Locally convex infinitesimal calculus. 2. Computations on Mackey \((\ell^{\infty})\)
type: Item
instance of: scholarly article
title: Locally convex infinitesimal calculus. 2. Computations on Mackey \((\ell^{\infty})\)
zbMATH Open document ID: 0533.46048
DOI: 10.1016/0022-1236(83)90043-5
published in: Journal of Functional Analysis
publication date: 1983-01-01T00:00:00Z
review text: 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).]
Mathematics Subject Classification ID: 46S20
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 91B99
Mathematics Subject Classification ID: 26E35
Mathematics Subject Classification ID: 58C20
zbMATH DE Number: 3846252
zbMATH Keywords: locally convex differentiation
zbMATH Keywords: Mackey topology
zbMATH Keywords: infinitesimal
zbMATH Keywords: nonstandard analysis
author: Keith Stroyan
reviewed by: W. A. J. Luxemburg
MaRDI profile type: Publication
full work available at URL: 0022-1236(83)90043-5
OpenAlex ID: W2028824670
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description: scientific article; zbMATH DE number 3846252