从Fredholm到Banach之算子理论历史研究

The development of the operator theory from Fredholm to Banach is a decisive stage in the establishment of functional analysis, which has drawn extensive attention and research from scholars in mathematical history. However, at present, many researchers have placed more emphasis on historical facts, and their works are lack of in-depth explorations of the mathematical ideas behind the facts. In this project, we will take the solving of integral equations as the main line, investigate the related works of Fredholm, Hilbert, Riesz, Banach and other mathematicians, reconstruct the idea maps of operator theory of each mathematician on the basis of interpreting the original documents, probe the inheritance, development and breakthrough of mathematical ideas among them in depth. With a comprehensive interpretation from new view, we try to outline a clear logical thinking diagram of the history of operator theory. Furthermore, this project will promote the understanding of the development of functional analysis from the perspective of operators, and explore the influence of operator theory on the development of analytics.

从Fredholm到Banach的算子理论发展，是泛函分析学科建立的一个决定性阶段，也得到数学史学者的广泛关注和研究。但目前已有研究成果更侧重陈述事实，缺乏对事实背后所隐藏数学思想方法的深入挖掘。本项目将以积分方程的求解为主线，研究Fredholm、Hilbert、Riesz、Banach等数学家的相关工作，在解读原始文献基础上，重构各个数学家的算子理论思想路线图，深入探讨他们之间数学思想方法的传承、发展与突破。力图通过全面有新意的解读，勾勒出一条内在思想逻辑清晰的算子理论思想脉络图。进而从算子角度进一步了解泛函分析的发展进程，探讨算子理论对整个分析学发展的影响。

算子理论是研究函数空间或更一般抽象空间之间的变换或算子的一种理论，是现今泛函分析学科主要理论之一。究其渊源，源自数学史上对具体积分方程求解问题的不断探索。本项目致力于探索从Fredholm到Banach的算子理论发展脉络，以“第二型积分方程求解问题”为主线，充分挖掘Fredholm、Hilbert、Riesz和Banach等重要数学家的积分方程工作，呈现算子理论发展的历史图景。这一理论的历史发展进程也体现了20世纪数学更抽象、更统一的发展特征。本项目的研究成果为一本《从Fredholm到Banach之算子理论历史研究》的专著，已与出版社签订合同，完成校稿，预计2022年4月份出版。通过对这一项目的研究不仅能理清算子理论的历史发展脉络，同时也能为抽象的泛函分析教学提供历史背景，为理解数学发展史提供一个窗口。