Weak-polynomial convergence on spaces \(\ell_{p}\) and \(L_{p}\)
type: Item
instance of: scholarly article
title: Weak-polynomial convergence on spaces \(\ell_{p}\) and \(L_{p}\)
zbMATH Open document ID: 1076.46034
author: Silvia Lassalle
author: Jose G. Llavona
published in: Positivity
publication date: 2005-03-15T00:00:00Z
review text: The authors consider various topologies on the real Banach spaces \(X=\ell_p\) or \(X=L_p[0,1]\), \(1 \leq p < \infty\). The following two topologies are in the center of investigations. The Banach space of all scalar-valued continuous \(n\)-homogeneous polynomials on \(X\) is denoted by \(P(^nX)\) and \(P_o(^nX)\) denotes the closed subspace of orthogonally additive polynomials, where the orthogonality results from the lattice structure of \(X\). For \(P \in P(^nX)\), define \(\widetilde{d}_P:X \times X \rightarrow {\mathbb R}\) by \[ \widetilde{d}_P(x,y) = \inf \{ | P(x-z_1)| ^{1/n} + | P(z_1-z_2)| ^{1/n} + \cdots + | P(z_{k-1}-z_k)| ^{1/n} + | P(z_k-y)| ^{1/n} \}, \] where the infimum is taken over all \(z_1,\dots, z_k \in X, k \geq 1 \), and put \(d_P(x)=\widetilde{d}_P(x,0)\). Then the topology \( \tau \) on \(X\) is defined by the family of seminorms \((d_P)_P\) when \(P\) varies over \(P_o(^nX)\) and \(n \geq 1\). A net \((x_\alpha)\) in \(X\) converges to \(x \in X\) with respect to the topology \(wp_0\) if \(P(x-x_\alpha) \rightarrow 0\) for all \(P \in P_o(^nX)\). The main results are the following. Theorem: Let \(X=\ell_p\), \(1\leq p < \infty\), and let \(k\) be the smallest integer with \(p \leq 2k\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) weakly and \(\| x_\alpha - x \| _{2k}\rightarrow 0\). Corollary: Let \(X=\ell_p\), \(p=1\) or \(p \in (2k-1,2k]\) for some integer \(k\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) w.r.t. \(wp_o\). Theorem: Let \(X=L_p[0,1]\), \(1\leq p < \infty\) and let \(k\) be the largest integer with \(2k \leq p\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) weakly and \(\| x_\alpha - x \…
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 2145886
zbMATH Keywords: polynomials on Banach spaces
zbMATH Keywords: weak-polynomial topologies
zbMATH Keywords: zeros of polynomials on \(\ell_p\) and \(L_p\)
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
full work available at URL: s11117-004-5008-x
OpenAlex ID: W2026712635
cites work: Q4218441
cites work: Q5748432
cites work: Zeros of real polynomials
cites work: Zeros of polynomials on Banach spaces: The real story
cites work: Polynomial compactness in Banach spaces
cites work: Q5519417
cites work: A Uniform Algebra of Analytic Functions on a Banach Space
cites work: Q4946767
cites work: A Theorem on Polynomial-Star Approximation
cites work: SEPARATING POLYNOMIALS ON BANACH SPACES
cites work: Polynomial continuity on $\ell _1$
cites work: Polynomially continuous operators
cites work: Q4136031
cites work: Q3784563
cites work: Q3996430
cites work: Q3974993
DOI: 10.1007/S11117-004-5008-X
description: scientific article; zbMATH DE number 2145886