On the definition of a topology in Hilbert spaces with applications to the white noise theory
type: Item
instance of: scholarly article
title: On the definition of a topology in Hilbert spaces with applications to the white noise theory
zbMATH Open document ID: 0584.60008
DOI: 10.1016/0016-0032(83)90090-X
published in: Journal of the Franklin Institute
publication date: 1983-01-01T00:00:00Z
review text: Let (\({\mathcal H},{\mathcal C},\mu_ G)\) be a cylinder-probability triple on the real separable Hilbert space \({\mathcal H}\) endowed with the finite additive measure \(\mu_ G\) having the characteristic functional \(\exp \{2^{-1}\| x\|^ 2\}\), \(x\in {\mathcal H}\). A measurable map \(F: {\mathcal H}\to {\mathcal B}\), \({\mathcal B}^ a \)Banach space is said to be uniformly S-continuous around the origin if for any \(\epsilon >0\) there exists a Hilbert-Schmidt (H.S.) operator \(L_{\epsilon}: {\mathcal H}\to {\mathcal H}\) such that \(\| L_{\epsilon}(x-x')\| <1\) implies \(\| F(x)-F(x')\| <\epsilon\) for all \(x,x'\in Q_ n=\{z\in {\mathcal H}: \| A_ nz\| <1\}\) where \(A_ n\), \(n=1,2,..\). is a sequence of H.S. operators such that \(\| A_ n\|_{H.S.}\to 0\) and \(\cup_{n}Q_ n={\mathcal H}.\) The authors prove that the uniform S-continuity around the origin is equivalent to the continuity w.r. to the topology \({\mathcal T}_ F\) generated by a sequence of seminorms \(\{\| L_ ix\| \}\), where \(L_ i\) is a sequence of H.S. operators. It follows from this result that the continuity in topology \({\mathcal T}_ F\) is sufficient for the map F to be a physical random variable. Some application to nonlinear differential equations driven by white noise is considered.
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 3936067
zbMATH Keywords: Hilbert-Schmidt operator
zbMATH Keywords: physical random variable
zbMATH Keywords: Hilbert space
zbMATH Keywords: finite additive measure
author: Alberto Gandolfi
author: A. Germani
MaRDI profile type: Publication
cites work: Q3051269
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cites work: Q4147364
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full work available at URL: 0016-0032(83)90090-x
OpenAlex ID: W2081431473
description: scientific article; zbMATH DE number 3936067