation between the two spaces? this is not clear in the book.

Am I right? what is the difference between $\mathbb R^m\times\mathbb R^n$ and the external direct sum $\mathbb R^m\oplus\mathbb R^n$ in this context?

user42912
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    The author probably means the same thing for these three objects, but the difference is in the structure that is emphasized on the set: $\times$ is usually used to denote the product of two topological spaces, and $\oplus$ is usually used to denote the direct sum of algebraic objects; in this case vector spaces. Therefore you can see $\mathbb{R}^m\times\mathbb{R}^n$ as a topological space which is indeed homeomorphic to $\mathbb{R}^{m+n}$, and $\mathbb{R}^m\oplus\mathbb{R}^n$ as a real vector space which is isomorphic to $\mathbb{R}^{m+n}$. – Uncountable Jan 16 '17 at 22:44

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