I recently started reading Linear Algebra Done Right by Axler, and I find it great up until Notation 1.23, where the first bullet point states:
If $S$ is a set then $F^S$ denotes the set of functions from $S$ to $F$
I would really like to know what this means; in other words, does this mean that for all $f \in S$, then $f \in F$, but the only difference is that $F$ is a vector field?
Best Answer
The question has already been answered, but here is a little context: the notation $Y^X$ is a little odd to denote the set of functions $X \rightarrow Y$. The origin of this notation lies in combinatorics, and more generally set theory.
Let $[n]$ be the set $\{1,\dots, n\}$. It is a classic motivational exercise in combinatorics to count the size of the set $\{f | f: [n] \rightarrow [m]\}$, and it turns out the answer is $m^n$ many functions (can you prove this?). So when $X$ and $Y$ are finite, we have the wonderful correspondence:
$$\left| Y^X\right| = |Y|^{|X|}$$
Moreover, once you have some cardinal arithmetic under your belt, we may extend this correspondence to infinite sets as well.