Basic Properties Of Dissecting Mappings

Closed graph theorem, open mapping theorem and equicontinuity theorem are the three basic principles of the theory of functional analysis. Working hard on the subject for more than sixty years to generalize and improve the three basic principles, we have turned it from the early works focusing on generalizations of spaces to relaxing the linearity of mappings. Recently, some scholars extend the linear operators to much larger families of mappings which include many nonlinear mappings, and discuss some basic problems of the mappings, which leads to substantial improvements of the three basic principles.Especially, Professor Li Ronglu established the new three basic principles for three very large families of mappings respectively. Each of the three families of mappings includes all linear mappings handled in the classical principles. Meanwhile, the family of dissecting mappings given by Professor Li is used to develop the theory of distributions. He also estiblishes a new pan-linear distribution theory. The theoretical value and field of applications of functional analusis are promoted and expanded to a new level. Based on the existing properties of dissecting mappings, this paper does some in-depth study of the dissecting mappings and gives several new properties.Firstly, the paper presents the continuous properties of dissecting mappings, points out that the dissecting mappings are continuous when and only when they are continuous in zero.Secondly, the paper improves an original sufficient and necessary condition for the dissecting mappings f :R→R, makes it link with the Lipschitz condition. We further study the structure of dissecting mappings, and points out the stong dissecting mappings are Lipschitz.Again, we define the norm on the dissecting mappings, and give the properties of the norm.Finally, this paper also studies the properties of the conjugate operator of dissecting mapping, points out the conjugate operator is linear.