Primary products of Banach spaces
type: Item
instance of: scholarly article
title: Primary products of Banach spaces
zbMATH Open document ID: 0859.46003
DOI: 10.1007/BF01781558
published in: Archiv der Mathematik
publication date: 1996-11-22T00:00:00Z
review text: A Fréchet space \(E\) is called primary if whenever \(E= F\oplus G\), then either \(F\) or \(G\) is isomorphic to \(E\). It is known that \(\omega\), \((c_0)^{\mathbb{N}}\) and \((\ell^p)^{\mathbb{N}}\), with \(1\leq p\leq \infty\), are primary Fréchet spaces without continuous norm. However, it is unknown whether there are other countable products of Banach spaces which are primary. It is the purpose of this paper to prove that the spaces \(X^{\mathbb{N}}\), with \(X\) a separable rearrangement invariant function space on \([0,1]\) with Boyd indices \(1<p_X\) and \(q_X<\infty\), are primary. The considered class of spaces contains many natural spaces like \(L^{\text{loc}}_p(\mathbb{R})\), with \(1<p<\infty\), and the analogous spaces \(L^{\text{loc}}_M(\mathbb{R})\) of Orlicz-type.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 938552
zbMATH Keywords: spaces of Orlicz-type
zbMATH Keywords: primary Fréchet spaces without continuous norm
zbMATH Keywords: separable rearrangement invariant function space
zbMATH Keywords: Boyd indices
author: Angela A. Albanese
MaRDI profile type: Publication
cites work: On complemented subspaces of the spaces \((l^p)^N\cap l^q(l^q)\)
cites work: Q4112460
cites work: Complemented Subspaces of Products of Banach Spaces
cites work: Complemented subspaces of sums and products of Banach spaces
cites work: On the Space lp+ = ∩lqq>p
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description: scientific article; zbMATH DE number 938552