<P> Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions . For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces . </P> <P> It is rather frequent that a function with domain X may be naturally extended to a function whose domain is a set Z that is built from X . </P> <P> For example, for any set X, its power set P (X) is the set of all subsets of X . Any function f: X → Y, (\ displaystyle f: X \ to Y,) may be extended to a function on power sets by </P> <Dl> <Dd> P (X) → P (Y) S ↦ f (S) (\ displaystyle (\ begin (aligned) & (\ mathcal (P)) (X) \ to (\ mathcal (P)) (Y) \ \ &S \ mapsto f (S) \ end (aligned))) </Dd> </Dl>

The relation y=x is a function