
TL;DR
This paper develops the theory of Gateaux derivatives for maps on Banach algebras over a commutative ring, establishing conditions for differentiability and Taylor series expansion.
Contribution
It introduces a formal definition of higher-order Gateaux derivatives in Banach algebras and derives the Taylor series expansion for such maps.
Findings
Defined Gateaux derivatives of arbitrary order in Banach algebras.
Established the Taylor series expansion for differentiable maps.
Provided a framework for analyzing differentiability in Banach algebra contexts.
Abstract
Let be Banach algebra over commutative ring . The map is called differentiable in the Gateaux sense, if where the Gateaux derivative of map is linear map of increment and is such continuous map that Assuming that we defined the Gateaux derivative of order , we define the Gateaux derivative of order of map . Since the map has all derivatives, then the map has Taylor series expansion
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Taxonomy
TopicsAdvanced Topics in Algebra
