We present projective descriptions of classical spaces of functions and distributions. More precisely, we provide descriptions of these spaces by seminorms, which are defined by a combination of ...classical norms and multiplication or convolution with certain functions. These seminorms are simpler than the ones given by a supremum over bounded or compact sets.
Directionally bounded sets in c0 with equivalent norms Castillo Santos, Francisco Eduardo; Fetter, Helga; Gamboa de Buen, Berta ...
Journal of mathematical analysis and applications,
11/2014, Letnik:
419, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We study the directionally bounded sets in c0 with respect to two different families of norms and conclude that in nonreflexive spaces directional boundedness is not necessarily preserved by ...isomorphisms.
Compactness criteria in fuzzy set spaces endowed with the Lp metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing ...existing compactness criteria in fuzzy set spaces endowed with the Lp metric with the Arzelà–Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the Lp metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy numbers space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the Lp metric, respectively. Constructing completions of fuzzy set spaces with respect to the Lp metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the Lp extension. We also clarify relationships among the ten fuzzy set spaces studied here—the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the Lp metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work.
Dual of an extended locally convex space Kumar, Akshay; Jindal, Varun
Journal of mathematical analysis and applications,
11/2023, Letnik:
527, Številka:
2
Journal Article
Recenzirano
Odprti dostop
This paper aims to study the dual of an extended locally convex space. In particular, we study the weak and weak⁎ topologies for an extended locally convex space and give necessary and sufficient ...conditions for the weak topology to be metrizable or normable. Finally, we display the appropriate modification of the conventional strong topology on the dual to the extended setting.
We study totally bounded sets in the spaces of variable integrability and summability. The full characterization of these sets is given. Furthermore, the Sudakov theorem in the setting of the mixed ...Lebesgue sequence spaces is proven.
In this paper we study the existence and uniqueness of mild solutions to integro-differential equations in terms of a resolvent operator on the interval 0,2π and on the real line. Moreover, we ...characterize the spectrum of the resolvent family that solves the Volterra equation u′=Au+(a⁎Au)+f in terms of their mild periodic solutions.
Infectious diseases have a significant impact on human life, and additional efforts are required to contain them. Treatment measure is very helpful to contain the epidemic and protect infected ...persons from disease-related mortality. Therefore, we consider a SIR epidemic model with nonlocal diffusion modeled by a convolution operator and treat-age effect. We show the well-posedness of the solution to the problem that is existence, uniqueness and positivity, and boundedness. Next, we determine the corresponding basic reproduction number R0 that depends on the structure of studied bounded domain Ω∈RN, and we show its threshold role in determining the asymptotic profiles of the solution. Moreover, it is proved that the solution map has a global compact attractor. Indeed, for R0<1, there exist a Lipschitz functions Sp such that (Sp,0,0) is globally stable, which is related to the extinction scenario of the epidemic. However, for R0>1, we show the solution map is uniformly persistent and there exists a unique endemic steady state denoted (S∗,I∗,T∗) that is globally stable. The influence of the treat-age on the spatiotemporal and threshold profiles is discussed through the research.
In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically ...bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.
On the bounded sets in Cc (X) Oubbi, Lahbib
Quaestiones mathematicae,
03/2022, Letnik:
45, Številka:
2
Journal Article
Recenzirano
If X is Hausdorff topological space and C
c
(X) is the topological algebra obtained by endowing the algebra C(X) of all continuous functions on X with the topology τ
c
of uniform convergence on the ...compact subsets of X, then the set Δ(ϕ) := {g ∈ C(X) : |g(x)| ≤ ϕ(x), x ∈ X} is bounded in C
c
(X), for every non-negative ϕ ∈ C(X). In this note we deal with the question whether the collection C
+
of all such sets constitutes a base of bounded sets in C
c
(X). We give instances, where the answer is in the affirmative, and others where even the collection S
+
of the sets Δ(µ), with µ upper semi-continuous, fails to constitute such a base. We nevertheless provide situations, including the local compact case, where S
+
is a base of bounded sets in C
c
(X).
Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper ...we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions.
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