Extension Theorems of Continuous Random Linear Operators on Random Domains

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 → 2 B a separable random domain with linear subspace values, and ƒ: Gr E → K a continuous random linear operator, where Gr E = {(ω, x) ∈ Ω × B| x ∈ E(ω)} denotes the graph of E. Then there exists a continuous random linear operator ƒ̃: Ω × B → K such that ƒ̃(ω, x) = ƒ(ω, x) ∀ ω ∈ Ω, x ∈ E(ω), and sup{|ƒ̃(ω, x)| | x ∈ B, ∥ x∥ ≤ 1} = sup{|ƒ(ω, x)| | x ∈ E(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.