Convex compactness property in certain spaces of measures

The strong convex compactness property is important for property persistence of operator semigroups under perturbations. It has been investigated in the linear setting. In this paper, we are concerned with the property in the nonlinear setting. We prove that the following spaces of (nonlinear) operators enjoy the strong convex compactness property: the space of compact operators, the space of completely continuous operators, the space of weakly compact operators, the space of conditionally weakly compact operators, the space of weakly completely continuous operators, the space of demicontinuous operators, the space of weakly continuous operators and the space of strongly continuous operators. Moreover, we prove the property persistence of operator semigroups under nonlinear perturbation.

Download Full-text

Locally convex topologies and the convex compactness property

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100048222 ◽

1974 ◽

Vol 75 (1) ◽

pp. 45-50 ◽

Cited By ~ 5

Author(s):

E. G. Ostling ◽

A. Wilansky

Keyword(s):

Compact Set ◽

Compactness Property ◽

Original Topology ◽

Krein’S Theorem ◽

Locally Convex Topologies ◽

The Subject ◽

Convex Compactness ◽

Convex Closure ◽

Bounded Sets ◽

Locally Convex

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

Download Full-text

Banach-Dieudonné Theorem Revisited

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700003487 ◽

2003 ◽

Vol 75 (1) ◽

pp. 69-83 ◽

Cited By ~ 6

Author(s):

Montserrat Bruguera ◽

Elena Martín-Peinador

Keyword(s):

Topological Group ◽

Locally Convex Spaces ◽

Compact Open Topology ◽

Compactness Property ◽

Character Group ◽

Abelian Topological Group ◽

Continuous Convergence ◽

Convergence Structure ◽

Convex Compactness ◽

Locally Convex

AbstractWe prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

Download Full-text

Probability logic: A model-theoretic perspective

Journal of Logic and Computation ◽

10.1093/logcom/exaa066 ◽

2020 ◽

Author(s):

M Pourmahdian ◽

R Zoghifard

Keyword(s):

Characterization Theorem ◽

Expressive Power ◽

Probability Logic ◽

Theoretic Analysis ◽

Compactness Property ◽

Probability Models ◽

Finitely Additive Probability ◽

Additive Probability ◽

Basic Probability ◽

Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

Download Full-text

DISCRETE COMPACTNESS FOR p AND hp 2D EDGE FINITE ELEMENTS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202503003070 ◽

2003 ◽

Vol 13 (11) ◽

pp. 1673-1687 ◽

Cited By ~ 15

Author(s):

DANIELE BOFFI ◽

LESZEK DEMKOWICZ ◽

MARTIN COSTABEL

Keyword(s):

Finite Element ◽

Finite Elements ◽

Model Problem ◽

Finite Element Approximation ◽

Stability Estimate ◽

Element Approximation ◽

Dimensional Model ◽

Compactness Property ◽

One Dimensional ◽

Discrete Compactness

In this paper we discuss the hp edge finite element approximation of the Maxwell cavity eigenproblem. We address the main arguments for the proof of the discrete compactness property. The proof is based on a conjectured L2 stability estimate for the involved polynomial spaces which has been verified numerically for p≤15 and illustrated with the corresponding one dimensional model problem.

Download Full-text