Riesz’s Contribution To Functional Analysis

Functional analysis is a much abstract discipline. Throughout its founding period and theperiod in which it developed into an independent discipline, the Hungarian mathematicianRiesz had made outstanding contributions. Not only was Riesz effected by abstract analysiswork of French scholars, but also by concret integral equation study led by Hilbert. Rieszmade significant results in the work on special function spaces and abstract operator. Heobtained the fundamental theory of functional analysis. Therefore, Riesz laid the importantfoundation for the axiomatization of functional analysis with Banach and othermathematicians.The important achivements Riesz got during different periods runs through the wholethesis on the basis of the study of documents. Connected with the developing history offunctional analysis, this dissertation explores the method how Riesz solved the specificproblem, and analysis the different work between Riesz and other mathematicians infunctional analysis. The main results were as follows:1The dissertation introduces Riesz’s life, and shows his elegant teaching attitude andrigorous ideas for pursuing studies.2The dissertation summarizes the academic background in founding period.⑴Thediscussion on abstract analysis of Hadamard and Frechét as well as the study about integralequations of Hilbert both gave birth to the concepts of function space and operator.⑵Thegeneration of Lebesgue integral unprecedentedly extented the study of function space. Basedon these academic background，Riesz had well prepared to unity the abstract theory andconcrete questions.3The dissertation explores the contribution Riesz has done during the founding periodof functional analysis. Riesz’s work is compared with Frechét’s and Fischer’s through fourfundamental theories. They are Riesz-Fischer theorem, Riesz representation theorem, dualitytheory and the theory of compact operator.4The dissertation expounds the significant contribution of Riesz in functional analysisduring the period when the discipline became independent.⑴He first presented the Riesz space;⑵He studied dependently the self-adjoint operator;⑶He extended the ergodic theory;⑷He accomplished the famous book Functional analysis. All of these consolidate hishistorical position in the discipline.