On the zero-set of 2-homogeneous polynomials in Banach spaces

2018

ABSTRACTGiving a partial answer to a conjecture formulated by Aron, Boyd, Ryan and Zalduendo, we show that if a real Banach space X is not linearly and continuously injected into a Hilbert space, t...

The Separable Complementation Property and Mrówka Compacta

2017

We study the separable complementation property for $C(K_{\cal A})$ spaces when $K_{\cal A}$ is the Mr\'owka compact associated to an almost disjoint family ${\cal A}$ of countable sets. In particular we prove that, if ${\cal A}$ is a  generalized ladder system,  then $C(K_{\cal A})$ has the separable complementation property ($SCP$ for short) if and only if it has the controlled version of this property. We also show that, when ${\cal A}$ is  a maximal generalized ladder system, the space $C(K_{\cal A})$ does not enjoy the $SCP$.

Rolle's Theorem Fails in l2

1996

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

2002

AbstractIn an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.

A property of connected Baire spaces

1997

Abstract We give a topological version of a classical result of F. Sunyer Balaguer's on a local characterization of real polynomials. This is done by studying a certain property on a class of connected Baire spaces, thus allowing us to obtain a local characterization of repeated integrals of analytic maps on Banach spaces.

El control de los roedores: revisión de los rodenticidas registrados en el ámbito de la sanidad ambiental en España

2004

Josefa Moreno Marí, Jesús López Ferrer y Ricardo Jiménez Peydró josefa (moreno@uv.es) Actualmente se precisa del empleo de los llamados rodenticidas, definidos como productos biocidas empleados para el control de roedores. En la reciente transposición de la Directiva de Biocidas a través del Real Decreto 1054/2002, los rodenticidas se incluyen en el Grupo Principal 3, Tipo de Producto 14. Se analiza la situación actual de los rodenticidas en el Registro Oficial de Plaguicidas. El estudio se ha realizado a partir de los datos que figuran en la base de datos del Registro Oficial de Plaguicidas de Uso en Salud Pública de España para los rodenticidas (ingredientes activos técnicos) y formulados…

A Short Proof that Some Mappings of the Unit Ball of ℓ2 Are Never Nonexpansive

2020

It is known that some particular self-mappings of the closed unit ball Bl2 of l2 with no fixed points cannot be nonexpansive with respect to any renorming of l2. We give here a short proof of this ...

ON TOPOLOGICAL SPACES WITH A UNIQUE QUASI-PROXIMITY

1994

Abstract Trying to solve the question of whether every T 1 topological space with a unique compatible quasi-proximity should be hereditarily compact, we show that it is true for product spaces as well as for locally hereditarily Lindelof spaces.

Quasi-pseudometric properties of the Nikodym-Saks space

2003

[EN] For a non-negative finite countably additive measure μ defined on the σ-field Σ of subsets of Ω, it is well known that a certain quotient of Σ can be turned into a complete metric space Σ (Ω), known as the Nikodym-Saks space, which yields such important results in Measure Theory and Functional Analysis as Vitali-Hahn-Saks and Nikodym's theorems. Here we study some topological properties of Σ (Ω) regarded as a quasi-pseudometric space.

On the Zero-Set of Real Polynomials in Non-Separable Banach Spaces

2007

We show constructively that every homogeneous polynomial that is weakly continuous on the bounded subsets of a real Banach space whose dual is not weak ∗ separable admits a closed linear subspace whose dual is not weak ∗ -separable either where the polynomial vanishes. We also prove that the same can be said for vectorvalued polynomials. Finally, we study the validity of this result for continuous 2homogeneous polynomials.

A Vector Approach to Euler's Line of a Triangle

1992

Among the many interesting properties that triangles possess there is one that quickly attracts our curiosity and stays easily in our mind: The centroid, circumcentre and orthocentre all lie in a common line (Euler's Line). An elementary simple proof can be obtained using metric and affine properties of the points involved, [1]. Our aim here is to illustrate a proof using vectors. We identify points in the plane with their position vectors. It is easy to see that the centroid G of the triangle ABC is given by the identity

Zeroes of real polynomials on C(K) spaces

2007

AbstractFor a compact Hausdorff topological space K, we show that the function space C(K) must satisfy the following dichotomy: (i) either it admits a positive definite continuous 2-homogeneous real-valued polynomial, (ii) or every continuous 2-homogeneous real-valued polynomial vanishes in a non-separable closed linear subspace. Moreover, if K does not have the Countable Chain Condition, then every continuous polynomial, not necessarily homogeneous and with arbitrary degree, has constant value in an isometric copy of c0(Γ), for some uncountable Γ.

A note on monolithic scattered compacta

2015

Abstract For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP, for short) is that the dual unit ball B E ⁎ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X turns out to be sufficient for C 0 ( X ) to enjoy the CSCP.

An approximate Rolle's theorem for polynomials of degree four in a Hilbert space

2005

We show that the fourth degree polynomials that satisfy Rolle’s Theorem in the unit ball of a real Hilbert space are dense in the space of polynomials that vanish in the unit sphere. As a consequence, we obtain a sort of approximate Rolle’s Theorem for those polynomials.

Almost disjoint families of countable sets and separable complementation properties

2012

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP …