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Mathematics
Geometry

Open Problems in the Geometry and Analysis of Banach Spaces

Profile image of Vicente MontesinosVicente Montesinos

2016

https://doi.org/10.1007/978-3-319-33572-8
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12 pages

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Abstract

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

FAQs

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What are some notable open problems in Banach space theory?add

The document identifies multiple open problems such as the nature of biorthogonal systems and fixed points in Banach spaces, underscoring ongoing gaps in understanding since 2023.

How do different methods of compact generating relate in Banach spaces?add

The collection hints that varying approaches to weak compact generating reveal disparities, with crucial implications noted in prior studies from 2014 and 2011.

What challenges remain regarding renormings in infinite-dimensional Banach spaces?add

The findings suggest that while techniques exist, new ideas are required to tackle longstanding issues around renormings, dating back several decades.

Why is the theory of Banach spaces considered a fertile research area?add

The text emphasizes its abundance of open problems and diverse applications, fostering interest among young researchers in functional analysis since 2023.

When were key methodologies in the topology of Banach spaces established?add

Essential methodologies and principles were established between 1997 and 2014, setting the groundwork for contemporary inquiries in Banach space analysis.

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References (42)

  1. Basic Linear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Chebyshev Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
  2. 2 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
  3. 3 Lifting Quotient Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
  4. 4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
  5. 5 Quasitransitive Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
  6. 6 Quasi-Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
  7. 7 Banach-Mazur Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
  8. 8 Rotund Renormings of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
  9. 9 More on the Structure of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
  10. 3 Biorthogonal Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1 Bases, Finite-Dimensional Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 51
  11. 2 Markushevich Bases, Auerbach Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
  12. 3 Weakly Compactly Generated Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . 55
  13. 4 Differentiability and Structure, Renormings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
  14. 2 Differentiability of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
  15. 3 Smooth Extension of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
  16. 4 Smooth Renormings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
  17. 4.1 Gâteaux Differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
  18. 4.2 Strongly Gâteaux Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
  19. 4.3 Fréchet Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
  20. 5 Strongly Subdifferentiable Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
  21. 6 Measure-Null Sets in Infinite Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
  22. 7 Higher Order Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
  23. 8 Mazur Intersection Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
  24. 9 Krein-Milman and Radon-Nikodým Properties . . . . . . . . . . . . . . . . . . . . . 86
  25. 10 Norm-Attaining Functionals and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 91
  26. Weak Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
  27. 12 Polyhedral Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
  28. Nonlinear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1 Lipschitz-Free Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
  29. 2 Lipschitz Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
  30. 3 Lipschitz Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
  31. 4 General Nonlinear Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
  32. 5 Some More Topological Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
  33. Some More Nonseparable Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.1 Schauder Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
  34. 2 Support Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
  35. 3 Equilateral Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
  36. Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.1 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
  37. 2 Riemann Integrability of Vector-Valued Functions . . . . . . . . . . . . . . . . . . 133
  38. 3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
  39. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
  40. List of Concepts and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
  41. Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
  42. Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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