arXiv subjects

Tommaso Russo

Publications and source records attributed to Tommaso Russo.

At least 19 recordsLinked to original sources

arXiv2026

Tilings and coverings by balls in $\ell_1$

A famous result of Klee from 1981 is that the Banach space $\ell_1(κ)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $κ$ with $κ^ω=κ$. Klee also observed that the smallest cardinal in which such a tiling might exist is $κ= 2^{\aleph_0}$, leaving open the question whether, for $κ< 2^{\aleph_0}$, $\ell_1(κ)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(κ)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable.

math.FA↗

arXiv2026

Packings in classical Banach spaces

We obtain several new results on the simultaneous packing and covering constant $γ(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $γ^*(\mathcal{X})$. These constants measure how efficient a (lattice) packing by unit balls in $\mathcal{X}$ can be, the optimal case being that $γ(\mathcal{X})= 1$ and the worst that $γ(\mathcal{X})= 2$. Our first main result is that $γ(\mathcal{X})> 1$ whenever $B_\mathcal{X}$ admits a LUR point, which leads us to a negative answer to a question of Swanepoel. We also develop general methods to compute these constants for a large class of spaces. As a sample of our findings: (i) $γ^*(\mathcal{X})= 1$ when $\mathcal{X}$ is a separable octahedral Banach space, or $\mathcal{X}= \mathcal{C}(\mathcal{K})$, where $\mathcal{K}$ is zero-dimensional; (ii) $γ(\ell_p(κ)\oplus_r \mathcal{X})= γ^*(\ell_p(κ)\oplus_r \mathcal{X})= \frac{2}{2^{1/p}}$, whenever $\rm{dens}(\mathcal{X})< κ$ and $1\leq r\leq p< \infty$; (iii) $γ(L_p(μ))= γ^*(L_p(μ))= \frac{2}{2^{1/p}}$ for $1\leq p\leq 2$ and every measure $μ$; (iv) there exist reflexive (resp. octahedral) Banach spaces $\mathcal{X}$ with $γ(\mathcal{X})= 2$. We leave a large area open for further research and we indicate several possible directions.

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arXiv2025

Norming Markushevich bases: recent results and open problems

We survey several results concerning norming Markushevich bases (M-bases, for short), focusing in particular on two recent examples of a weakly compactly generated Banach space with no norming M-basis and of an Asplund space with norming M-basis that is not weakly compactly generated. We highlight the context for these problems and state several open problems in different directions that arise from these results.

math.FA↗

arXiv2025

Lattice tilings of Hilbert spaces

We construct a bounded and symmetric convex body in $\ell_2(Γ)$ (for certain cardinals $Γ$) whose translates yield a tiling of $\ell_2(Γ)$. This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain the first example of an infinite-dimensional reflexive Banach space that admits a tiling with balls (of radius $1$). Further, our tiling has the property of being point-countable and lattice (in the sense that the set of translates forms a group). The same construction performed in $\ell_1(Γ)$ yields a point-$2$-finite lattice tiling by balls of radius $1$ for $\ell_1(Γ)$, which compares to a celebrated construction due to Klee. We also prove that lattice tilings by balls are never disjoint and, more generally, each tile intersects as many tiles as the cardinality of the tiling. Finally, we prove some results concerning discrete subgroups of normed spaces. By a simplification of the proof of our main result, we prove that every infinite-dimensional normed space contains a subgroup that is $1$-separated and $(1+\varepsilon)$-dense, for every $\varepsilon>0$; further, the subgroup admits a set of generators of norm at most $2+\varepsilon$. This solves a problem due to Swanepoel and yields a simpler proof of a result of Dilworth, Odell, Schlumprecht, and Zsák. We also give an alternative elementary proof of Steprāns' result that discrete subgroups of normed spaces are free.

math.FA↗

arXiv2024

Tilings of the Hyperbolic Space and Lipschitz Functions

We use a special tiling for the hyperbolic $d$-space $\mathbb{H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(P)\oplus\mathcal{F}(\mathcal{N})$ where $P$ is a polytope in $\mathbb{R}^d$ and $\mathcal{N}$ a net in $\mathbb{H}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(\mathbb{R}^{d})\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in $\mathbb{H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}(\mathbb{H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}(\mathbb{H}^{d})$ and $\mathrm{Lip}(\mathbb{R}^d)$.

math.FA↗

arXiv2023

Banach spaces of continuous functions without norming Markushevich bases

We investigate the question whether a scattered compact topological space $K$ such that $C(K)$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević, and the authors, to an open problem from the Nineties, due to Godefroy. Our prime tool consists in proving that $C([0,ω_1])$ does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact $K$ for $C(K)$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that $K$ is a $0$-dimensional compact space with a P-point, or a compact tree of height at least $ω_1 +1$. In particular, this allows us to answer the said question in the case when $K$ is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than $ω_2$ are Valdivia.

math.FA↗

arXiv2023

Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely lineable in $\ell_\infty$, thus complementing a recent result of Papathanasiou who proved the same for the sequences with continuum many accumulation points. We also prove that these sets are spaceable. We then consider the same problems for the set of bounded non-convergent sequences with a finite number of accumulation points. We prove that such set is densely lineable in $\ell_\infty$ and that it is nevertheless not spaceable. The said problems are also studied in the setting of ideal convergence and in the space $\mathbb{R}^ω$.

math.FA↗

arXiv2022

Smooth and polyhedral norms via fundamental biorthogonal systems

Let $\mathcal{X}$ be a Banach space with a fundamental biorthogonal system and let $\mathcal{Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal{Y}$ admits a $C^\infty$-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal{Y}$ admits locally finite, $σ$-uniformly discrete $C^\infty$-smooth and LFC partitions of unity and a $C^1$-smooth LUR norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every WLD Banach space (hence, all reflexive ones), $L_1(μ)$ for every measure $μ$, $\ell_\infty(Γ)$ spaces for every set $Γ$, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum {\sf MM}, all Banach spaces of density $ω_1$ are covered by our result.

math.FA↗

arXiv2022

Projecting Lipschitz functions onto spaces of polynomials

The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${{\rm Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate whether $\mathcal{P}({}^2X)$ is a complemented subspace of ${{\rm Lip}_0}(B_X)$. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $\mathcal{P}({}^1X)=X^*$ is complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$. Our main result asserts that $\mathcal{P}({}^2X)$ is not complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$ with non-trivial type.

math.FA↗

arXiv2022

Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(Γ)$ comprising all elements $y$ such that $y \in \ell_q(Γ)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $\left\Vert\cdot \right\Vert_p $-norm. This provides examples of dense subspaces of $\ell_p(Γ)$ with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $p>1$ or $Γ$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(Γ)$ admits a $C^1$-smooth norm.

math.FA↗

arXiv2022

Weakly Corson compact trees

We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every chain complete, rooted tree into a Fréchet--Urysohn, countably compact topological space. We show the rôle of such topology in the theory of weakly Corson and weakly Valdivia compacta. In particular, we give the first example of a compact space $T$ whose every closed subspace is weakly Valdivia, yet $T$ is not weakly Corson. This answers a question due to Ondřej Kalenda.

math.GN↗

arXiv2021

An Asplund space with norming Markuševič basis that is not weakly compactly generated

We construct an Asplund Banach space $\mathcal{X}$ with a norming Markuševič basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman.

math.FA↗

arXiv2021

Small semi-Eberlein compacta and inverse limits

We study properties of semi-Eberlein compacta related to inverse limits. We concentrate our investigation on an interesting subclass of small semi-Eberlein compacta whose elements are obtained as inverse limits whose bonding maps are semi-open retractions.

math.GN↗

arXiv2020

Symmetrically separated sequences in the unit sphere of a Banach space

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 non-trivial cotype, we obtain quantitative estimates for the said $\varepsilon$. Certain renorming results are also presented.

math.FA↗

arXiv2020

Some remarks on smooth renormings of Banach spaces

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.

math.FA↗

arXiv2020

An uncountable version of Pták's combinatorial lemma

In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Pták. We show that the validity of the result for $ω_1$ can not be decided in ZFC alone. We also provide a sufficient condition, for a class of larger cardinals.

math.FA↗

arXiv2020

Separated sets and Auerbach systems in Banach spaces

The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains an uncountable $(1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established that happen to be sharp for the class of WLD spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $c_0(ω_1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $(1+\varepsilon)$-separated subset of any regular cardinality not exceeding the density of $X$; should the space $X$ be super-reflexive, the unit sphere of $X$ contains such a subset of cardinality equal to the density of $X$. The said problem is studied for other classes of spaces too, including the RNP spaces or strictly convex ones.

math.FA↗

arXiv2020

On densely isomorphic normed spaces

In the first part of our note we prove that every Weakly Lindelöf Determined (WLD) (in particular, every reflexive) non-separable Banach $X$ space contains two dense linear subspaces $Y$ and $Z$ that are not densely isomorphic. This means that there are no further dense linear subspaces $Y_0$ and $Z_0$ of $Y$ and $Z$ which are linearly isomorphic. Our main result (Theorem B) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of $\ell_2(ω_1)$ (or more generally every WLD space of density $ω_1$) which contains no uncountable biorthogonal system. This result lies between two fundamental results concerning biorthogonal systems, namely the construction of Kunen (under CH) of a non-separable Banach space which contains no uncountable biorthogonal system, and the construction of Todorucević (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space.

math.FA↗