Topological classification of compact nonlinear mappings

In this paper, two kinds of approximate dimensions are introduced, namely one is the approximate dimension of the compact nonlinear mappings on infinite dimensional topological vector spaces, and the other is the approximate dimension of the compact nonlinear mappings on finite dimensional topological vector spaces. In the infinite dimensional case, it is shown that the approximate dimension of a compact nonlinear bijective mapping f is closely related 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 compact 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 mapping.