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I know there are several different ways to define the dimension. What is the correct point of view to say that $\mathbb{R}^n$ has dimension $n$ in mathematical analysis?

Anna
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    Because it takes $n$ real numbers to uniquely specify a point, perhaps? – user132181 Aug 08 '14 at 16:07
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    The analyst sees an $n$-dimensional manifold in it. But that boils down to the tangent space (again $\mathbb R^n$) being an $n$ dimensional vector space over the field $\mathbb R$. – Hagen von Eitzen Aug 08 '14 at 16:12
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    @user132181 I can uniquely specify any point in $\mathbb{R}^{\left(10^{100}\right)}$ with a single real number. Analysis is the key here; the notion of continuity (as expressed via the tangent space in particular, as Hagen notes) is critical. – Steven Stadnicki Aug 08 '14 at 16:23

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