Functions as vector spaces

The logic behind ${\mathbb{F}}^S$

Definition:

1. Say $S$ is a set. Then ${\mathbb{F}}^S$ denotes all functions from $S\to F$.
2. Addition is defined as such: For two functions $f,g\in F^S$ the function $f+g\in F^S$ is defined by $(f+g)(x)=f(x)+g(x)$ for $x\in S$
3. For scaler multiplication: For $\lambda \in F$ and $f\in F^S$ we have it so $(\lambda f)(x)=\lambda f(x)$ for $x\in S$

$F^n$ and $F^S$

You can think of $F^n$ as another version of $F^{\{1,...,n\}}$ as, for $n=3$, any $x\in F^n$ can be thought of as a function taking $x:(1,2,3)\to F$. Meaning, $(5,3,8)$ is just a function taking 1 → 5, 2→3, 3→8. The function has the exact same number of inputs and outputs (as its only continuous over the set {1,…,n}).