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The almost fixed point property is not invariant under isometric renormings
In the present note we prove the non set-stability of the AFPP under isometric renormings in the setting of Banach spaces containing a complemented...
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A concise list of coordinates for some Relationships
We collect here a number of examples, counterexamples, and results, classified according to some basic features, and formulated in a telegraphic way....
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Locally uniformly convex renorming of nonseparable spaces
This chapter deals with renormings —for the main part, LUR renormings— of the more “accessible” nonseparable Banach spaces (weakly compactly...
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Renormings in Banach Spaces A Toolbox
This monograph presents an up-to-date panorama of the different techniques and results in the large field of renorming in Banach spaces and its...
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Some structural properties of Banach Spaces
In this chapter we shall introduce, for later references, some structures in a Banach space that allow for vector representation and computations —a...
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Checking renormability in some classical Spaces
This chapter provides a list of classical spaces with some possible and impossible renormings. Risking some redundancies, it is maybe better to...
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A remark on totally smooth renormings
E. Oja, T. Viil, and D. Werner showed, in Totally smooth renormings , Archiv der Mathematik, 112 , 3, (2019), 269–281, that a weakly compactly...
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Totally smooth renormings
We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We...
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Some renormings of Banach spaces with the weak fixed point property for nonexpansive map**s
In 2013, Jiménez–Melado and Llorens–Fuster proved that the renorming of ℓ 2 , x = max{‖ x ‖ 2 , p ( x )}, where p is a seminorm on ℓ 2 satisfying certain...
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Strictly convex renorming
This chapter follows in part the excellent introduction and reproduces some results in [OriSmTr12], where a more complete information is provided. It...
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Examples on Rotundity
Spaces with a URED norm have normal structure (Theorem 387 below), and the notion of URED can be described by using Chebyshev centers (Theorem 383)....
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Tools for renorming
This chapter focuses on separable Banach spaces. Despite this, some non-separable results are included, mostly because they provide basic tools for...
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Weakly compactly generated spaces and their relatives III
Projectional resolutions of the identity were a fundamental tool for elucidating the structure of WCG spaces, and one of the fundamental...
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Miscellaneous applications
The following results are very important for understanding the relation of linear and metric structures of Banach spaces.
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The Banach–Saks property
It was asked if the existence of a C∞-smooth norm on a space can guarantee its WBS (see [GuiMoZi16, Problem 177]).
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Lipschitz functions II
One of the most important results regarding Lipschitz functions is the following one. The reader is advised to review Lemma 48 above and Theorem 443...
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Nonlinear transfer techniques
Inspired by Troyanski’s fundamental idea from his original [Tr71], R. Deville proved a powerful renorming theorem —reproduced below as Lemma 426—...
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The \( \mathcal{L}_{\infty} \) spaces
The following type of spaces were introduced by J. Lindenstrauss, A. Pełczynski, and H. P. Rosenthal in the papers [LiPe68] and [LiRo69] in attempt...
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Three-space properties
Related to (iv) and (xiv) in Theorem 655 above, we may mention that it is not known if renorming by a Fréchet smooth norm is a three-space property.
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Higher-order smoothness
It confirms an old good Czech mathematical saying: The utmost important is a good definition. It took D. Preiss a few minutes to show this new...