The S-Property and Best Approximation

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Sawsan Azmi Sabri Al-Dwaik
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The problem of best approximation is the problem of finding , for a given point x and a given set G in a normed space (X,||.||) , a point go in G which should be nearest to x among all points of the set G . However , in our study , we shall mainly take as X not an arbitrary normed space but Orlicz space , we shall denote by P(x,G) , the set of all elements of best approximants of x in G. i.e P(x,G)= { gₒ є G llx- g= inf{||x-g||: g є G } The problem of best approximation began , in 1853 , with P. L. Chebyshev who considered the problem in the space of all real valued continuous function defined on [a,b] , a closed real interval in R . My theses consist of four chapters. Each chapter is divided into sections. A number like 2.1.3 indicates item (definition, theorem, corollary or lemma) number 3 in section 1 of chapter 2. Each chapter begins with a clear statement of the pertinent definitions and theorems together with illustrative and descriptive material. At the end of this thesis we present a collection of references. In chapter (1) we introduce the basic results and definitions which shall be needed in the following chapters. The topics include projection, normed space, compactness , Hilbert space and measure theory . This chapter is absolutely fundamental. The results have been stated without proofs, for theory may be looked up in any standard text book in Functional Analysis. A reader who is familiar with these topics may skip this chapter and refer to it only when necessary. Chapter (2) will be devoted to give an introduction to fundamental ideas of Best Approximation in Normed Space. We will start by introducing the definition of best approximants of x є X in a closed subspace G of X . We denote the set of all best approximation of x in G by P(x,G) . In section (2) we study the properties of P(x,G) . In section (3) we define proximinal set and Chebyshev subspace , and we mention some conditions that can assure that G is proximinal in X . Finally , we define Lᵖ- summand and give a simpler proof for the fact that “every a closed subspace of a Hilbert space is proximinal ". Chapter (3) has two purposes .First, we review the properties of Orlicz spaces. Second, we introduce some ofthe basic theory ofproximinality in Orlicz space . This material was designed to meet the needs of chapter (4). W. Deeb and R. Khalil proved the following results. (1) If G is 1-complemented in X, then G is proximinal in X. [1, p.529] . (2) If Lᶲ(μ,G) is proximinal in Lᶲ(μ,X) ,then G is proximinal in X. [3 , p.8] , [2 ,p.297] , [4 , p.37] (3) If L¹(μ,G) is proximinal in L¹(μ,X), then L∞(μ,G) is proximinal in L∞(μ,X) . [1 , p.528] Some questions about proximinality in Lᶲ (μ,G now suggest themselves. (1) Let X be a Banach space and let G be proximinal in X. Under what conditions can it be asserted that G is l-complemented in X? (2)If G is proximinal in X, Under what conditions can it be concluded that Lᶲ (μ,G) is proximinal in Lᶲ(μ,X)? In particular, is the proximinatily of G in X a sufficient condition? (3)If L∞(μ,G) is proximinal in L∞(μ,X). Under what condition can be asserted that L¹(μ,G) is proximinal in L¹(μ,X). These questions are addressed in the section (1) of chapter (4). The answer depends on the S-property. Some interesting results have been achieved. Among of which it is shown that if G has the S-property then L∞ (μ,G) has the S-property . lt is also proved that if G has the S-property then Lᶲ (μ P⁻¹ɢ (0))= P⁻¹Lᶲ (μ,G) (0) I ask our God to be our assistant to continue our efforts so as to achieve the hopes and desires of all scholars in mathematics.
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