The Resonance Theorem And Operator Space In F～* Spaces

In this dissertation, we mainly discuss the resonance theorems for families of linear and nonlinear operators, and the theory of bounded linear operator space in F~* spaces. The main includes four sections as follows :In chapter 1, we dreview the definition of F~* space, and give a characterization of it by the " standard generating sequence of quasi-norms" ρ of F~* space. And then, we introduce the concepts of ρ-bounded sets, ρ-semi-bounded sets and ρ-unbounded sets in a F~* space, and study some of thier properties.In chapter 2, by using the characterization of F~* space, we es-taiblish the resonance theorems for family of lower semi-continuous linear operators and family of a kind of quasi-homogeneous operators of pointwise ρ-bounded, pointwise ρ-semi-bounded and point-wise ρ-non-unbounded in such spaces.In chapter 3, as the direct application of the resonance theorems what discussed in chapter 2, we estaiblish corresponding the resonance theorems in Menger probabilistic normed spaces and fuzzy normed spaces.In chapter 4, we give the definition of quasi-norm of linear operator with respect to a given bounded set in F* space and study some of its properties. On the basis of this, we estaiblish a theory of the space of bounded linear operators on F* spaces.