Smooth convergence of a series from derivative bounds

Statement

Let $f_j\in C^\infty(U)$ , where $U\subseteq\mathbb R^n$ is open. Suppose that for every compact $K\subset U$ and every integer $k\ge0$ ,

\sum_j p_{K,k}(f_j)<\infty,

with $p_{K,k}$ the compact derivative seminorm. Then $f=\sum_j f_j$ is smooth and

\partial^\alpha f=\sum_j\partial^\alpha f_j

locally uniformly for every multi-index $\alpha$ .

Proof

The Weierstrass M-test gives uniform convergence of each derivative series on compact sets. Inside any closed rectangular box contained in $U$ , apply the one-variable differentiation theorem along coordinate segments. Iterating identifies all the derivative limits. Such boxes cover $U$ .

A diagonal sufficient condition

Let $(K_j)$ exhaust $U$ , with every compact subset eventually contained in $K_j$ . It suffices to arrange $p_{K_j,j}(f_j)\le2^{-j}$ for all sufficiently large $j$ . For any fixed compact set and derivative order, the tail is then dominated by a geometric series; finitely many early terms cause no difficulty.