One property of families of imbedded Banach spaces
type: Item
instance of: scholarly article
title: One property of families of imbedded Banach spaces
zbMATH Open document ID: 0897.46004
DOI: 10.1007/BF01102495
published in: Journal of Soviet Mathematics
publication date: 1992-09-26T00:00:00Z
review text: Let \(\{E_\alpha: \alpha\in[0, 1]\}\) be a family of imbedded Banach spaces, i.e., for each pair of numbers \(\alpha<\beta\) \((\alpha,\beta\in [0,1])\) there is a one-to-one bounded linear mapping (in what follows, imbedding) \(T_{\beta\alpha}: E_\beta\to E_\alpha\), and the following conditions are satisfied: 1) \(\text{cl }T_{\beta\alpha} E_\beta= E_\alpha\), i.e., \(T_{\beta\alpha}\) is a dense imbedding; 2) \(T_{\alpha\gamma} T_{\beta\alpha}= T_{\beta\gamma}\), \(0\leq \gamma< \alpha<\beta\leq 1\); 3) \(T_{\beta\alpha}^{-1}\) is an unbounded mapping. Under these conditions, the author proves two theorems including Theorem 2. Let \(\{E_\alpha: \alpha\in [0,1]\}\) be a family of imbedded separable Banach spaces. Then, for every \(\alpha\in [0,1]\), the space \(E_\alpha\) contains a subspace \(G_\alpha\) with basis \(\{g_i\}\) such that 1) \(G_\alpha\cap \bigcup_{\beta> \alpha} T_{\beta\alpha} E_\beta= 0\); 2) for all \(\gamma< \alpha\) the system \(\{T_{\alpha\gamma} g_i\}_{i= 1}^\infty\) is an \(M\)-basis of the space \(E_\gamma\); in particular, the image \(T_{\alpha\gamma} G_\alpha\) is dense in \(E_\gamma\).
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46A13
zbMATH DE Number: 56619
zbMATH Keywords: imbedded separable Banach spaces
zbMATH Keywords: \(M\)-basis
author: V. P. Fonf
MaRDI profile type: Publication
cites work: Q4722647
full work available at URL: https://doi.org/10.1007/bf01102495
OpenAlex ID: W2065047203
description: scientific article; zbMATH DE number 56619