本文選題：積分方程　切入點：無窮線性方程組　出處：《西北大學》2016年博士論文　論文類型：學位論文

【摘要】：對偶空間理論是泛函分析的核心內(nèi)容之一,與眾多數(shù)學分支聯(lián)系緊密,亦有著廣泛應用。本文通過歷史分析和文獻考證的方法,以“為什么數(shù)學”為指導,以“積分方程和線性方程組的求解”為主線,在研讀相關原始文獻和研究文獻的基礎上,對對偶空間理論的歷史進行了較為深入細致的研究,并對其上重要定理——弱*緊定理的形成與發(fā)展脈絡進行了探討,挖掘了蘊涵在相關數(shù)學家工作中的深邃思想,探究了數(shù)學家之間的思想傳承。主要取得如下成果：1.通過分析希爾伯特在積分方程方面的三篇重要文獻,追溯其產(chǎn)生無限二次型理論的根源及對積分方程工作的影響,還原了他求解有限線性方程組的方法以及通過內(nèi)積將積分方程轉(zhuǎn)化為無窮線性方程組的代數(shù)化求解過程,揭示出這些工作中蘊含的對偶思想以及希爾伯特對對偶空間理論形成所做出的奠基性貢獻。2.在對連續(xù)線性泛函概念產(chǎn)生和弗雷歇泛函表示工作分析的基礎上,深入細致地研究了里斯在具體空間上的積分方程和線性方程組工作,探尋出里斯求解積分方程和無窮線性方程組的思想淵源,挖掘出其積分方程和線性方程組求解問題與相應空間上連續(xù)線性泛函表示之間的聯(lián)系,勾勒出具體對偶空間的形成過程,揭示出隱藏在其工作中的統(tǒng)一化和抽象化思想以及這些思想對對偶空間抽象理論形成的影響。也分析了斯坦豪斯的具體對偶空間工作,揭示出其工作與前人工作的不同之處。3.深入細致地分析了對偶空間抽象理論形成之際重要數(shù)學家們的相關研究工作。通過探討黑利在凸理論思想下的序列賦范線性空間中的工作,漢恩在泛函方程思想指導下的一般賦范線性空間中的工作,巴拿赫在算子思想指導下的巴拿赫空間中的工作,還原了他們抽象理論建立背后的具體問題來源,探索了他們對偶空間理論的形成過程,建立起以泛函延拓定理為主的對偶空間理論形成的完整思想脈絡。4.深入細致分析了弱*緊定理形成過程中一些數(shù)學家們所做的變革和發(fā)展。圍繞“緊,,和“弱收斂”兩個核心概念,探討了弱*緊定理的前史。透過希爾伯特、里斯在積分方程方面的工作揭示了引入“弱收斂”概念的必要性以及其在有限過渡到無限過程中所起的關鍵作用。從對偶的角度揭示了巴拿赫在對偶空間上引入弱收斂理論的緣由,最后從弱拓撲的深度歸結(jié)到弱*緊定理。5.系統(tǒng)考察了巴拿赫之后對偶空間理論的發(fā)展狀況,特別是在這門學科形成之后,測度理論、拓撲理論對其產(chǎn)生的深遠影響。同時探討了對偶空間理論的思想和方法對20世紀數(shù)學發(fā)展的影響。
[Abstract]:Dual space theory is one of the core contents of functional analysis, which is closely related to many branches of mathematics and is also widely used. Taking the solution of integral equations and linear equations as the main line, the history of the dual space theory is studied in detail on the basis of the study of the original literature and the research literature. The formation and development of its important theorem, weak * compactness theorem, are discussed, and the profound ideas contained in the work of relevant mathematicians are excavated. This paper probes into the ideological heritage among mathematicians. The main achievements are as follows: 1.Through analyzing three important papers on integral equations, Hilbert traces the origin of the theory of infinite quadratic form and its influence on the work of integral equations. His method of solving finite linear equations and the algebraic solution process of converting integral equations into infinite linear equations by inner product are reduced. The dual thought contained in these works and Hilbert's fundamental contribution to the formation of dual space theory are revealed. 2. On the basis of the analysis of the concept of continuous linear functional and the representation of Freichet functional, Rhys' work on integral equations and linear equations in specific spaces is studied in detail, and the origin of Reese's ideas for solving integral equations and infinite linear equations is explored. The relation between the integral equation, the system of linear equations and the continuous linear functional representation in the corresponding space is excavated, and the forming process of the concrete dual space is outlined. This paper reveals the unification and abstraction thought hidden in his work, and their influence on the formation of the abstract theory of dual space, and also analyzes the concrete dual space work of Stannhaus. This paper reveals the difference between his work and his predecessors' work. 3. The relevant research work of important mathematicians at the time of the formation of the abstract theory of dual space is deeply and meticulously analyzed. By discussing the sequence normed line of Hailey's thinking in convex theory, Work in the sex space, Hann's work in the general normed linear space under the guidance of functional equations, and Barnach's work in the Barnabian space under the guidance of operator thought, have reduced the specific problem sources behind the establishment of their abstract theory. Explored the formation of their dual space theory, The complete thought of dual space theory, which is based on functional extension theorem, is established. 4. The transformation and development of some mathematicians in the forming process of weak * compact theorem are analyzed in detail. The two core concepts of convergence, The prehistory of weak * compact theorem is discussed. Reese's work on integral equations reveals the necessity of introducing the concept of "weak convergence" and its key role in the transition from finite to infinite. The reason for the weak convergence theory, Finally, from the depth of weak topology to weak * compact theorem .5.We systematically investigate the development of dual space theory after Barnach, especially after the formation of this subject, measure theory, The influence of the theory of topology on the development of mathematics in 20th century is also discussed.
【學位授予單位】：西北大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O177
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本文編號：1598080