On copies of the null sequence Banach space in some vector measure spaces

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

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The bounded vector measure associated to a conical measure and pettis differentiability

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002251 ◽

2001 ◽

Vol 70 (1) ◽

pp. 10-36

Author(s):

L. Rodriguez-Piazza ◽

M. C. Romero-Moreno

Keyword(s):

Vector Measure ◽

Locally Convex Space ◽

Finite Intersection ◽

Vector Measures ◽

Weak Space ◽

Countably Additive Vector Measure ◽

Differentiable Vector ◽

Locally Convex ◽

Additive Vector ◽

Additive Vector Measure

AbstractLet X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

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Unconditionally converging polynomials on Banach spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007314x ◽

1995 ◽

Vol 117 (2) ◽

pp. 321-331 ◽

Cited By ~ 2

Author(s):

Manuel Gonz´lez ◽

Joaquín M. Gutiérrez

Keyword(s):

Banach Space ◽

Weak Convergence ◽

Banach Spaces ◽

Weak Topology ◽

Linear Operators ◽

Null Sequence ◽

Good Tool ◽

Image Position ◽

Weakly Unconditionally Cauchy Series ◽

Cauchy Series

In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

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Integral Representation by Boundary Vector Measures

Canadian Mathematical Bulletin ◽

10.4153/cmb-1982-022-4 ◽

1982 ◽

Vol 25 (2) ◽

pp. 164-168 ◽

Cited By ~ 1

Author(s):

Paulette Saab

Keyword(s):

Banach Space ◽

Integral Representation ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Linear Subspace ◽

Vector Measure ◽

Vector Measures ◽

Boundary Vector

AbstractIn this paper we show that if X is a compact Hausdorff space, A is an arbitrary linear subspace of C(X, C), and if E is a Banach space, then each element L of (A ⊗ E)* can be represented by a boundary E*-valued vector measure of the same norm as L.

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Almost-Lp-projections and Lp isomorphisms

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023921 ◽

1989 ◽

Vol 113 (1-2) ◽

pp. 13-25 ◽

Cited By ~ 1

Author(s):

Michael Cambern ◽

Krzysztof Jarosz ◽

Georg Wodinski

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Dimensional Case ◽

Geometric Condition ◽

Finite Dimensional ◽

Measure Spaces ◽

Finite Dimensional Case

SynopsisLp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.

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Non-complemented Spaces of Operators, Vector Measures, and co

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-084-0 ◽

2012 ◽

Vol 55 (3) ◽

pp. 548-554 ◽

Cited By ~ 1

Author(s):

Paul Lewis ◽

Polly Schulle

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Compact Operators ◽

Vector Measures ◽

Spaces Of Operators ◽

Space Of Compact Operators

AbstractThe Banach spaces L(X, Y), K(X,Y), Lw* (X*, Y), and Kw* (X*, Y) are studied to determine when they contain the classical Banach spaces co or ℓ∞. The complementation of the Banach space K(X, Y) in L(X, Y) is discussed as well as what impact this complementation has on the embedding of co or ℓ∞ in K(X, Y) or L(X, Y). Results of Kalton, Feder, and Emmanuele concerning the complementation of K(X, Y) in L(X, Y) are generalized. Results concerning the complementation of the Banach space Kw* (X*, Y) in Lw* (X*, Y) are also explored as well as how that complementation affects the embedding of co or ℓ∞ in Kw* (X*, Y) or Lw* (X*, Y). The ℓp spaces for 1 = p < ∞ are studied to determine when the space of compact operators from one ℓp space to another contains co. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

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Nonstandard Methods in Measure Theory

Abstract and Applied Analysis ◽

10.1155/2014/851080 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Grigore Ciurea

Keyword(s):

Banach Spaces ◽

Nonstandard Analysis ◽

Measure Theory ◽

New Approach ◽

Vector Measures ◽

Measure Spaces

Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give a new approach to the study of the extension of vector measures. Applications of our results lead to simple new proofs for theorems of classical measure theory. The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandard proof) to obtain in an unified way some notable theorems which have been obtained by Fox, Brooks, Ohba, Diestel, and others. The methods of proof are quite different from those used by previous authors, and most of them are realized by means of nonstandard analysis.

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On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

Annales Mathematicae Silesianae ◽

10.2478/amsil-2020-0022 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Piotr Mikusiński ◽

John Paul Ward

Keyword(s):

Banach Space ◽

Bounded Variation ◽

Vector Measure ◽

Measurable Space ◽

Unique Extension ◽

Natural Conditions ◽

Vector Measures ◽

Nikodym Property

AbstractIf \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

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Nonlinear Weakly Sequentially Continuous Embeddings Between Banach Spaces

International Mathematics Research Notices ◽

10.1093/imrn/rny181 ◽

2018 ◽

Vol 2020 (18) ◽

pp. 5506-5533 ◽

Cited By ~ 1

Author(s):

B M Braga

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Uniform Convexity ◽

Null Sequence ◽

Uniform Embedding ◽

Spreading Model ◽

Continuous Map ◽

Weakly Sequentially Continuous

Abstract In these notes, we study nonlinear embeddings between Banach spaces that are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $$ \|e_1+\ldots+e_k\|_{\overline{\delta}_Y}\lesssim\|e_1+\ldots+e_k\|_S,$$where $\overline{\delta }_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.

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Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽

10.3390/axioms10030150 ◽

2021 ◽

Vol 10 (3) ◽

pp. 150

Author(s):

Andriy Zagorodnyuk ◽

Anna Hihliuk

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Entire Functions ◽

Dimensional Algebra ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Linear Subspaces ◽

Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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