Diffeomorphism invariant norms on \(C^k(X)\)
The main result of the paper is the following: If \(X\) is a compact manifold without boundary of class \(C^k\) with \(k>0\) and \(\dim(X)>1\), then every \(C^k\)-invariant semi-norm on the space \(C^k(X)\) depends for \(f\in C^k(X)\) only on \(\sup(f)\) and \(\inf(f)\) and is either equivalent to the sup-norm or is a multiple of the semi-norm \(\sup(f) -\inf(f)\). The statement also holds for \(k=0\) if the semi-norm is continuous to start with. The author also obtains results for non-connected manifolds and manifolds with boundary. In an appendix it is shown that \(C_0(S^1)\) is not finitely generated as module over \(\text{Homeo}_0(S^1)\).
scientific article; zbMATH DE number 1240928
scientific article