# Why can we think of $F^n$ as $F^{\{1,2,…,n\}}$?

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.

• What is $\Bbb R^3$ for you? – Asaf Karagila Dec 22 '17 at 10:27
• More importantly, if $\Bbb R^3$ is a space of $3$-tuples, how can a tuple be a number? – Asaf Karagila Dec 22 '17 at 10:34
• @Asaf Karagila I always thought of $\mathbb{R}$ as the space containing all possible combinations of 3 real numbers. – Surzilla Dec 22 '17 at 11:12
• 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". – Asaf Karagila Dec 22 '17 at 11:16

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 $$F$$.

• Thank you, I think I get it. You use the output of the function, not the function itself. – Surzilla Dec 22 '17 at 11:14
• @Surzilla If my answer was useful, perhaps that you could mark it as the accepted one. – José Carlos Santos Dec 22 '17 at 11:15

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 $\mathbb R$.