Multi-Hilbertian Spaces and their Duals
NORTH CAROLINA UNIV AT CHAPEL HILL CENTER FOR STOCHASTIC PROCESSES
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In the last several years there has been a remarkable amount of work in probability on infinitely dimensional spaces, in particular on nuclear spaces. Although the most of work has been done in nuclear spaces, some of the basic theorems for instance, Itos regularization theorem, are given in a much more general context of multi-Hilbertian spaces. In this paper we study topological properties of multi-Hilbertian spaces and their duals, hoping that this will serve as an introduction to a study of probability problems on these spaces. We tried to clearly distinguish properties that are consequences of nuclearity from those that hold on non-nuclear spaces. In section 5, we propose a non-standard completion theorem, removing the condition of compatibility of norms, a condition that seems to be overlooked in most probability papers in this area. Also, we give a detailed account on open, bounded and compact sets. Elaborated proofs are left for appendices, and, as a result, appendices occupy a considerable space. This is mostly due to results related to seminorms that we wanted to make rigorous. The results that are given in this paper are selected with a purpose to serve as a basis for probability investigation the topology alone was not the aim. As a continuation of this work, we plan to investigate sigma- algebras and probability measures, and weak convergence of measures in a general context of multi-Hilbertian spaces. Also, we plan to investigate examples of interest in applications.
- Statistics and Probability