The very-strong C\(^\infty\) topology on C\(^{\infty}(M,N)\) and \(K\)-equivariant maps
type: Item
instance of: scholarly article
title: The very-strong C\(^\infty\) topology on C\(^{\infty}(M,N)\) and \(K\)-equivariant maps
zbMATH Open document ID: 1054.57033
published in: Osaka Journal of Mathematics
publication date: 2003-11-10T00:00:00Z
review text: A main result of the paper is Theorem A asserting that the set \(C^{\omega}(M,N)\) of real analytic maps between real analytic manifolds \(M,N\) is dense in the space \(C^{\infty}_{vS}(M,N)\) of all infinitely differentiable maps between \(M\) and \(N\), endowed with so-called very strong \(C^\infty\)-topology. This theorem also has an equivariant counterpart proved in Theorem C. Namely, if the manifolds \(M,N\) are endowed with smooth actions of a compact Lie group \(K\), then the space \(C^{\omega,K}(M,N)\) of real analytic \(K\)-equivariant maps from \(M\) to \(N\) is dense in the subspace \(C^{\infty,K}_{vS}(M,N)\) of \(C^\infty_{vS}(M,N)\) consisting of \(K\)-equivariant smooth maps, provided the number of \(K\)-isotropy types in \(N\) is finite. (The latter condition allows to find an equivariant embedding of \(N\) into a Euclidean \(K\)-space and to reduce the problem to maps into a suitable \({\mathbb R}^m\)). A substantial part of the paper is devoted to discussing a true topology on the space \(C^r(M,N)\) of \(C^r\)-maps from \(M\) to \(N\), where \(1\leq r\leq\infty\). In case of finite \(r\) such a true topology on \(C^r(M,N)\) is the Whitney topology. For infinite \(r\) there are two natural choices for a true topology on \(C^\infty(M,N)\): the \textit{strong topology}, which is the projective limit of the induced Whitney \(C^r\)-topologies with \(r<\infty\); and the \textit{very strong topology} which is the strongest topology making the restriction operators \(C^\infty(M,N)\to C^\infty(B,N)\) open and continuous for any compact subset \(B\subset M\), where \(C^\infty(B,N)\) is endowed with the strong topology. Thus two functions \(f,g:M\to N\) are close in the very strong topology if they have near derivatives \(D^\alpha f(x)\), \(D^\alpha g(x)\) up to some order \(| \alpha| \leq r\) growing as \(x\in M\) tends to infinity. The author proves many basic properties of the very strong topology, in particular, Glue Lemma B asserting that for any \(f\in C^…
Mathematics Subject Classification ID: 57R35
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 58K20
Mathematics Subject Classification ID: 58K70
zbMATH DE Number: 2002265
zbMATH Keywords: smooth map
zbMATH Keywords: real analytic map
zbMATH Keywords: equivariant map
zbMATH Keywords: function space
zbMATH Keywords: smooth manifold
zbMATH Keywords: Whitney topology
zbMATH Keywords: Mather topology
author: Soeren Illman
reviewed by: Taras Banakh
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2002265