Discontinuous non-linear mappings on locally convex direct limits

Abstract

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the space of functions supported in K is smooth (and thus continuous), for each compact subset K of the line. More generally, we construct mappings with analogous pathological properties on spaces of compactly supported smooth sections in vector bundles over non-compact bases. The results are useful in infinite-dimensional Lie theory, where they can be used to analyze the precise direct limit properties of test function groups and groups of compactly supported diffeomorphisms.