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On Banach spaces of regulated functions

100%
Drewnowski L.
Commentationes Mathematicae
|
2017
|
tom 57
|
nr 2
EN
For a relatively compact subset \(S\) of the real line \(\BR\), let \(R(S)\) denote the Banach space (under the sup norm) of all regulated scalar functions defined on \(S\). The purpose of this paper is to study those closed subspaces of \(R(S)\) that consist of functions that are left-continuous, right-continuous, continuous, and have a (two-sided) limit at each point of some specified disjoint subsets of \(S\). In particular, some of these spaces are represented as \(C(K)\) spaces for suitable, explicitly constructed, compact spaces \(K\).
2
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Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property

100%
Drewnowski L.
Studia Mathematica
|
1993
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tom 104
|
nr 2
125-132
EN
It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.
3
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Arrangements of series preserving their convergence or boundedness

100%
Drewnowski L.
Commentationes Mathematicae
|
2007
|
tom 47
|
nr 1
EN
For a map \(\rho\) of \(\mathbb{N}\) into itself, consider the induced transformation \(\sum_{n} x_n \mapsto \sum_{n} x_{\rho_(n)}\) of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of ρ which reduces to those found by Agnew and Pleasants (in the case of permutations) and Wituła (in the general case) as necessary and sufficient conditions for the above transformation to preserve convergence of scalar series. In the paper, the scalar case is treated first using simple Banach space methods, and then the result is easily extended to the general setting.
4
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On vector measures which have everywhere infinite variation or noncompact range

69%
Drewnowski L., Lipecki Z.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Abstract Let X be an infinite-dimensional Banach space, let Σ be a σ-algebra of subsets of a set S, and denote by ca(Σ,X) the Banach space of X-valued measures on Σ equipped with the uniform norm. We say that a nonzero μ ∈ ca(Σ,X) is everywhere of infinite variation [has everywhere noncompact range] if |μ|(A) = ∞ or 0 [{μ(E): E ∈ Σ, E ⊂ A} is not relatively compact or equals {0}] for every A ∈ Σ. Let λ be a nonatomic probability measure on Σ, and denote by ca(Σ,λ,X) the closed subspace of ca(Σ,X) consisting of λ-continuous measures. Analogously as above, we define measures in ca(Σ,λ,X) that are λ-everywhere of infinite variation or have λ-everywhere noncompact range. Using the Dvoretzky-Rogers theorem, we give two constructions of an absolutely convergent series of λ-simple measures in ca(Σ,λ,X) such that the sum of each of its subseries is λ-everywhere of infinite variation. In particular, the normed space P(λ,X) of Pettis λ-integrable functions with values in X lacks property (K), and so is incomplete. These results refine and improve some earlier results of E. Thomas, and L. Janicka and N. J. Kalton. One of the constructions also yields the existence of an infinite-dimensional closed subspace in ca(Σ,λ,X) all of whose nonzero members are λ-everywhere of infinite variation. Moreover, modifying some ideas of R. Anantharaman and K. M. Garg, we prove that the measures that are λ-everywhere of infinite variation form a dense $G_δ$-set in ca(Σ,λ,X). From this we derive an analogous result on measures that are everywhere of infinite variation and the closed subspace of ca(Σ,X) consisting of nonatomic measures. Similar results concerning measures that have [λ-] everywhere noncompact range are also established. In this case, the existence of X-valued measures with noncompact range must, however, be postulated. We also prove that the measures of σ-finite variation form an $F_{σδ}$-, but not $F_σ$-, subset of ca(Σ,λ,X), and the same is true for P(λ,X) provided that X is separable. Finally, we consider the special case when X is a Banach lattice and, for X nonisomorphic to an AL-space, we note analogues of some of the results above for positive X-valued measures on Σ.
5
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Generalized Helly spaces, continuity of monotone functions, and metrizing maps

64%
Drewnowski L., Michalak A.
Fundamenta Mathematicae
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2008
|
tom 200
|
nr 2
161-184
EN
Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes from the fact that there is a natural one-to-one correspondence between increasing functions f: [0,1] → C(K,E) (with countably many discontinuities) and continuous maps F: K → H(E) (with metrizable ranges). It leads to the investigation of general continuous metrizing maps (those with metrizable ranges), and especially of the so called separately metrizing maps, and the results obtained are then used to derive some permanence properties of the class of spaces C(K,E) with property (λ). For instance, it is shown that if K is the product of compact spaces $K_{j}$ (j ∈ J) such that each of the spaces $C(K_{j},E)$ has property (λ), so does C(K,E); and, for any compact space K, if both C(K) and a Banach lattice E have property (λ), so does C(K,E).
6
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Vector series whose lacunary subseries converge

64%
Drewnowski L., Labuda I.
Studia Mathematica
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2000
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tom 138
|
nr 1
53-80
EN
The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series $∑_n x_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries $∑_k x_{n_k}$ (i.e. those with $n_{k+1} - n_k → ∞$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence Property, or ZCP, is defined similarly though of lesser importance here. It is shown that for every ℒ-convergent series the set of all its finite sums is metrically bounded; however, it need not be topologically bounded. Next, a space with the LCP contains no copy of the space $c_0$. The converse holds for Banach spaces and, more generally, sequentially complete locally pseudoconvex spaces. However, an F-lattice of measurable functions is constructed that has both the Lebesgue and Levi properties, and thus contains no copy of $c_0$, and, nonetheless, lacks the LCP. The main (and most difficult) result of the paper is that if a Banach space E contains no copy of $c_0$ and λ is a finite measure, then the Bochner space $L_0$ (λ,e) has the LCP. From this, with the help of some Orlicz-Pettis type theorems proved earlier by the authors, the LCP is deduced for a vast class of spaces of (scalar and vector) measurable functions that have the Lebesgue type property and are "metrically-boundedly sequentially closed" in the containing $L_0$ space. Analogous results about the convergence of ℒ-convergent positive series in topological Riesz spaces are also obtained. Finally, while the LCP implies the ZCP trivially, an example is given that the converse is false, in general.
7
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Uncomplementability of the spaces of norm continuous functions in some spaces of "weakly" continuous functions

51%
Domański P., Drewnowski L.
Studia Mathematica
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1990
|
tom 97
|
nr 3
245-251
8
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On symmetric bases in nonseparable Banach spaces

32%
Drewnowski L.
Studia Mathematica
|
1987
|
tom 85
|
nr 2
157-161
9
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On uncountable unconditional bases in Banach spaces

32%
Drewnowski L.
Studia Mathematica
|
1988
|
tom 90
|
nr 3
191-196
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