Best Approximation in General Normed Spaces

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Date
2001
Authors
Mu'tas Hasan Mahmoud Al-Sayed
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Let X=(X,||.||) be a normed space and suppose that any given x in X is to be approximated by an element Y in Y, where Y is a fixed subspace of X. We let d denoted the distance from x to Y. By definition, d = d(x, Y) = infyєY ||x – y|| .Clearly, d depends on both x and Y, which we keep, fixed, so that the simple notation d is in order. If there exists a YoєY such that ||x - Yo|| =d. then Yo is called a best approximation of Y to x or a best approximant of x in Y. We see that a best approximation Yo is an element of minimum distance from the given x. Such a YoєY mayor may not exist; this raises the problem of existence. The problem of uniqueness is of practical interest, too, since for a given x and Y there may be more than one best approximation. My thesis consists of three chapters. In chapter one we summarize some of the essential and basic concepts which shall be needed in the following chapters, this chapter consists of two sections; in the first one we present metric, normed, Banach spaces, and the last one we present inner product, and Hilbert spaces. This chapter is absolutely fundamental. In chapter two, we define best approximations in section one. In section two we study some properties of the set of all best approximations P(x,Y). In section three we study some properties of the proximinaI set and show that compact subspace and finite-dimensional subspace are proximal. In section four we consider the problem of uniqueness of best approximation. In section five we review the properties of OrIicz spaces in which we introduce some of the basic theory of proximinality In chapter three, which is the main body of our thesis, we, in section one, study the main characterizations and properties of best approximations and some consequences of the characterization in arbitrary normed linear spaces. In sections two and three we gives some application in several spaces like L ¹(T,v), C(K) and CR(K).
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