A generalization of a result of Hille on analytic implicit functions

TL;DR

This paper extends Hille's 1959 analytic implicit function theorem to lattice-normed linear spaces, broadening its applicability to Banach and Riesz spaces through a geometric interpretation.

Contribution

It generalizes Hille's implicit function result to lattice-normed spaces, including Banach and Riesz spaces, using a geometric approach.

Findings

Generalized implicit function theorem to lattice-normed spaces

Included cases for Banach and Riesz spaces

Provided geometric interpretation of Hille's condition

Abstract

Hille in 1959 gave a stronger form for the classical implicit function theorem of Cauchy for equations of the type $y = \psi(x,y)$, $x,y$ real or complex, $\psi$ analytic with a condition which determines a radius of convergence of the resulting series solution. Based on a geometrical interpretation of Hille's condition, this report generalizes his result to the case when x and y are elements of lattice-normed linear spaces in the sense of Kantorovich. This formulation includes the special cases when x and y belong to Banach or Riesz spaces.