Separate and Joint Continuity presents and summarises the main ideas and theorems that have been developed on this topic, which lies at the interface between General Topology and Functional Analysis (and the geometry of Banach spaces in particular). The book offers detailed, self-contained proofs of many of the key results.
Although the development of this area has now slowed to a point where an authoritative book can be written, many important and significant problems remain open, and it is hoped that this book may serve as a springboard for future and emerging researchers into this area. Furthermore, it is the strong belief of the authors that this area of research is ripe for exploitation. That is to say, it is their belief that many of the results contained in this monograph can, and should be, applied to other areas of mathematics. It is hoped that this monograph may provide an easily accessible entry point to the main results on separate and joint continuity for mathematicians who are not directly working in this field, but who may be able to exploit some of the deep results that have been developed over the past 125 years.
Features
Provides detailed, self-contained proofs of many of the key results in the area
Suitable for researchers and postgraduates in topology and functional analysis
Is the first book to offer a detailed and up-to-date summary of the main ideas and theorems on this topic
1. Introduction. 1.1. Background. 1.2. Baire and Related Spaces. 1.3. Quasicontinuous Functions. 1.4. Set-Valued Mappings. 1.5. Basics of Function Spaces. 1.6. Concepts in Banach Spaces. 1.7. Commentary and Exercises. 2. Fundamental Results. 2.1. Fundamental Questions. 2.2. First Countable Spaces. 2.3. q-Spaces. 2.4. Second Countable Spaces. 2.5. Separately Quasicontinuous Functions. 6. Piotrowski’s Theorem. 2.7. Talagrand’s Problem. 2.8. Commentary and Exercises. 3. Continuity of Group Actions and Operations. 3.1. Semitopological and Paratopological Groups. 3.2. Δ-Baire Spaces. 3.3. Continuity of Group Actions. 3.4. Some Counterexamples. 3.5. Miscellaneous Applications. 3.6. Commentary and Exercises. 4. Namioka Theorem and Related Spaces. 4.1. Namioka Theorem. 4.2. Namioka Theorem - a Functional Analytic Proof. 4.3. Namioka Spaces. 4.4. Co-Namioka Spaces. 4.5. Commentary and Exercises. 5. Various Applications. 5.1. Point of Continuity Properties. 5.2. Minimal USCO Mappings. 5.3. Ryll-Nardzewski Fixed-Point Theorem. 5.4. Differentiability of Continuous Convex Functions. 5.5. Applications in Variational Analysis. 6. Future Directions and Open Problems. 6.1. Topologies of Separate and Joint Continuity. 6.2. Semitopological and Paratopological Groups. 6.3. Namioka Spaces. 6.4. Co-Namioka and Related Spaces. 6.5. Baire Measurability of Separately Continuous Functions. 6.6. Sets of Discontinuity Points of Separately Continuous Functions. 6.7. Various Maslyuchenko Spaces.
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