Топологические и категорные свойства пространства нелинейных ס -гладких функционалов

Аннотация

Subject of the inquiry: <t-smooth weakly additive, order- preserving, normed functionals and their space and functor. Aim of the inquiry: to investagc topological and categorical properties of the spaces of cr -smooth order- preserving functionals. Methods of inquiry: methods of general topology, covariant functors theory and functional analysis have been used. The results achieved and their noveltyj. results obtained in the thesis arc new and consist of the following: It is proven that the construction On is a covariant functor, acting in the category of Tychonoff spaces and their continuous maps. A criteria is given for cr -smoothness of weakly additive, order- preserving functionals. It is shown that if f : X —> Y is a z -embedding between Tychonoff spaces, then the map Oa (f) : On (X) —> On (%) is an embedding. It is shown that the functor Oa forms a monada. It is proven that the space O^iX) is Hewitt complete for every Tychonoff space X. A description is given for the Hewitt completions of Tychonoff spaces in terms of the spaces of a -smooth order-preserving functionals. We give a condition for coincidence of the spaces of r-smooth weakly additive and cr-smooth weakly additive functionals. It is shown that weight of the space O^X) of a -smooth weakly additive functionals is between the weight of Hewitt completions and z -weight of the given Tychonoff space X. The practical value: the results of the thesis have a theoretical character. Degree of application and economic effectivity: Results and methods introduced in the work can be used in special courses on general topology, functional analysis and theory of covariant functors. Sphere of usage: the results of the thesis may be used in general topology, covariant functors theory and functional analysis.