Nonstandard Hull And Its Application

In this paper, the structure and its basic properties of the nonstandard hull of metric space are given by using the basic theory of nonstandard analysis, and then, the nonstandard hull and its basic properties of several other common spaces are given, too. Using these has given theoretical knowledge, from the point of nonstandard analysis, the application in Banach algebra and Hilbert algebra has been researched and developed. The theory of reversible element and generalized reversible of Banach algebra which constructed by using nonstandard hull, the theory of spectrum and generalized spectrum of the element of Banach algebra are mainly explained. Finally, the definations of the internal product algebra and Hilbert algebra are given, and its properties are discussed preliminarily.In the first and second chapter, the nonstandard analysis theory is briefly introduced. Using the axioms in nonstandard analysis, the nonstandard analysis is axiomatic approached. On this basis, some properties of nonstandard model and nonstandard saturation model are discussed. And several equivalent conditions of saturation are given.In the third chapter, the structure and its basic properties of the nonstandard hull ofmetric space are introduced firstly, and from its basic property-completeness, we cansee the advantages of a structured nonstandard hull of space. Several other spaces followed by a discussion of the structure and the basic properties of the nonstandard hull. Finally, the use of nonstandard hull can be embedded into the hyperfinite space characteristics of a finite dimensional space, Hilbert space is studied by the spectral decomposition conjugate operator.In the fourth chapter, how we construct a Banach algebra by using nonstandard hull method is given firstly. Then, the properties of its reversible element are discussed. On this basis, the concept of reversible is extended to generalized reversible, and some important relationships between reversible and generalized reversible are discussed. After that, the spectrum theory of element in Banach algebra constructed by using nonstandard hull method is discussed mainly. Finally, on the basis of generalized reversible, given the generalized spectrum concepts of elements in Banach algebra, and discussed its basic property.In the final chapter, we give the concepts of internal product algebra and Hilbert algebra, and discuss its basic properties. On this basis, we use nonstandard methods to construct Hilbert algebra and study its basic properties preliminarily.