Existence of continuous right inverses to linear mappings in finite-dimensional geometry
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Existence of continuous right inverses to linear mappings in finite-dimensional geometry Christer Oscar Kiselman · Erik Melin
Received: 27 May 2020 / Accepted: 29 July 2020 © The Author(s) 2020
Abstract A linear mapping of a compact convex subset of a finite-dimensional vector space always possesses a right inverse, but may lack a continuous right inverse, even if the set is smoothly bounded. Examples showing this are given, as well as conditions guaranteeing the existence of a continuous right inverse. Keywords Linear mapping · Continuous right inverse · Projection · Finitedimensional vector space
Existenz von stetigen Rechtsinversen zu linearen Abbildungen in der endlich-dimensionaler Geometrie Zusammenfassung Eine lineare Abbildung einer kompakten konvexen Menge in einem endlich-dimensionalen Vektorraum hat immer eine Rechtsinverse, aber nicht notwendigerweise eine stetige Rechtsinverse, auch dann nicht wenn der Rand der Menge glatt ist. Wir geben Beispiele dafür, sowie hinreichende Voraussetzungen für die Existenz einer stetigen Rechtsinversen.
1 Introduction A mapping has a right inverse if and only if it is surjective—at least if we allow the use of the axiom of choice. However, it is of interest to have not only a right inverse C. O. Kiselman () Department of Information Technology, Uppsala University, P.O. Box 337, 751 05 Uppsala, Sweden E-Mail: [email protected]; [email protected] E. Melin Eksoppsvägen 16, 756 46 Uppsala, Sweden E-Mail: [email protected]
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C. O. Kiselman, E. Melin
to a surjective mapping, but also a right inverse with some additional properties, like continuity. In this note we shall study the question of continuity for linear mappings in finite-dimensional vector spaces. An indication of the interest for this question is the book with its many references by Repovš & Semenov [4] on the selection of a continuous mapping inside a given multivalued mapping. Let F W Rn ! Rm be a linear mapping. Let A be a subset of Rn and B D fF .x/I x 2 Ag its image in Rm under F . We denote by f W A ! B the restriction of F to A and ask whether f has a right inverse which is continuous for the usual vector space topologies on Rn and Rm , thus whether there exists a continuous mapping v W B ! A such that f ı v D idB . The answer is no in general as we shall see in Sect. 2, even if we assume that A is compact and convex with smooth boundary. Then we prove some results in the positive direction. If A is convex, then so is its image B. If A is compact, then so is B as the image of a compact set under a continuous mapping. If A is open and if we choose as the codomain the smallest vector space containing the image of F , then also B is open. In studying this problem we may assume that F is a projection .x1 ; :::; xn / 7! .x1 ; :::; xm / for some numbers 0 6 m 6 n. If m D n, then f W A ! B D A is the identity and equal to its own inverse. If A is compact and convex and m 6 1, then the existence of such an inverse is also clear. The problem is therefore of interest only if 1 < m < n; there is a counterexamp
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