Improvements Of Fundamental Propositions Of Functional Analysis

Closed graph theorem, open mapping theorem and equicontinuity theorem (imply-ing uniform boundedness principle, i.e., resonance theorem) are the three basic principlesof the theory of functional analysis. Functional analysis greatly depends on Baire cate-gory theorem and Hahn-Banach extension theorem. But the former is a proposition of thetheory of general topology and the latter, as W. Orlicz and a scholar of our country XiaDaoxing explicitly say, belongs to pure algebra.There is a common property among classical three basic principles of functionalanalysis, that is, treated mappings are only linear operators which are too ideal. So thethree basic principles are not applicable to nonlinear mappings. This greatly makes thetheoretical value and applications of three basic principles limited. So there have beenmore than sixty years for our hard work to generalize and improve the three basic prin-ciples. However, the three basic principles are still as before with extensions and im-provements of more than sixty years. In this dissertation, we substantially give all-aroundimprovements of the three basic principles of functional analysis. In fact, we establish thenew three basic principles for three very large families of mappings respectively. Each ofthe three families of mappings includes all linear mappings handled in the classical prin-ciples. Especially, the number of nonlinear mappings in each of the three families are notfewer than that of linear mappings. Henceforth, the three basic principles of functionalanalysis are promoted to the basic principles of pan-linear analysis with general mean-ing which make linear analysis to be a special case. The theoretical value and field ofapplications of three basic principles are promoted and expanded to a new level. The lin-ear dual theory, based on the new three basic principles, has been extended to pan-lineardual theory. Especially, the theory of pan-linear distributions has given some very newresults, and new open mapping theorem also yields some results which have very strongtheoretical value and practical significance.Our guided thought is to try to reflect real life. For example, each of linear operatorsmakes absolutely precise dissection: f(x + tz) = f(x) + tf(z), (?)x, z∈X, t∈C,where absoluteness means"(?) x, z∈X, (?)t∈C"and precision means that the coeffi- cients 1 and t accurately appear in the dissecting result. But dissecting phenomenons inreal life are not often so absolutely precise. To re?ect this unideal real life, we definitedissecting mappings: for each x∈X, f(x + tu) = rf(x) + sf(u), where u belongsto a neighborhood of 0, t satisfies |t|≤1, and |r - 1| and |s - t| are simply controlledby |t|. Hence, the family of dissecting mappings is a very natural large extension of thefamily of linear operators. The new equicontinuity theorem is just established for thefamily of dissecting mappings. And each of linear operators has absolutely precise addi-tivity: f(x) + f(z) = f(x + z). This is very ideal too. But f(x) + f(z) = f(u) where is fairly common. Following this fact, we obtain the new openmapping theorem. The new closed graph theorem is obtained as the same idea.New three basic principles not only greatly promote the theoretical level, but alsoexpand the field of applications of three basic principles. In the last chapter, we give someapplications of new basic principles in dual theory, distributions and the theory of Banachspaces.