对偶空间理论的形成与发展

发布时间：2018-03-11 12:03

  本文选题：积分方程　切入点：无穷线性方程组　出处：《西北大学》2016年博士论文　论文类型：学位论文

【摘要】：对偶空间理论是泛函分析的核心内容之一,与众多数学分支联系紧密,亦有着广泛应用。本文通过历史分析和文献考证的方法,以“为什么数学”为指导,以“积分方程和线性方程组的求解”为主线,在研读相关原始文献和研究文献的基础上,对对偶空间理论的历史进行了较为深入细致的研究,并对其上重要定理——弱*紧定理的形成与发展脉络进行了探讨,挖掘了蕴涵在相关数学家工作中的深邃思想,探究了数学家之间的思想传承。主要取得如下成果：1.通过分析希尔伯特在积分方程方面的三篇重要文献,追溯其产生无限二次型理论的根源及对积分方程工作的影响,还原了他求解有限线性方程组的方法以及通过内积将积分方程转化为无穷线性方程组的代数化求解过程,揭示出这些工作中蕴含的对偶思想以及希尔伯特对对偶空间理论形成所做出的奠基性贡献。2.在对连续线性泛函概念产生和弗雷歇泛函表示工作分析的基础上,深入细致地研究了里斯在具体空间上的积分方程和线性方程组工作,探寻出里斯求解积分方程和无穷线性方程组的思想渊源,挖掘出其积分方程和线性方程组求解问题与相应空间上连续线性泛函表示之间的联系,勾勒出具体对偶空间的形成过程,揭示出隐藏在其工作中的统一化和抽象化思想以及这些思想对对偶空间抽象理论形成的影响。也分析了斯坦豪斯的具体对偶空间工作,揭示出其工作与前人工作的不同之处。3.深入细致地分析了对偶空间抽象理论形成之际重要数学家们的相关研究工作。通过探讨黑利在凸理论思想下的序列赋范线性空间中的工作,汉恩在泛函方程思想指导下的一般赋范线性空间中的工作,巴拿赫在算子思想指导下的巴拿赫空间中的工作,还原了他们抽象理论建立背后的具体问题来源,探索了他们对偶空间理论的形成过程,建立起以泛函延拓定理为主的对偶空间理论形成的完整思想脉络。4.深入细致分析了弱*紧定理形成过程中一些数学家们所做的变革和发展。围绕“紧,,和“弱收敛”两个核心概念,探讨了弱*紧定理的前史。透过希尔伯特、里斯在积分方程方面的工作揭示了引入“弱收敛”概念的必要性以及其在有限过渡到无限过程中所起的关键作用。从对偶的角度揭示了巴拿赫在对偶空间上引入弱收敛理论的缘由,最后从弱拓扑的深度归结到弱*紧定理。5.系统考察了巴拿赫之后对偶空间理论的发展状况,特别是在这门学科形成之后,测度理论、拓扑理论对其产生的深远影响。同时探讨了对偶空间理论的思想和方法对20世纪数学发展的影响。
[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.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177
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本文编号：1598080