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 and , the full function space from to is

Why the Exponent Has This Order

A function chooses one value for every . It can therefore be viewed as an -indexed family

If and are finite, the number of such functions is

Thus the codomain is the base and the domain is the exponent.

Examples

Real-valued sequences

A real-valued sequence is a function from to . The set of all such sequences is therefore