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To be more precise, in what sense is $\mathbb R^N$ a function space?

I quote from page number 3, in the first chapter of "Introduction to Hilbert Spaces with Applications" by Debnath and Mikusinski

"Note that spaces $\mathbb R^N$ and $\mathbb C^N$ can be defined as function spaces: $\mathbb R^N$ is the space of all real valued functions defined on $\{1,\dots ,N\}$ ..."

Could somebody please elaborate?

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2 Answers 2

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The vector $v=(a,b,c) \in \mathbb{R}^3$ can be seen as the function $v: \{1,2,3\} \to \mathbb{R}$ defined by

$f(1) = a$, $f(2)=b$, $f(3)=c$

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  • $\begingroup$ This is somewhat redundant compared to my answer. $\endgroup$
    – ThorbenK
    Apr 29, 2015 at 15:00
  • $\begingroup$ And both were posted at the same time... $\endgroup$
    – Tryss
    Apr 29, 2015 at 15:17
  • $\begingroup$ I'm pretty sure that my answer was there before. Like a 100% sure! $\endgroup$
    – ThorbenK
    Apr 29, 2015 at 15:57
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Define $f \colon \mathrm{Map}(\{ 1, \dotsc ,N \},\mathbb{R},) \to \mathbb{R}^N$ by $f(g)=(g(1),g(2),\dotsc ,g(N))$. This gives you the desired isomorphism. This is basically the choice of a basis.

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