I understand how in 1.23 Axler states that Fs is the set of all functions that map from the set S to F.
He later states that we can think of Fn as F{1,2,3,….,n}
Although an explanation in the book is provided, I'm still confused as to how we can represent it that way.
For example, according to the previous statement, R4 can be written as R{1,2,3,4}. However wouldn't that say R4 is the set of all functions that map from {1,2,3,4} to a real number R? Wouldn't R4 just be the all the sets that have 4 elements from R or am I misunderstanding something?
Best Answer
$\mathbb{R}^4$ is not the set of all sets that contain $4$ elements from $\mathbb{R}$. For instance, $\{1,5,10,12\}=\{12,5,1,10\}$, but for elements of $\mathbb{R}^4$, $(1,5,10,12)\ne (12,5,1,10)$. The distinction is that in sets order of elements make no difference, but elements of $\mathbb{R}^4$ are $4$-tuples where order matters.
An $n$-tuple can be thought of as a function that maps each index $i=1~..~n$ to its corresponding value. For instance $(1,5,10,12)$ can be thought of as the function $f:\{1,2,3,4\}\to\mathbb{R}$ given by $$1\mapsto 1,~~2\mapsto 5,~~3\mapsto 10,~~4\mapsto 12.$$