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I understand why $F^S$ (where S is a set) is a vector space. I just don't see why we can think of $F^n$ as $F^{\{1,2,...,n\}}$. I saw someone saying that we can think of $\mathbb{R}^3$ as the set of functions from ${\{1,2,3\}}$ to $\mathbb{R}$, but how can a function be a number in $\mathbb{R}$? Or do they mean the result of the function? I'm self-learning math so I apologise if the answer is obvious.

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  • $\begingroup$ What is $\Bbb R^3$ for you? $\endgroup$ – Asaf Karagila Dec 22 '17 at 10:27
  • $\begingroup$ More importantly, if $\Bbb R^3$ is a space of $3$-tuples, how can a tuple be a number? $\endgroup$ – Asaf Karagila Dec 22 '17 at 10:34
  • $\begingroup$ @Asaf Karagila I always thought of $\mathbb{R}$ as the space containing all possible combinations of 3 real numbers. $\endgroup$ – Surzilla Dec 22 '17 at 11:12
  • $\begingroup$ Okay, so either a combination of numbers is a number, or your comment about "how can a function be a number in $\Bbb R$" is irrelevant here. Because $(0,0,1)$ and $(1,0,0)$ are not the same vector, even though it's "almost the same combination". $\endgroup$ – Asaf Karagila Dec 22 '17 at 11:16
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If $f$ is a function from $\{1,2,3\}$ into $\mathbb R$, then $\bigl(f(1),f(2),f(3)\bigr)\in\mathbb{R}^3$. And if $(a,b,c)\in\mathbb{R}^3$, then you can define a function $f$ from $\{1,2,3\}$ into the reals by $f(1)=a$, $f(2)=b$, and $f(3)=c$. So, there is a bijection between $\mathbb{R}^3$ and the set of all functions from $\{1,2,3\}$ into $\mathbb R$.

By the same argument, there is a natural way of identifying $F^n$ with the set of all functions from $\{1,2,\ldots,n\}$ into $F$.

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  • $\begingroup$ Thank you, I think I get it. You use the output of the function, not the function itself. $\endgroup$ – Surzilla Dec 22 '17 at 11:14
  • $\begingroup$ @Surzilla If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Dec 22 '17 at 11:15
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Mathematically speaking, the @José Carlos Santos is the right one, and is the one teached in universtity classes.

What you probably need is a more "semantic" tranlsation of that answer. Let's try:

$\mathbb R^3$ is the set of triples $(x_1,x_2,x_3)$ where $x_1,x_2,x_3$ are real numbers. Right? Now, what does it mean "$x_1$"? "$x_1$" is the number that corresponds to the place $1$. So $x_2$ is the number that corresponds to the place $2$, and so on.

Therefore, $(x_1,x_2,x_3)$ is in fact a correspondence from the set of places $1,2,3$ to the set of real numbers, that is to say a function from $\{1,2,3\}$ to $\mathbb R$.

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