Extension Theorems of Continuous Random Linear Operators on Random Domains
Extension Theorems of Continuous Random Linear Operators on Random Domains
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DOI：
10.1006/jmaa.1995.1221
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发表时间：
1995-07
影响因子：
1.3
通讯作者：
T. Guo
中科院分区：
文献类型：
--
作者：
T. Guo
本文的主要目的是证明如下定理：设（Ω，σ，u）是完备概率空间，（B，ω·E）是标量场K上的赋范线性空间，E：Ω → 2 B是具有线性子空间值的可分随机域，ω：Gr E → K是连续随机线性算子，其中Gr E = {（ω，x）∈ Ω × B| x ∈ E（ω）}表示E的图。则存在连续随机线性算子<$：Ω × B → K使得<$（ω，x）=<$（ω，x）<$ω ∈ Ω，x ∈ E（ω），且sup{|n（ω，x）||x ∈ B，≤ 1} = sup{|n（ω，x）||x ∈ E（ω），<$x <$≤ 1}.对于E不可分的情形，给出了一个类似于上述定理的结果，推广和改进了Hahn-Banach定理随机推广的许多结果.
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 ƒ: Ω × B → K such that ƒ(ω, x ) = ƒ(ω, x ) ∀ ω ∈ Ω, x ∈ E (ω), and sup{|ƒ(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.