Function Space

functional-analysis analysis

Definition

Function Space

A function space is a set whose elements are functions with a common domain and codomain. For sets $X$ and $Y$, the full function space from $X$ to $Y$ is

$$Y^X:=\{f\mid f:X\to Y\}.$$

Why the Exponent Has This Order

A function $f\in Y^X$ chooses one value $f(x)\in Y$ for every $x\in X$. It can therefore be viewed as an $X$-indexed family

$$(f(x){)}_{x\in X}\in \prod _{x\in X}Y.$$

If $X$ and $Y$ are finite, the number of such functions is

$$|Y^X|=|Y{|}^{|X|}.$$

Thus the codomain $Y$ is the base and the domain $X$ is the exponent.

Examples

Real-valued sequences

A real-valued sequence is a function from $\mathbb{N}$ to $\mathbb{R}$. The set of all such sequences is therefore

$${\mathbb{R}}^{\mathbb{N}}=\{a\mid a:\mathbb{N}\to \mathbb{R}\},a⟷(a(0),a(1),\dots ).$$