This book discusses the process by which Ulam's conjecture is proved, aptly detailing how mathematical problems may be solved by systematically combining interdisciplinary theories. It presents the state-of-the-art of various research topics and methodologies in mathematics, and mathematical analysis by presenting the latest research in emerging research areas, providing motivation for further studies. The book also explores the theory of extending the domain of local isometries by introducing a generalized span. For the reader, working knowledge of topology, linear algebra, and Hilbert space theory, is essential. The basic theories of these fields are gently and logically introduced. The content of each chapter provides the necessary building blocks to understanding the proof of Ulam's conjecture and are summarized as follows: Chapter 1 presents the basic concepts and theorems of general topology. In Chapter 2, essential concepts and theorems in vector space, normed space, Banach space, inner product space, and Hilbert space, are introduced. Chapter 3 gives a presentation on the basics of measure theory. In Chapter 4, the properties of first- and second-order generalized spans are defined, examined, and applied to the study of the extension of isometries. Chapter 5 includes a summary of published literature on Ulam's conjecture; the conjecture is fully proved in Chapter 6.

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Soon-Mo Jung is a professor of mathematics at Hongik University in the Republic of Korea. His research interests include measure theory, number theory, and classical analysis. He received his bachelor's, master's, and doctoral degrees in 1988, 1992 and 1994, respectively, from the Department of Mathematics at the University of Stuttgart, Germany. One of the themes of his doctoral dissertation is closely related to the subject of this book, Ulam's conjecture. He has been a professor at Hongik University since 1995 and has published numerous papers and books in the fields of measure theory, fractal geometry, number theory, classical analysis, discrete mathematics, differential equations, and functional equations.