Formation And Development Of The Theory Of Dual Space

Dual space theory is one of the core contents in functional analysis, not only does it have close contacts with many branches of mathematics, but also has wide applications. By way of historical analysis and literature research, guided by "Why mathemaiics was done", with "Solving integral equations and linear equations" as the main line, based on extensive investigation of original sources and research papers on this subject, the dissertation more intensively studies the formation and development of dual space and weak* compactness theorem, excavates the deep thought contained in the works of related mathematicians, explores the ideological heritage among them. Main results are the following:1. By analyzing Hilbert's three important documents in the integral equation, traces the root of generating infinite quadratic theory and its influence on the work of integral equations, reverts his approach of solving linear equations with finite variables, reconstructs his algebralization process transforming the integral equation into an infinite system of linear equations by inner product, reveals Hilbert's dual thought underlying his works and his fundamental contribution to the forming of dual space theory.2. Analyzes the creation of continuous linear functional concept and Frechet's work on functional representation, based on this, thoroughly researches Riesz's work on integral equations and linear equations in different spaces, penetrates his ideological origins of solving integral equations and linear equations, digs the link between the solution of integral equations and linear equations and representation of continuous linear functional on the corresponding space, reconstructs the formation of the specific dual spaces, reveals the deepest unity and abstract thought in his work, exposes the influence of these ideas on the nurture of dual space abstract theory. Analyzes Steinhaus's work on dual space and reveals the difference with the previous work.3. Deeply analyzes the related research work of important mathematicians during the process that the abstract theory of dual space completes. Through the investigation of Helly's work on the sequence of normed linear space with the struction of convex theory, Hahn's work on general normed linear space under the guidance of functional equations, Banach's work on Banach space under the influence of the operator theory, restores the source of the specific problem concealing in their abstract theory, restructures the forming process of their dual space theory, establishes whole ideological system on the formation of dual space theory by the main line of the functional extension theorem.4. Deeply looks into the reform and development that some mathematicians have done during the construction of weak* compactness theorem. Around the two core concepts of "compactness" and "weak convergence", discusses the pre-history of weak* compactness theorem. From Hilbert and Riesz's work on integral equations reveals the necessity of introducing the "weak convergence" concept, and comprehends the key role during the transition from the finite to the infinite. Discloses the cause that Banach introduced the theory of weak convergence in the dual space from the perspective of duality, and raises to weak* compactness theorem from the depth of the weak topology.5. Systematically investigates the development condition of dual space theory and the far-reaching impact that measure theory and set topology theory have to it after Banach, especially after the formation of the subject. At the same time, discusses the influence of the ideas and methods of dual space theory to the development of mathematics in twentieth century.