Locally convex infinitesimal calculus. 2. Computations on Mackey \((\ell^{\infty})\)
This paper deals with an application of a general locally convex differentiation theory for infinite dimensional spaces to the space \(\ell^{\infty}\) in the Mackey topology. The point of view of the author is that of the theory of infinitesimal encountered in nonstandard analysis. After a thorough discussion of the Mackey topology from this point of view the author then clarifies the notions of the calculus for, in general non-linear, Mackey continuous functions on \(\ell^{\infty}\). It is pointed out that the need for such a calculus is to a large extend dictated by the examples occurring in mathematical economics. A typical example of functions of \(\ell^{\infty}\) into \(\ell^{\infty}\) that can be treated by the methods of the paper are the following: \(F(x)=\sum^{\infty}_{t=1}a(t)\phi(x)(t)\), where \(x\in \ell^{\infty}\), \(a\in \ell^ 1\) and \(\phi\) is a smooth numerically valued function. [For part 1 see Trans. Am. Math. Soc. 240, 363-383 (1978; Zbl 0389.46032).]
scientific article; zbMATH DE number 3846252
scientific article