Extension Theorems of Continuous Random Linear Operators on Random Domains

✍ Scribed by T.X. Guo

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 570 KB
• Volume: 193
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1995.1221

✦ Synopsis

The central purpose of this paper is to prove the following theorem: let ((\Omega, \sigma), (u)) be a complete probability space, ((B,|\cdot|)) a normed linear space over the scalar field (K, E: \Omega \rightarrow 2^{B}) a separable random domain with linear subspace values, and (f: \operatorname{Gr} E \rightarrow K) a continuous random linear operator, where (\operatorname{Gr} E={(\omega, x) \in \Omega \times) (B \mid x \in E(\omega)}) denotes the graph of (E). Then there exists a continuous random linear operator (\tilde{f}: \Omega \times B \rightarrow K) such that (\tilde{f}(\omega, x)=f(\omega, x) \forall \omega \in \Omega, x \in E(\omega)), and (\sup {|f(\omega, x)| \mid x \in B,|x| \leq 1}=\sup {|f(\omega, x)| \mid x \in E(\omega),|x| \leq 1}), for each (\omega) in (\Omega). For the case where (E) is not separable, a result similar to the above-stated theorem is also given, which generalizes and improves many previous results on random generalizations of the Hahn-Banach Theorem. O 1995 Academic Press, Inc.