The Theory Of Pan-Linear Equicontinuity

Equicontinuity theorem is one of the three basic principles of functional analysis, and is also the essential part of this discipline. The Development of functional analysis has been promoted by equations of mathematical physics and quantum mechanics. It changes specific analytical problems into purer ones of algebra and topology. However, with the development of mathematics, traditional functional analysis cannot solve more general problems. For many nonlinear uncountable mappings,such as x , x~2,sinx and e x-1, equicontinuity theorem is not applicable. Fortunately, some mathematicians extend the family of linear mappings to much larger families, these families include some nonlinear mappings. And on the basis of this extension, they get comprehensive improvements of the three basic principles. With these improvements, some linear theories are extended to pan-linear ones. Especially, Professor Li Ronglu defines a new mapping named the dissecting mapping,and gives some properties of anatomical mapping, and for the family of such mapping on a special space, he establishes new equicontinuity theorem and uniform boundedness principle. Applications of functional analysis will be more extensive and closer to our lives.This paper makes some further improvements on the new equicontinuity theorem, gets a beautiful conclusion of this theorem on topological linear space, and gives some inferences and applications, which allow us to get many conclusions without very strong conditions and services life better. Then we discuss the equicontinuity of general mapping system in the last chapter of this paper, and get a related equicontinuity theorem and its application for the spaces of all mappings.