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Start Over Descriptor "Weak topology" Topic banach space
300 results on '"Weak topology"'

1. Weakly compact sets in Orlicz–Bochner sequence spaces.

Author

Gong, Wanzhong, Shi, Siyu, and Shi, Zhongrui

Abstract

In this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces l(Φ)(X)$l_{(\Phi)}(X)$ without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from X$X$ to l(Φ)(X)$l_{(\Phi)}(X)$, for example, l∞(X)$l_{\infty }(X)$. [ABSTRACT FROM AUTHOR]

Published
2024
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2. Weakly Compact Sets and Riesz Representation Theorem in Musielak Sequence Spaces.

Author

Gong, Wan Zhong, Shi, Si Yu, and Shi, Zhong Rui

Subjects
*SEQUENCE spaces, *ORLICZ spaces, *COMPACT operators, *FUNCTION spaces, *BANACH spaces
Abstract

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

Published
2024
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3. On a Cauchy Problem in Banach Spaces.

Author

Cheng, Lixin and Zhang, Wen

Subjects
BANACH spaces, EXISTENCE theorems, COMMERCIAL space ventures, ORDINARY differential equations, CAUCHY problem, DIFFERENTIABLE functions
Abstract

In this paper, we consider solvability of the Cauchy initial problem d x / dt = f (t , x) , x (0) = x 0 in a Banach space X. As a result, we generalize Peano's existence theorem in the following manner: For every Banach space X, the problem always has a solution x ∈ C 1 ([ 0 , a ] , X) for all a > 0 under the assumption that f : R + ⊕ X → X is weak-to-weak continuous on some bounded set with a relatively weakly compact range. We also show that for any infinite dimensional reflexive Banach space X with an unconditional basis, in particular, ℓ p and L p (Ω , ∑ , μ) ( 1 < p < ∞ and (Ω , ∑ , μ) is σ -finite), and for all a > 0 there is a bounded nowhere locally Lipschitz function f : R + ⊕ X → X which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution x ∈ C 1 ([ 0 , a ] , X) . [ABSTRACT FROM AUTHOR]

Published
2023
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4. A note on Banach spaces E for which Ew is homeomorphic to Cp(X)

Author

Ka̧kol, Jerzy, Leiderman, Arkady, and Michalak, Artur

Abstract

C p (X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by E w . It is unknown whether C p (K) and C (L) w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that E w is homeomorphic to the space C p (X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets X n , n ∈ ω , where at least one component X n is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ 1 . [ABSTRACT FROM AUTHOR]

Published
2022
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5. Weakly compact sets in Orlicz sequence spaces.

Author

Shi, Siyu, Shi, Zhongrui, and Wu, Shujun

Abstract

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of , we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set A in Riesz spaces lp (1 < p < ∞) is relatively sequential weakly compact if and only if it is normed bounded, that says sup . The result again confirms the conclusion of the Banach-Alaoglu theorem. [ABSTRACT FROM AUTHOR]

Published
2021
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6. Duality

Author

Bogachev, V. I., Smolyanov, O. G., Gallagher, Isabelle, Editor-in-chief, Kim, Minhyong, Editor-in-chief, Axler, Sheldon, Series editor, Braverman, Mark, Series editor, Chudnovsky, Maria, Series editor, Güntürk, C. Sinan, Series editor, Le Bris, Claude, Series editor, Massart, Pascal, Series editor, Pinto, Alberto A, Series editor, Pinzari, Gabriella, Series editor, Ribet, Ken, Series editor, Schilling, René, Series editor, Souganidis, Panagiotis, Series editor, Süli, Endre, Series editor, Weinberger, Shmuel, Series editor, Zilber, Boris, Series editor, Bogachev, V.I., and Smolyanov, O.G.

Published
2017
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11. Basic Functional Analysis

Author

Peypouquet, Juan, Pardalos, Panos M., Series editor, Pintér, János D., Series editor, Robinson, Stephen M., Series editor, Terlaky, Tamás, Series editor, Thai, My T., Series editor, and Peypouquet, Juan

Published
2015
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12. Introduction

Author

Blanchard, Philippe, Brüning, Erwin, Boutet de Monvel, Anne, Editor-in-chief, Kaiser, Gerald, Editor-in-chief, Blanchard, Philippe, and Brüning, Erwin

Published
2015
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14. Descent Methods

Author

Reich, Simeon, Zaslavski, Alexander J., Alladi, Krishnaswami, Series editor, Farkas, Hershel M., Series editor, Reich, Simeon, and Zaslavski, Alexander J.

Published
2014
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15. Best Approximation

Author

Reich, Simeon, Zaslavski, Alexander J., Alladi, Krishnaswami, Series editor, Farkas, Hershel M., Series editor, Reich, Simeon, and Zaslavski, Alexander J.

Published
2014
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16. Consequences of Convexity

Author

Bowers, Adam, Kalton (deceased), Nigel J., Axler, Sheldon, Series editor, Capasso, Vincenzo, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczynski, Wojbor A., Series editor, Bowers, Adam, and Kalton, Nigel J.

Published
2014
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19. On the fixed point property for orbital contractions in Banach spaces

Author

Najeh Redjel and Abdelkader Dehici

Subjects
Mathematics::Functional Analysis, Pure mathematics, Algebra and Number Theory, Weak topology, Functional analysis, Convergence in measure, Applied Mathematics, Banach space, Regular polygon, Fixed point, Fixed-point property, Special functions, Geometry and Topology, Analysis, Mathematics
Abstract

In this paper, we investigate some fixed point properties of nonexpansive mappings with respect to orbits in Banach spaces X equipped with abstract linear topologies $$\tau $$ for which the norm $$\Vert .\Vert _X$$ is sequentially lower semicontinuous relative to $$\tau $$ . The important situations where $$\tau $$ is the weak topology on X or $$\tau $$ is the topology of the convergence in measure on $$L^{1}(\mu )$$ will be considered. Next, we study the existence of common fixed points for actions of jointly continuous left amenable semitopological semigroups of nonexpansive and Kannan mappings with respect to orbits on compact convex subsets of Banach spaces.

Published
2021

20. The Tubby Torus as a Quotient Group

Author

Sidney A. Morris

Subjects
torus, tubby torus, separable quotient problem, locally convex space, nuclear space, banach space, pontryagin duality, weak topology, Mathematics, QA1-939
Abstract

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach−Mazur separable quotient problem.

Published
2020
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22. Preliminaries

Author

Migórski, Stanisław, Ochal, Anna, Sofonea, Mircea, Migórski, Stanisław, Ochal, Anna, and Sofonea, Mircea

Published
2013
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23. Banach spaces

Author

Clarke, Francis, Axler, Sheldon, Series editor, Ribet, Kenneth, Series editor, and Clarke, Francis

Published
2013
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26. Overview

Author

Kąkol, Jerzy, Kubiś, Wiesław, López-Pellicer, Manuel, Kąkol, Jerzy, Kubiś, Wiesław, and López-Pellicer, Manuel

Published
2011
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28. Weakly Analytic Spaces

Author

Kąkol, Jerzy, Kubiś, Wiesław, López-Pellicer, Manuel, Kąkol, Jerzy, Kubiś, Wiesław, and López-Pellicer, Manuel

Published
2011
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36. Introduction

Author

Moltó, Aníbal, Orihuela, José, Troyanski, Stanimir, Valdivia, Manuel, Morel, J.-M., Series Editor, Takens, F., Series Editor, Teissier, B., Series Editor, Moltó, Aníbal, Orihuela, José, Troyanski, Stanimir, and Valdivia, Manuel

Published
2009
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37. σ-Continuous and Co-σ-continuous Maps

Author

Moltó, Aníbal, Orihuela, José, Troyanski, Stanimir, Valdivia, Manuel, Morel, J.-M., Series Editor, Takens, F., Series Editor, Teissier, B., Series Editor, Moltó, Aníbal, Orihuela, José, Troyanski, Stanimir, and Valdivia, Manuel

Published
2009
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41. On the sum of simultaneously proximinal sets

Author

Wen Zhang, Yuqi Sun, Zheming Zheng, and Longfa Sun

Subjects
Statistics and Probability, Matematik, Algebra and Number Theory, Weak topology, Regular polygon, Banach space, Linear subspace, Functional Analysis (math.FA), Mathematics - Functional Analysis, Combinatorics, Compact space, FOS: Mathematics, simultaneous strong proximinality,simultaneous approximative compactness,compact sets,Banach space, Geometry and Topology, Mathematics, Analysis
Abstract

In this paper, we show that the sum of a compact convex subset and a simultaneously $\tau$-strongly proximinal convex subset (resp. simultaneously approximatively $\tau$-compact convex subset) of a Banach space X is simultaneously $\tau$-strongly proximinal (resp. simultaneously approximatively $\tau$-compact ), and the sum of a weakly compact convex subset and a simultaneously approximatively weakly compact convex subset of X is still simultaneously approximatively weakly compact, where $\tau$ is the norm or the weak topology. Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.

Published
2021

47. Linear Functionals and Linear Operators

Author

Başar, Tamer, editor, Åström, Karl Johan, editor, Chen, Han-Fu, editor, Helton, William, editor, Isidori, Alberto, editor, Kokotović, Petar V., editor, Kurzhanski, Alexander, editor, Vincent Poor, H., editor, Soner, Mete, editor, Kurdila, Andrew J., and Zabarankin, Michael

Published
2005
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48. A countable dense homogeneous topological vector space is a Baire space

Author

Witold Marciszewski, Mikołaj Krupski, and Tadeusz Dobrowolski

Subjects
Pure mathematics, Primary: 54C35, 54E52, 46A03, Secondary: 22A05, Weak topology, Function space, Applied Mathematics, General Mathematics, General Topology (math.GN), Banach space, Mathematics::General Topology, Baire space, Topological space, Topological vector space, Cantor set, Mathematics::Logic, Metrization theorem, FOS: Mathematics, Mathematics - General Topology, Mathematics
Abstract

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products., Comment: slightly modified and expanded version

Published
2021

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