Topological classification of compact nonlinear mappings

In this paper, two kinds of approximate dimensions are introduced, namely o ne is the approximate dimension of the compact nonlinear mappings on infini te dimensional topological vector spaces, and the other is the approximate dimension of the compact nonlinear mappings on finite dimensional topologic al vector spaces. In the infinite dimensional case, it is shown that the ap proximate dimension of a compact nonlinear bijective mapping f is closely r elated to the degree of continuity of f(-1). In the finite dimensional case , it is shown that if the dimension of the domain space on which the compac t nonlinear mappings are defined is equal to n, then the semigroup together with the superposition operation, which consists of all compact nonlinear mappings whose approximate dimensions are less than n, has no identity mapp ing.