On the definition of a topology in Hilbert spaces with applications to the white noise theory
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.
scientific article; zbMATH DE number 3936067