Compact derivative seminorm

For open $U\subseteq\mathbb R^n$ , nonempty compact $K\subset U$ , and integer $k\ge0$ , the compact derivative seminorm on $C^\infty(U)$ is

p_{K,k}(f)=\max_{|\alpha|\le k}\sup_{x\in K}|\partial^\alpha f(x)|.

It is finite by continuity and compactness, and satisfies absolute homogeneity and the triangle inequality. It is generally only a seminorm: a nonzero function supported away from $K$ can have $p_{K,k}(f)=0$ .

Smooth convergence

A sequence converges in $C^\infty(U)$ when every such seminorm of its difference from the limit tends to zero. A countable compact exhaustion and increasing derivative orders suffice to describe this topology, which makes $C^\infty(U)$ a Fréchet space.

For vector-valued functions, replace absolute value by a fixed finite-dimensional norm; the resulting notions of convergence agree.