A study on order bounded linear mappings from a linear lattice E into A linear lattice F

Abstract

필자는 일반 선형위상공간의 이론을 L(E,F)에 적용하여 다음 결과를 얻었다. 정리1. L₁, L₂를 L-공간이라고 하자 L_(0)가 L₁에서 Closed ideal이면 Natural restriction map θ : L(L₁,L₂) -> L(L_(0), L₂)는 L(L_(0), L₂)상에의 사상이고 θ(L(L₁,L₂)^(+)) = L(L_(0), L₂)^(+)이다 정리2. L^(t)(L_(0), L₂) = {p^(t)｜p∈(L₁, L₂)}는 Linear lattice 정리3. [f, g]가 L^(b)_(2)에서 Order interval p^(t)[f, g]는 W(M₁, L₁) Compact 이다.;When E and F are linear lattices, the set of order bounded linear mappings from E into F, which we denote by L(E,F), has many beautiful properties. One of recent results concerning about L(E,F) is the one found by F. Riesz which states that; Theorem L(E,F) is a Dedekind complete linear lattice if F is a Dedekind complete linear lattice. Most important Dedekind complete linear lattice in L-space intriduced by Kakutani [1]. Motivated b, the above fact we shall study the case when E and F are L-space L₁ and L₂. The first chapter includes the fundamental conceptions and theorems which can be found in Vulikh [3]. In chapter 2, the extension problems similar to the Hahn-Banach extension theorem is settled. In chapter 3, two theorems on the dual transformation which recently developped is stated and proved. The auther tried to combine the general topological character of L-space with the order character in L(L₁,L₂) according to the development in Kelley [2] and Vulikh [3].