I.M.Gelfand And The Establishment Of Normed Rings Theory

Based on the concept of mathematical structure created in the 20 th century,around the center of Normed rings theory,this thesis summarizes the life and related works of some major mathematicians such as Gelfand and systematically sorts out the historical process of related concepts and theory.Then the article summarizes the development of the theory after its establishment and its influence on mathematics especially on abstract harmonic analysis.It is expected to contribute to the history of mathematics in the 20 th century and provide reference for related research work.The thesis is composed of four parts.The first part introduces Gelfand’s life and scientific work,which is the key content of the thesis.His life history,growing environment,mathematical writings,discussion classes,reports during three mathematicians’ congresses,awards and birthday greetings are included.In particular,the author studied some new source materials and historical materials,analyzed the social background of the former Soviet Union and the style characteristics of discussion classes from the perspective of mathematical sociology,revealed Gelfand’s contribution to mathematical theories such as index theorem,and expounded Gelfand’s philosophy of "mathematical unity",which provided rich reference for a comprehensive understanding of Gelfand.The second part introduces the development of Fourier analysis,set theory,Lebesgue measure and integral,general topology,abstract algebraic structure,functional analysis and other branches related to normed ring theory at the turn of the 19 th and 20 th centuries.Especially,it explains the background of structural mathematics at the turn of the century,which paves the way for an overall understanding of the general situation of mathematical development before the birth of normed ring theory.The third part is the core content of the thesis,which fully clarifies the development of the normed ring theory and answers the historical questions of the origin and development of the theory.The author summarized the establishment process of the normed ring theory in detail,including the research foundation of predecessors,the establishment process of the theory and its further development.After the establishment of the abstract theory of "Barnach Space",it has become the starting point of functional analysis and more general space research.Because Barnach space is a complete normed linear space,it has a topological structure defined by norm and an algebraic structure of linear space.As the source is function transformation,mathematicians started to study around the analytical structure rather than the algebraic structure,and the reasoning methods adopted were all analytical.Later mathematicians gradually noticed the potential value of multiplication inequality and ring structure.At the end of 1930 s,Gelfand and his school created the word "normed ring",and put forward the basic concepts such as maximum ideal and a series of theorems,thus creating the theory of "normed ring" which shocked the mathematics world.This theory not only explains a large number of famous old problems in analytical fields using Algebraic technique,such as Tauber’s type theorem and Wiener’s theorem,but also creates a series of new fields,which is the perfect unity of analytical structure and algebraic structure.The origin of the concept of "Normed Ring" is also the process of mathematicians’ understanding of mathematical objects from shallow to deep,and finally evolved into the name of "Barnach Algebra" under the transformation of American mathematicians.The fourth part introduces the influence of normed ring theory,including Gelfand’s application of normed ring theory to create a series of new fields such as general spectrum theory and C*-algebra.In particular,Gelfand established the abstract harmonic analysis theory by using the normed ring theory.The author extended to the development of harmonic analysis theory.After going through the development of harmonic analysis from the Group perspective,which confirmed the great role of group structure in mathematical unity.Finally,a brief introduction of non-commutative harmonic analysis and classical harmonic analysis is stated.