Differentiation in modules over topological *-algebras
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.
scientific article; zbMATH DE number 150652
scientific article