A characterization of almost continuity and weak continuity

Chrisostomos Petalas; Theodoros Vidalis

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Displaying similar documents to “A characterization of almost continuity and weak continuity”

On the class of positive almost weak $^{☆}$ Dunford-Pettis operators

Abderrahman Retbi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce and study the class of almost weak $^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP $^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak $^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

Induced almost continuous functions on hyperspaces

Alejandro Illanes (2006)

Colloquium Mathematicae

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For a metric continuum X, let C(X) (resp., $2^{X}$ ) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^{f} : 2^{X} \to 2^{Y}$ be the induced functions given by $C (f) (A) = c l_{Y} (f (A))$ and $2^{f} (A) = c l_{Y} (f (A))$ . In this paper, we prove that: • If $2^{f}$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1]...

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

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We recall some classical results relating normality and some natural weakenings of normality in $Ψ$ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$ -sets, $λ$ -sets and $σ$ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $ℵ_{0}$ -separated. This new class fits between $λ$ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being...

Asymmetric tie-points and almost clopen subsets of $ℕ^{*}$

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

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A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of $ℕ^{*}$ . One especially important application, due to Veličković, was to the existence of nontrivial involutions on $ℕ^{*}$ . A tie-point of $ℕ^{*}$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost...

The almost Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Colloquium Mathematicae

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Let G be a metrizable, compact abelian group and let Λ be a subset of its dual group Ĝ. We show that $C_{Λ} (G)$ has the almost Daugavet property if and only if Λ is an infinite set, and that $L ¹_{Λ} (G)$ has the almost Daugavet property if and only if Λ is not a Λ(1) set.

Almost Prüfer v-multiplication domains and the ring $D + X D_{S} [X]$

Qing Li (2010)

Colloquium Mathematicae

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This paper is a continuation of the investigation of almost Prüfer v-multiplication domains (APVMDs) begun by Li [Algebra Colloq., to appear]. We show that an integral domain D is an APVMD if and only if D is a locally APVMD and D is well behaved. We also prove that D is an APVMD if and only if the integral closure D̅ of D is a PVMD, D ⊆ D̅ is a root extension and D is t-linked under D̅. We introduce the notion of an almost t-splitting set. $D^{(S)}$ denotes the ring $D + X D_{S} [X]$ , where S is a multiplicatively...

On some representations of almost everywhere continuous functions on $ℝ^{m}$

Ewa Strońska (2006)

Colloquium Mathematicae

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It is proved that the following conditions are equivalent: (a) f is an almost everywhere continuous function on $ℝ^{m}$ ; (b) f = g + h, where g,h are strongly quasicontinuous on $ℝ^{m}$ ; (c) f = c + gh, where c ∈ ℝ and g,h are strongly quasicontinuous on $ℝ^{m}$ .

Characterising weakly almost periodic functionals on the measure algebra

Matthew Daws (2011)

Studia Mathematica

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Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{W A P}$ . In this paper, we investigate properties of $K_{W A P}$ . We present a short proof that $K_{W A P}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens...

On weakly $θ$ -continuous functions

F. Commaroto, T. Noiri (1986)

Matematički Vesnik

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Functionals on Banach Algebras with Scattered Spectra

H. S. Mustafayev (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let A be a complex, commutative Banach algebra and let $M_{A}$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space $M_{A}$ is scattered (i.e., $M_{A}$ has no nonempty perfect subset). (b) Weakly almost periodic...

A note on an approximative scheme of finding almost homoclinic solutions for Newtonian systems

Robert Krawczyk (2014)

Banach Center Publications

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In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system $q ̈ + \nabla_{q} V (t, q) = f (t)$ , where t ∈ ℝ, q ∈ ℝⁿ. It is assumed that a potential V: ℝ × ℝⁿ → ℝ is C¹-smooth and its gradient map $\nabla_{q} V : ℝ \times ℝ ⁿ \to ℝ ⁿ$ is bounded with respect to t. Moreover, a forcing term f: ℝ → ℝⁿ is continuous, bounded and square integrable. We will show that the approximative scheme due to J. Janczewska (see [J2]) for a time periodic potential extends to our case.

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

Remotely $c$ -almost periodic type functions in $ℝ^{n}$

Marco Kostić, Vipin Kumar (2022)

Archivum Mathematicum

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In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$ -almost periodic functions in $ℝ^{n},$ slowly oscillating functions in $ℝ^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$ -almost periodic...

Characterizing matrices with $𝐗$ -simple image eigenspace in max-min semiring

Ján Plavka, Sergeĭ Sergeev (2016)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability)...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...