arXiv subjects

Daniel L. Rodríguez-Vidanes

Publications and source records attributed to Daniel L. Rodríguez-Vidanes.

14 recordsLinked to original sources

arXiv2026

The super Alternative Daugavet property, unconditional bases and SCD geometry

We answer negatively the 1-unconditional part of Question 6.4, by Langemets, Lõo, Martín, Perreau and Rueda Zoca, and, more generally, prove that no infinite-dimensional Banach space with a $K$-unconditional basis, for $1\leq K<3/2$, can satisfy the super Alternative Daugavet property. We also address two recent questions posed by Lõo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that the weak unit ball of every Banach space with a 1-unconditional basis admits a countable $π$-base, thereby giving a positive answer to Question 5.1. We further show that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable $π$-base for its relative weak topology. We complement this result with two permanence principles: one for shrinking Schauder decompositions whose finite partial sums have countable weak $π$-bases, and another for unconditional sums over boundedly complete Banach sequence spaces. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.

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arXiv2026

Note on infinite-dimensional $L^p$-spaces

We prove that, for every $1\leq p<\infty$, the $L^{p}$-space of Baker's measure on $\mathbb{R}^{\mathbb{N}}$ is isometrically isomorphic to $\ell^p(\mathfrak{c},L^{p}[0,1])$ in ZFC. This solves in a negative manner the main problem stated in [Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure, Banach J. Math. Anal. 20 (2026), no. 1, Paper No. 7].

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arXiv2026

Two notes on valued fields

This paper studies two questions on valued fields: the metric dimension induced by an absolute value, and the uniform openness of multiplication. For nontrivial non-archimedean absolute values, we prove that the metric dimension equals the density character. In the archimedean case, the metric dimension is 2 for subfields of $\mathbb{R}$, while for non-real subfields of $\mathbb{C}$ it is either 2 or 3, according to whether the field contains a non-real element together with its complex conjugate. We also show that multiplication is uniformly open on every valued field. Finally, we prove that this property is genuinely metric, not purely topological, even on $\mathbb{R}$ with a suitable compatible choice of metric.

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arXiv2026

On a problem of Johnson and Wolfe

In 1979, Johnson and Wolfe proved that norm-attaining operators are dense in $L(C(K),C(S))$ when $K$ and $S$ are compact Hausdorff spaces in the real setting. The corresponding complex case has remained open since then, mainly because the real proof relies on order and sign-decomposition arguments that are no longer available for complex measures. In this paper, we settle the complex case. We prove that, for arbitrary compact Hausdorff spaces $K$ and $S$, the set of norm-attaining operators from the complex space $C(K)$ into the complex space $C(S)$ endowed with the supremum norm is dense in $L(C(K),C(S))$. The proof replaces the real order-theoretic mechanism by a measure-theoretic phase-correction argument, based on polar decompositions, unimodular approximation, and a semicontinuity principle for weighted total variation. This yields a complex defect-reduction procedure which recovers the Johnson-Wolfe density theorem in full generality for complex $C(K)$-spaces.

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arXiv2026

Linear structures in the set of non-norm-attaining operators on Banach spaces

We study large linear structures inside sets arising in the theory of norm-attaining operators. We provide several results in the context of lineability, spaceability, maximal-spaceability, and $(α, β)$-spaceability for sets of non-norm-attaining bounded linear operators whenever such sets are nonempty. To be more specific, we show that if $Y$ is a strictly convex renorming of $c_0 (Γ)$, then the set $$ \mathcal{L}(c_0 (Γ),Y)\setminus \overline{\text{NA} (c_0 (Γ),Y)} $$ is $2^{|Γ|}$-spaceable. We also prove that $$ \mathcal{L}(d_* (w,1) ,\ell_p )\setminus \overline{\text{NA} (d_* (w,1),\ell_p )} $$ is maximal-spaceable. Finally, we establish that whenever the set of non-norm-attaining operators from a Banach space $X$ into $\ell_p (Γ)$ (respectively, $c_0 (Γ)$) is nonempty, it contains a subspace linearly isometric to $\ell_p(Γ)$ (respectively, $c_0 (Γ)$). These results extend and complement several known results in the literature concerning large linear structures in sets of non-norm-attaining operators. Our results are obtained in a more general framework involving group-invariant operators, which allows us to treat classical spaces of operators as special cases.

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arXiv2026

Algebraic structures featuring graph dimensions, Hölder regularity, and fractional differentiability

We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, Hölder regularity, and fractional differentiability. Our first main result shows that for every $s \in (1,s]$, the set of continuous functions on $[0, 1]$ whose graph has Hausdorff and box dimensions equal to s is strongly $\mathfrak{c}$-algebrable, thereby tackling an open question by Bonilla et al., and complementing recent findings by Liu et. al and Carmona et al. We then extend the analysis to Hölder spaces: although the pointwise Hölder exponent of a generic function in $C^α[0, 1]$ is constant, we prove that the collection of functions realizing this behavior is $\mathfrak{c}$-lineable but cannot form an algebra. Nevertheless, we construct strongly $\mathfrak{c}$-algebrable families of functions that exhibit Hölder exponent $α$ outside a set of Hausdorff dimension zero. Finally, as a consequence of the relation between strongly monoHölder functions and fractional differentiability, we analyze the strong $\mathfrak{c}$-algebrability of nowhere (Riemann-Liouville) fractional differentiable functions.

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arXiv2025

Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure

We investigate the isometric structure of $L^{p}$-spaces for the infinite-dimensional Lebesgue measure $(\mathbb{R}^{\mathbb{N}},μ)$. Under the continuum hypothesis (CH) we prove $L^{p}(μ)\cong \ell^{p}(\mathfrak{c},L^{p}[0,1])$, where $\mathfrak{c}$ denotes the cardinality of the continuum, and without CH we obtain an isometric, complemented copy of $\ell^{p}(\mathfrak{c},L^{p}[0,1])$ inside $L^{p}(μ)$. In a general framework, we characterize precisely when $L^{p}(ν)\cong \ell^{p}(κ,L^{p}[0,1])$ and classify all such isometries.

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arXiv2024

Twisted Hilbert spaces defined by bi-Lipschitz maps

We obtain an infinite-dimensional cone of singular twisted Hilbert spaces $Z(φ)$ which are isomorphic to their duals but not to their conjugate duals. We do that by showing that the subset of all bi-Lipschitz maps from $[0, \infty)$ to $\mathbb{R}$ is coneable. We also provide a characterization of the Kalton-Peck space among all twisted Hilbert spaces of the form $Z(φ)$, which gives a partial answer to a conjecture of F. Cabello Sánchez and J. Castillo.

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arXiv2024

Searching for linear structures in the failure of the Stone-Weierstrass theorem

We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal{C}(L, \mathbb{K})$ that vanishes nowhere on $L$ but does not necessarily separate the points of $L$. We address the problem of finding the precise codimension of $\overline{\mathcal{A}}$ in a broad setting, which allows us to describe the lineability of $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $(\mathcal{C}(\partial{D}, \mathbb{C}) \setminus \overline{\text{Pol}(\partial{D})}) \cup \{0\}$, where $\text{Pol}(\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account.

math.FA↗

arXiv2024

New Results in Analysis of Orlicz-Lorentz spaces

In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $Λ_{φ,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions $(φ_n)_{n=1}^{\infty}$ satisfies certain order relations with respect to a given Orlicz function $φ$, the subset of the order-continuous subspace $(Λ_{φ,w})_a$ whose elements do not belong to $\bigcup_{n=1}^{\infty}Λ_{φ_n,w}$ is spaceable, and even maximal-spaceable when $φ$ satisfies the $Δ_2$-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being $(α, β)$-spaceable are provided. A similar analysis is also performed on the subset $Λ_{φ,w} \setminus (Λ_{φ,w})_a$ when $φ$ does not satisfy the $Δ_2$-condition. The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs $(φ,w)$. For a fixed Orlicz function that satisfies the $Δ_2^{\infty}$-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on $[0,1]$ for which lineability problem is not valid. Moreover, various types of $(α,β)$-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces.

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arXiv2024

Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points

We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is weakly dense; (3) The set of ccw $Δ$-points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming. We also show that there are points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $Δ$-points, meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

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arXiv2024

Algebraic genericity of certain families of nets in Functional Analysis

In Functional Analysis, certain conclusions apply to sequences, but they cannot be carried over when we consider nets. In fact, some nets, including sequences, can behave unexpectedly. In this paper we are interested in exploring the prevalence of these unusual nets in terms of linearity. Each problem is approached with different methods, which have their own interest. As our results are presented in the contexts of topological vector spaces and normed spaces, they generalize or improve a few ones in the literature. We study lineability properties of families of (1) nets that are weakly convergent and unbounded, (2) nets that fail the Banach-Steinhaus theorem, (3) nets indexed by a regular cardinal $κ$ that are weakly dense and norm-unbounded, and finally (4) convergent series which have associated nets that are divergent.

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arXiv2023

Lineability on nets and uncountable sequences of functions in measure theory

In general, some of the well known results of measure theory dealing with the convergence of sequences of functions such as the Dominated Convergence Theorem or the Monotone Convergence Theorem are not true when we consider arbitrary nets of functions instead of sequences. In this paper, we study the algebraic genericity of families of nets of functions that do not satisfy important results of measure theory, and we also analyze the particular case of uncountable sequences.

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arXiv2022

Some properties of differentiable p-adic functions

In this paper, using the tools from the lineability theory, we distinguish certain subsets of $p$-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order $r$ but not strictly differentiable of order $r+1$, (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order $α>1$, (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum.

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