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Let J be some index set.

I was watching a lecture by Mikhail Gromov and he made a passing comment about why $\mathbb{R}^J$ makes sense because J is a set, but $\mathbb{R}^n$ does not because $n$ is a number.

I thought we just used $n$ because it was a convenient way of saying we are taking $n$ Cartesian products of $\mathbb{R}$ with itself.

Can some explain what this means?

Asaf Karagila
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o0BlueBeast0o
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2 Answers2

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If $A,B$ are sets, one often writes $A^B$ for the set of all functions $B\to A$. Some motivation for this may be that $A^B$ has $a^b$ elements if $A$ ahas $a$ and $B$ has $b$ elements. In a way you again take $|B|$ cartesian products of $A$ with itself, but for infinite $B$ this mental image breaks down.

You can view $\mathbb R^n$ as a special case of this if you identify $n$ with a standard $n$-element set such as $\{0,1,\ldots, n-1\}$.

Hagen von Eitzen
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  • So if we could talk about this in more detail... The definition is just a definition and that is fine (also I have seen it before). So let me make sure I understand. We have $$\mathbb{R}^J := { f : J \rightarrow \mathbb{R} ; | ; f ; \text{a function} }$$ So for $J = { 0 , \cdots ,n }$, the number of functions from $J \rightarrow \mathbb{R}$ is $\mathbb{R}^n$? I am not understanding fully. Since $|\mathbb{R}| = c$ and $c^n = c$ (just using some naive set theory, I think maybe wrong). – o0BlueBeast0o Jul 03 '14 at 16:47
  • @Actually, you should take $J={0,\ldots,n-1}$ or ${1,\ldots,n}$. A function from an $n$-element set is just a list (of length $n$) of values, so not much different from an $n$-tuple. The cardinality of $\mathbb R^n$ happens to be the same as that of $\mathbb R$, but that is not the main point - the finite case is the motivation only. – Hagen von Eitzen Jul 03 '14 at 17:09
  • I meant $J = { 0, \cdots , n-1 }$ just a typo, but thank you I am starting to understand – o0BlueBeast0o Jul 03 '14 at 17:50
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This is a symbol for the family of all functions defined on $J$, taking values in $\mathbb{R}$. Technically, you can consider $n$-tuples or real numbers as functions defined on $\{1,2, \ldots, n\}$ which take values in $\mathbb{R}$.

This is a special case of a Cartesian product of sets.

Tomasz Kania
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    I would rather call it a symbol of the set of all families in $\mathbb{R}$ that are indexed by $J$. This in the sense that a family in $\mathbb{R}$ indexed by $J$ is exactly a function defined on $J$ (domain) and taking values in $\mathbb{R}$ (codomain). In my view the symbol $\mathbb{R}^{J}$ is not the symbol for a family. Have a look at http://math.stackexchange.com/q/35462/75923 – drhab Jul 03 '14 at 16:03