Infinite products of relations, set-valued series and uniform openness of multifunctions
type: Item
instance of: scholarly article
title: Infinite products of relations, set-valued series and uniform openness of multifunctions
zbMATH Open document ID: 0843.54027
DOI: 10.1007/BF01033638
author: Jean-Paul Penot
author: Chin Cheng Chou
published in: Set-Valued Analysis
publication date: 1996-08-15T00:00:00Z
review text: The authors, using the notations of convergent series of sets and convergent products of relations, present general open mapping theorems. Let \(X\) be a topological vector space. A series of nonempty subsets of \(X\), \(\sum X_n\), is said to be Kuratowski (resp. Mosco) convergent if the sequence of partial sums \(S_n= X_0+ \dots+ X_n\) converges in the sense of Kuratowski (resp. Mosco). The uniform structure of \(X\) leads the authors to the use of hemiconvergence. \(\sum X_n\) is said to be hemiconvergent with sum \(S\) if the sequence \((S_n)\) hemiconverges to \(S\): \[ \forall U\in N_X (0)\;\exists n_U\in N:\;\forall n\geq n_U,\;S_n\subset S+U,\;S\subset S_n+U. \] \(N_X (0)\) denotes the family of neighborhoods of 0. \(\sum X_n\) is called selectionwise convergent (in short convergent) if for any selection \((x_n)\) of \((X_n)\) the series \(\sum x_n\) is convergent. \(\sum X_n\) is a Cauchy series if the following condition is satisfied: \[ \forall V\in N_X (0)\;\exists n_V\in N:\;p\geq n\geq n_V \Rightarrow X_n+ \dots+ X_p \subset V. \] It is proved that any convergent series is a Cauchy series. If the space \(X\) is sequentially complete the converse holds. It is also proved that any convergent series is hemiconvergent. Also a number of results valid for metric spaces extend to the framework of uniform spaces which is adapted to uniform notions which also appers when one deals with topological vector spaces or, more generally, with topological groups. To achieve this, the authors give the following definition: Given a sequence \((R_n )_{n\geq 0}\) of relations of a topological space \(X\) we say that the product \(\prod_{n\geq 0} R_n\) is pointwise convergent if the sequence \((x_n )_{n\geq 0}\) of \(X\) such that \(x_{n+1}\in R_n (x_n)\) for each \(n\geq 0\) is convergent. Then, the product relation \(\prod_{n\geq 0} R_n\) is given by: for \(x\in X\) the image \(R(x)= \prod_{n\geq 0} R_n (x)\) is the set of limits of such sequences with \(x_0 =x\). When \(…
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 773786
zbMATH Keywords: convergence of sets
zbMATH Keywords: series of sets
zbMATH Keywords: open mapping theorems
zbMATH Keywords: hemiconvergence
zbMATH Keywords: quasi-uniform spaces
reviewed by: B. K. Papadopoulos
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01033638
OpenAlex ID: W2060052339
description: scientific article; zbMATH DE number 773786