Regular Article
Extension Theorems of Continuous Random Linear Operators on Random Domains

https://doi.org/10.1006/jmaa.1995.1221Get rights and content
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Abstract

The central purpose of this paper is to prove the following theorem: let (Ω, σ, u) be a complete probability space, (B, ∥·∥) a normed linear space over the scalar field K, E: Ω → 2B a separable random domain with linear subspace values, and ƒ: GrEK a continuous random linear operator, where GrE = {(ω, x) ∈ Ω × B|xE(ω)} denotes the graph of E. Then there exists a continuous random linear operator ƒ̃: Ω × BK such that ƒ̃(ω, x) = ƒ(ω, x) ∀ ω ∈ Ω, xE(ω), and sup{|ƒ̃(ω, x)| |xB, ∥x∥ ≤ 1} = sup{|ƒ(ω, x)| |xE(ω), ∥x∥ ≤ 1}, for each ω in Ω. 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.

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