Some Properties Of Approximately Compact Chebyshev Sets

As is known to all, approximation theory is one of the most important branchsof modern mathematics. It is very important in the study of mathematical theoriesand practical applications. In the late eighteenth, Russian mathematician Chebyshevproposed the conception of optimal approximation, and established thecharacterizing theorem which used to determine weather the polynomial is the bestapproximation or not. Since then, the research of the nonlinear approximate theorieshas developed rapidly. Especially in recent years, nonlinear approximate theory isnot only limited in the applied mathematics, for instance, mathematics ofcomputation, probabilistic satistics, equation of mathematical physics and so on, butalso expanded to the basic mathematics, for example functional analysis, topologyalgebra and so on. Therefore, it has important theoretical significance and practicalvalue to investigate the nonlinear approximate theories in Banach spaces.In this paper, the solarity of the approximately compact Chebyshev sets inlocally uniformly convex spaces, the approximate compactness in strongly nearlyconcave spaces and their internal relations are discussed. The content of this paper isdivided into three parts.In the first part, this dissertation introduces the background and simple historyof approximation theory and their close relations between approximate compactnessand some properties in Banach spaces, which provide the preparation and basis forthe study of this paper.In the second part, the concept of the sun set is strengthened. The solarity of theapproximately compact Chebyshev sets in locally uniformly convex spaces isstudied. That the approximately compact Chebyshev set is equivalent to the sun setin locally uniformly convex spaces is proved.In the third part, the definition of strongly nearly concave space is proposed.The relationship between the approximate compactness and the strongly nearconcavity is studied. That X is approximately-compact iff that X is stronglynearly concave in Banach spaces is proved.