Theorem N23.1. If , pointwise, and uniformly, then and (and uniformly).
Corollary N23.2. is a complete metric space.
Proof. Use completeness of (uniform topology). If is Cauchy in , then and are Cauchy in , so there are uniform limits . By the theorem, and . Now and uniformly, so in . This proves the case . Repeat the same argument for successive derivatives for .∎
Corollary N23.3. If is a power series with radius of convergence , then is on and .
Integration
The definite integral of a continuous function is a linear operator,, for each , satisfying …