Probabilistic characterization of certain Banach spaces
type: Item
instance of: scholarly article
title: Probabilistic characterization of certain Banach spaces
zbMATH Open document ID: 0595.46020
author: Jun Kawabe
published in: Kodai Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The author studies the relation between mean square convergence for random elements with values in a Banach space and the structure of the Banach space. One of the results is as follows. For a real separable Banach space X, the following assertions are equivalent: (a) X is isomorphic to a Hilbert space. (b) Let (\(\Omega\),\({\mathcal A},P)\) be a probability measure space and \((\xi_ n)_{n\geq 1}\) be a sequence of mean-zero random elements in \(L^ 2(\Omega;X)\). Then \(\xi_ n\) converges in mean square if (and only if) the following two conditions are satisfied; (i) for each \(f\in X^*(=the\) dual of X), \(<\xi_ n,f>\) converges in mean square, (ii) the set \(\{R_{\xi_ n}\}\) of covariance operators of \(\xi_ n\) is relatively compact with respect to the nuclear norm. The author also shows that assertion (b) holds only for Gaussian random elements if and only if X is of type 2, provided X has the approximation property.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 3957968
zbMATH Keywords: convergence in probability
zbMATH Keywords: uniform tightness covariance operator
zbMATH Keywords: nuclear operator
zbMATH Keywords: cotype
zbMATH Keywords: mean square convergence for random elements with values in a
zbMATH Keywords: Banach space
zbMATH Keywords: nuclear norm.
zbMATH Keywords: type 2
zbMATH Keywords: approximation property
zbMATH Keywords: mean square convergence for random elements with values in a Banach space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037151
OpenAlex ID: W2044805966
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DOI: 10.2996/KMJ/1138037151
description: scientific article; zbMATH DE number 3957968