The Integral Equation Theory Of Hilbert

Integral equation is an important mathematical tool. Not only does it have a close relationship with many branches of mathematics, but also it has a variety of applications. Based on an extensive investigation of original sources and research papers on this subject, and using "Why mathematics was done" as the starting point and main purpose, by means of historical analysis and literature review, the dissertation studies systematically Hilbert’s works on integral equations and explores Hilbert’s idea on integral equations, and explores the inheritance between Hilbert and his followers. The main contributions of this study are as follows:1.Some intensive investigations about Fredholm’s theory of integral equations are conducted, the origin of Fredholm’s idea on integral equations is analyzed, the early history of system of infinitely many equations in infinitely many unknowns is sorted, and the perspective and process to establish his theory is examined.2.Hilbert’s works on integral equations with symmetric kernel are studied in detail, the train of thought to establish his theory is sorted, the problems which his theory concerned are pointed out, the key to successfully establish his theory is analyzed, which is generalized principal axis theorem. The idea and process to establish this theorem is examined, the applications of this theorem to prove the existence of the problem and to establish expansion theorem are discussed and described, and his idea on Hilbert space is elaborated.3. Spectral theory which Hilbert established in the language of infinite quadratic forms is thoroughly discussed, and the applications of spectral theory to integral equations are investigated in detail. The thought and purpose which established spectral theory is analyzed. This paper discusses how Hilbert defined point spectrum and continuous spectrum of infinite quadratic forms and how Hilbert established spectral decomposition of bounded quadratic forms. The importance of the concept of completely continuous is noted. Through the investigations about his applications of spectral theory to integral equations, Hilbert’s idea on function space and operator theory is expound.4. Some intensive investigations about followers’contributions to the theory of Hilbert spaces. This paper analyzes Schmidt’s early works in which he simplified Hilbert’s eigenvalue theory, and points out that he took a step closer to Hilbert space. His works to establish Hilbert sequence space and introduce geometric language into this space is studied in detail. Using Lebesgue integral, Riesz extended Hilbert’s works and established Riesz-Fisher theorem. This paper analyzes the related works of Riesz, which can help us to better understand his later works.5.The significant impact of Hilbert’s works during the evolution process of abstract space theory of is investigated. When Riesz extended the space of Lebesgue square integrable functions, he found concrete Banach space. Then he introduced norms to study the function instead of inner product, gave the three axioms of norms, and established his theory of compact operators based on the concept of completely continuous of Hilbert. To generalize the theory of integral equations, Banach developed the theory of Banach space in an abstract environment. To satisfy the need for development of quantum mechanics and follow the trend of development of abstract, John von Neumann reformulated and developed Hilbert’s theory and created abstract theory of Hilbert space.