Closed ideals of \(A^\infty\) and a famous problem of Grothendieck
type: Item
instance of: scholarly article
title: Closed ideals of \(A^\infty\) and a famous problem of Grothendieck
zbMATH Open document ID: 1242.46003
DOI: 10.4171/RMI/660
author: Juan-Miguel Gracia
published in: Revista Matemática Iberoamericana
publication date: 2011-10-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1312906785
review text: In the paper the author constructs another examples of nuclear Fréchet spaces without bases. The Fréchet algebra techniques play an important role. The paper consists of four sections. In the introductory part the author gives a brief history of the problem. The second part recalls basic definitions and well-known facts. The main result is contained in Theorem 2.7. It establishes examples of nuclear Fréchet spaces without a basis which are quotients of some Fréchet algebras. In Section 4 the author gives more concrete examples of nuclear Fréchet spaces without a basis. The first of them deals with the space \(A^{\infty}(\mathbb{T})\) and \(I(E):=\{f\in A^{\infty}(\mathbb{T}):\,\,f\equiv0\,\,\text{on}\,\,E\}\) for a closed, disconnected infinite \(E\subset\mathbb{T}\) such that \(I(E)\) is a proper, closed ideal of \(A^{\infty}(\mathbb{T})\). Then in Section 3 it is shown that for some \(E\) the ideal \(I(E)\) is indeed proper.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46J05
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 5965421
zbMATH Keywords: nuclear Fréchet space
zbMATH Keywords: basis
zbMATH Keywords: Fréchet algebra
zbMATH Keywords: power series generator
zbMATH Keywords: analytic functions
zbMATH Keywords: \(C^{\infty}\)-functions
MaRDI profile type: Publication
cites work: Fréchet algebras and formal power series
cites work: Boundary values of absolutely convergent Taylor series
cites work: Frechet Spaces with Nuclear Kothe Quotients
cites work: Q4130520
cites work: Nuclear Frechet spaces without bases. III: Every nuclear Frechet space not isomorphic to omega admits a subspace and a quotient space without a strong finite dimensional decomposition
cites work: On Fréchet algebras of power series
cites work: Q4487424
cites work: Fréchet algebras of power series
cites work: Q4857475
cites work: Modified construction of nuclear Frechet spaces without basis
cites work: The space of real-analytic functions has no basis
cites work: Quotient Spaces Without Bases in Nuclear Frechet Spaces
cites work: Fréchet spaces with quotients failing the bounded approximation property
cites work: Zur Theorie der Systeme linearer Gleichungen
cites work: Q4071668
cites work: Fréchet Spaces without continuous Norms and without Bases
cites work: Fréchet algebras, formal power series, and automatic continuity
cites work: Fréchet algebras, formal power series, and analytic structure
cites work: Ideals in Rings of Analytic Functions with Smooth Boundary Values
cites work: Q3437707
description: scientific article; zbMATH DE number 5965421