Differentiation in modules over topological *-algebras
type: Item
instance of: scholarly article
title: Differentiation in modules over topological *-algebras
zbMATH Open document ID: 0810.46049
DOI: 10.1016/0022-247X(92)90018-9
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: The author develops a method of differentiation (called total differentiation) on topological modules \(\text{Mod}(A)\) over a unital topological *-algebra \(A\). Here is the definition: Let \(M, N\in \text{Mod}(A)\), \(x\in M\) and let \(U\) be a neighborhood of \(x\). A map \(f: U\to N\) is called totally differentiable at \(x\), if there exists a continuous \(A\)-linear map \(Df(x): M\to N\) and a continuous skew- linear map \(Sf(x): M\to N\) such that the element \(f(x+ h)- f(x)- Df(x)(h)- Sf(x)(h)\) is infinitesimal. The map \(Tf(x):= Df(x)+ Sf(x)\) is called the total differential of \(f\) at \(x\). The author proves the chain rule property and shows that total differentiation is stronger than the Fréchet, Hadamard and Gâteaux differentiations. The paper also contains some interesting examples of totally differentiable maps.
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 46K05
zbMATH DE Number: 150652
zbMATH Keywords: \(A\)-bundle
zbMATH Keywords: total differentiation
zbMATH Keywords: topological modules
zbMATH Keywords: unital topological *-algebra
zbMATH Keywords: total differential
zbMATH Keywords: chain rule property
author: M. H. Papatriantafillou
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 150652