There is no natural topology on duals of locally convex spaces
type: Item
instance of: scholarly article
title: There is no natural topology on duals of locally convex spaces
zbMATH Open document ID: 1041.46502
DOI: 10.1007/BF01196437
author: Klaus Floret
author: Heinz König
published in: Archiv der Mathematik
publication date: 1994-01-01T00:00:00Z
review text: The authors show that there is no ``natural topological duality'' on the category of locally convex spaces. That is, there does not exist any functor from the category of locally convex spaces into the category of topological spaces which associates to each \((X,\tau)\) a topology \(\tau\sphat\) on \((X,\tau)'\) such that the following two conditions hold: (1) For any pair of locally convex spaces \((X,\tau)\) and \((Y,\tilde\tau)\), a linear map \(f\colon X\to Y\) is \(\tau\)-\(\tilde\tau\)-continuous if and only if its linear adjoint \(f^*\) maps \((Y,\tilde\tau)'\) into \((X,\tau)'\) and the restriction of \(f^*\) to \((Y,\tilde \tau)'\) is \(\tilde \tau \sphat\)-\(\tau \sphat\)-continuous. (2) If \(\tau\) is the norm topology of any normed space \((X,\|\cdot\|)\), then \(\tau \sphat\) equals the topology induced by the dual norm\break \(\|\cdot\|^*\).
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 46M15
zbMATH DE Number: 590076
reviewed by: Klaus Dieter Bierstedt
MaRDI profile type: Publication
cites work: Q5663077
cites work: Q5817725
cites work: Q3227025
cites work: Q4132979
cites work: Q4722576
description: scientific article; zbMATH DE number 590076