In a paper from 1987, Gruenhage defined a class of topological spaces that now bear his name, and used it to solve a problem of Talagrand on the existence of dense $G_δ$ metrizable subsets of Gul'ko compact spaces. Gruenhage's paper became highly influential among researchers in renorming theory, a branch of Banach space theory. In this paper we survey Gruenhage and related spaces, and their interactions with renorming theory.
Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.
The problems connected with equivalent norms lie at the heart of Banach space theory. This is a short survey on some recent as well as classical results and open problems in renormings of Banach spaces.
We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$-topology (has a G$_δ$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).
We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.
We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We also consider renormings for $(a,B,c)$-ideals.
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any…
We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its …
We introduce a new diametral notion for points of the unit sphere of Banach spaces, that naturally complements the notion of Delta-points, but is weaker than the notion of Daugavet points. We prove that this notion can be used to provide a new geometric characterization of the Daugavet property, as well as to recover -- and even to provide new -- results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Finally, we show that this notion leads to powerful new ideas for renorming questions, and that those ideas can be combined with pr…
We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which mo…
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of $\ell_1$ if and only if there is an equivalent norm $|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and every $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq (1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and every $r\in\mathbb R$. In this paper we generalize this result to larger cardinals, showing that if $κ$ is an uncountable cardinal then a Banach space $X$ contains a copy of $\ell_1(κ)$ if and only if there is an equivalent norm $|||\cdot|||$ on $X$…
A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
We are concerned about improvements of the modulus of convexity by renormings of a super-reflexive Banach space. Typically optimal results are beyond Pisier's power functions bounds $t^p$, with $p \geq 2$, and they are related to the notion of generalized cotype. We obtain an explicit upper bound for all the modulus of convexity of equivalent renormings and we show that if this bound is equivalent to $t^2$, the best possible, then the space admits a renorming with modulus of power type $2$. We show that a UMD space admits a renormings with modulus of convexity bigger, up to a multiplicative co…
We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and summable Szlenk index (property $\textsf{A}_\infty$) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also …
We continue with the study of greedy-type bases in quasi-Banach spaces started in [3]. In this paper, we study partially-greedy bases focusing our attention in two main results: -Characterization of partially-greedy bases in quasi-Banach spaces in terms of quasi-greediness and different conservative-like properties. -Given a $C$-partially-greedy basis in a quasi-Banach space, there exists a "renorming" such that the basis is $1$-partially-greedy.
Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert\rightarrow 1+\Vert x\Vert$ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert$ for every $x\in X$. A result of V. Kadets, V. Shepelska and D. Werner asserts that a Banach space contains an isomorphic copy of $\ell_1$ if and only if there exists an equivalent renorming with an $L$-orthogonal sequence, whereas a result of G. Godefroy claims that containing an isomorphic copy of…
We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of …
We prove that a Banach space of continuous functions $C(K)$ has a renorming that is uniformly rotund in every direction (URED) if and only if the compact space $K$ supports a strictly positive measure
Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blo…
The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($β$). As applications we prove that up to renorming, property ($β$) is stable under coarse Lipschitz embeddings and coarse quotients.
We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements $x,y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\| \geqslant 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as no…
In the first part of our paper, we show that $\ell_\infty$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as $\ell_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0(ω_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty}$-smooth norm. Finally, we …
We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets o…
