Search results for "Complete metric space"

Injective spaces of real-valued functions with the baire property

1995

Generalizing the technique used by S.A. Argyros in [3], we give a lemma from which certain Banach spaces are shown to be non-injective. This is applied mainly to study the injectivity of spaces of real-valued Borel functions and functions with the Baire property on a topological space. The results obtained in this way do not follow from previous works about this matter.

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.

Berinde mappings in orbitally complete metric spaces

2011

Abstract We give a fixed point theorem for a self-mapping satisfying a general contractive condition of integral type in orbitally complete metric spaces. Some examples are given to illustrate our obtained result.

VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE

2003

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Kothe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions ‖f‖X : Ω → [0,∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the pvariation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE(X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X))∗ = VE′ (X ∗), that VE(X) can be identified with space of cone absolutely summing operators fr…

On the stability of the Bohl — Brouwer — Schauder Theorem

1996

A note on the closed graph theorem

1977

A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

2018

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we prove a new Cartan-type property for the fine topology in the case $p=1$. Then we use this property to prove the existence of $1$-finely open \emph{strict subsets} and \emph{strict quasicoverings} of $1$-finely open sets. As an application, we study fine Newton-Sobolev spaces in the case $p=1$, that is, Newton-Sobolev spaces defined on $1$-finely open sets.

Covering by discrete and closed discrete sets.

2008

Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $\sigma$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.

On complete metric spaces containing the Sierpinski curve

1998

It is proved that a complete metric space topologically contains the Sierpiński universal plane curve if and only if it has a subset with so-called bypass property, i.e. it has a subset K K containing an arc such that for each a ∈ K a\in K and for each open arc A ⊂ K A\subset K with a ∈ A a\in A , there exists an arbitrary small arc in K ∖ { a } K\setminus \{a\} joining the two components of A ∖ { a } A\setminus \{a\} .

Measures with predetermined regularity and inhomogeneous self-similar sets

2016

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding self-similar set $E$ and the condensation set $C$. If the Assouad dimension of $C$ is strictly smaller than the Assouad dimension of $E$, then the upper regularity dimens…