## Open Problems in the Geometry and Analysis of Banach SpacesThis is an collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help in convincing young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems are longstanding open problems, some are recent, some are more important and some are only local problems. Some would require new ideas, some may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area. |

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### Contents

1 | |

2 Basic Linear Geometry | 37 |

3 Biorthogonal Systems | 51 |

4 Differentiability and Structure Renormings | 58 |

5 Nonlinear Geometry | 103 |

6 Some More Nonseparable Problems | 125 |

7 Some Applications | 129 |

References | 135 |

List of Concepts and Problems | 149 |

157 | |

159 | |

### Other editions - View all

Open Problems in the Geometry and Analysis of Banach Spaces Antonio J. Guirao,Vicente Montesinos,Václav Zizler No preview available - 2016 |

Open Problems in the Geometry and Analysis of Banach Spaces Antonio J Guirao,Vicente Montesinos,Vaclav Zizler No preview available - 2018 |

### Common terms and phrases

admits a Fréchet admits a norm admits an equivalent Anal approximation property Asplund space Assume biorthogonal systems bounded linear operator bump function C1-smooth closed convex compact space convex functions convex norm convex set copy of c0 countable DeGoZi93 dense equivalent norm exist finite following problem Fréchet differentiable Fréchet differentiable norm Funct Gâteaux differentiable Gâteaux differentiable norm Godefroy GoLaZi14 Hájek Hilbert space hypercyclic infinite-dimensional Banach space infinite-dimensional subspace isometric isomorphic isomorphic copy Koszmider Lindenstrauss linearly isomorphic LinTza77 Lipschitz homeomorphic Lipschitz quotient LUR norm Markushevich basis Math metric space Montesinos N. J. Kalton nonseparable Banach space norm k k open problem Pełczy´nski Proc proved refer to BenLin00 reflexive Banach space reflexive space renormed result Schauder basis separable Banach space sequence strictly convex superreflexive theorem took this problem topology Troyanski uncountable uniformly convex unit ball W. B. Johnson WCG space weak Hilbert space weakly compact Zizler