Search

3 results

10 - Back to Banach Space Theory
Félix Cabello Sánchez, Universidad de Extremadura, Spain, Jesús M. F. Castillo, Universidad de Extremadura, Spain
Book:

Homological Methods in Banach Space Theory

Published online:

19 January 2023

Print publication:

26 January 2023, pp 468-520
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

The final chapter of the book returns to the place the journey started: classical Banach space theory, with a twist. We can now provide solutions, or at least a better understanding, for a number of open problems. Among the topics covered, the reader will encounter vector-valued forms of Sobczyk’s theorem, isomorphically polyhedral $\mathscr L_\infty$-spaces, Lipschitz and uniformly homeomorphic $\mathscr L_\infty$-spaces, properties of kernels of quotient operators from $\mathscr L_1$-spaces, sophisticated 3-space problems, the extension of $\mathscr L_\infty$-valued operators, Kadec spaces, Kalton-Peck spaces and, at last, the space $Z_2$. All these topics can be easily considered as part of classical Banach space theory, even if the techniques we employ involve most of the machinery developed throughout the book.

Ramification conjecture and Hirzebruch’s property of line arrangements
Part of
D. Panov, A. Petrunin
Journal:

Compositio Mathematica / Volume 152 / Issue 12 / December 2016

Published online by Cambridge University Press:

25 October 2016, pp. 2443-2460

Print publication:

December 2016
- Article
- - You have access
- PDF
- Export citation
The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.

Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls
Vladimir P. Fonf, Clemente Zanco
Journal:

Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014

Published online by Cambridge University Press:

20 November 2018, pp. 42-50

Print publication:

14 March 2014
- Article
- - You have access
- PDF
- Export citation
We prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.