Implicit Functions From Topological Vectors Spaces To Banach Spaces

Abstract

We show that there exist a ball of non zero radius within which one can express a certain subset of variables , in a system of analytic equation , as analytic functions of the remaining variables . We derive a nontrivial lower bound on the radius of such a ball on the domain of validity of the implicit function theorem . We obtain implicit function theorem for mappings from arbitrary topological vector spaces to Banach spaces. We prove Newton approximation with parameters , as the basis , for the inverse function theorem . Also we prove inverse and implicit function theorems for SCk-map over complete valued fields, which parallel to the classical theorems for continuously Fre'chet differentiable mapping in the real case.