Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube
- 2023
- Buch
- Verfasst von
- Soon-Mo Jung
- Buchreihe
- Frontiers in Mathematics
- Verlag
- Springer Nature Switzerland
Über dieses Buch
Über dieses Buch
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.
Inhaltsverzeichnis
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Frontmatter
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Chapter 1. Topology
Soon-Mo JungAbstractIn this chapter, we will briefly introduce the basic concepts and theorems of general topology necessary to understand the subject matter of this book. Among many other literatures listed in the References section, we mainly refer to the book [13] by R. H. Kasriel and the book [19] by G. F. Simmons for this purpose. -
Chapter 2. Hilbert Spaces
Soon-Mo JungAbstractIn this chapter, we briefly introduce basic concepts and theorems in vector space, normed space, Banach space, and Hilbert space that are essential to prove Ulam’s conjecture, the main subject of this book. -
Chapter 3. Measure Theory
Soon-Mo JungAbstractThe concept of a measure is a generalization and formalization of length, area, volume, and other common notions such as mass and probability of events. These seemingly different concepts have many similarities and can often be unified by the concept of measure. Measures are fundamental to probability theory and integration theory. -
Chapter 4. Extension of Isometries
Soon-Mo JungAbstractIn this chapter, we define the first- and second-order generalized spans and the index set, examine their properties, and apply them to the study of the extension of isometries. To this end, we develop a theory that extends the domain of local isometries to the generalized spans, where we call an isometry defined in a subset of a Hilbert space a local isometry. -
Chapter 5. History of Ulam’s Conjecture
Soon-Mo JungAbstractA conjecture of Ulam states that the standard product probability measure π on the Hilbert cube Iω is da-invariant when the sequence \(a = \{ a_i \}_{i \in \mathbb {N}}\) of positive numbers satisfies the condition \(\sum \limits _{i=1}^\infty a_i^2 < \infty \). In 1974 and 1977, J. Mycielski published the first papers on this topic. Indeed, he proved the conjecture of Ulam affirmatively under the additional assumption that the sets are open. -
Chapter 6. Ulam’s Conjecture
Soon-Mo JungAbstractThe conjecture of Ulam states that the standard product probability measure π on the Hilbert cube Iω is invariant under the induced metric da when the sequence \(a = \{ a_i \}_{i \in \mathbb {N}}\) of positive numbers satisfies condition (4.1). This conjecture was proved in [6] when E1 is a non-degenerate subset of Ma. In this chapter, we will completely prove Ulam’s conjecture to be true by considering both non-degenerate as well as degenerate cases. -
Backmatter
- Titel
- Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube
- Verfasst von
-
Soon-Mo Jung
- Copyright-Jahr
- 2023
- Verlag
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-031-30886-4
- Print ISBN
- 978-3-031-30885-7
- DOI
- https://doi.org/10.1007/978-3-031-30886-4
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