An analog of the Hille theorem for hypercomplex functions in a finite-dimensional commutative algebra

TL;DR

This paper extends the classical Hille theorem to hypercomplex functions within finite-dimensional commutative Banach algebras, establishing new differentiability equivalences.

Contribution

It proves an analog of the Hille theorem for hypercomplex functions in finite-dimensional commutative Banach algebras, linking Gateaux and Lorch differentiability.

Findings

Locally bounded Gateaux differentiable functions are also Lorch differentiable.

The result generalizes classical complex analysis theorems to hypercomplex settings.

Provides a foundation for further analysis of hypercomplex functions in Banach algebras.

Abstract

We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.