The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ($R^r$)

V. Farmaki

Displaying similar documents to “The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ( $R^{r}$ )”

The Lindelöf property and σ-fragmentability

B. Cascales, I. Namioka (2003)

Fundamenta Mathematicae

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In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space ${[- 1, 1]}^{D}$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of ${[- 1, 1]}^{D}$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is...

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Radii of p-valence of certain analytic functions of order $α$ with negative coefficients

M. K. Aouf (1989)

Matematički Vesnik

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Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $C^{n}$

J. Siciak (1969)

Annales Polonici Mathematici

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On the structure of non-dentable subsets of $C (ω^{ω^{k}})$

Pericles D. Pavlakos, Minos Petrakis (2011)

Studia Mathematica

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It is shown that there is no closed convex bounded non-dentable subset K of $C (ω^{ω^{k}})$ such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C (ω^{ω^{k}})$ .

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

On Hattori spaces

A. Bouziad, E. Sukhacheva (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a subset $A$ of the real line $ℝ$ , Hattori space $H (A)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H (A)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...

On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class $R^{λ} (a, c, A, B)$ of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ${\tilde{R}}^{λ} (a, c, A, B)$ of $R^{λ} (a, c, A, B)$ and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

On the complexity of subspaces of $S_{ω}$

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

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Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^{X}$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{ω}$ is $F_{σ δ}$ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{ω}$ . We show that $S_{ω}$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

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.

Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝒫_{n}$ denote the class of analytic functions $p (z)$ of the form $p (z) = 1 + c_{n} z^{n} + c_{n + 1} z^{n + 1} + \dots$ in the open unit disc $𝕌$ . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p (z)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

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We study extension operators between spaces of continuous functions on the spaces $σ ₙ (2^{X})$ of subsets of X of cardinality at most n. As an application, we show that if $B_{H}$ is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator $T : C (λ B_{H}) \to C (μ B_{H})$ .

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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Divisors in global analytic sets

Francesca Acquistapace, A. Díaz-Cano (2011)

Journal of the European Mathematical Society

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We prove that any divisor $Y$ of a global analytic set $X \subset ℝ^{n}$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$ . We also prove that there are functions with arbitrary multiplicities along $Y$ . The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X / Y$ is not connected and $Y$ represents the zero class in $H_{q - 1}^{\infty} (X, ℤ_{2})$ then the divisor $Y$ is globally principal.

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

Connectedness of some rings of quotients of $C (X)$ with the $m$ -topology

F. Azarpanah, M. Paimann, A. R. Salehi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this article we define the $m$ -topology on some rings of quotients of $C (X)$ . Using this, we equip the classical ring of quotients $q (X)$ of $C (X)$ with the $m$ -topology and we show that $C (X)$ with the $r$ -topology is in fact a subspace of $q (X)$ with the $m$ -topology. Characterization of the components of rings of quotients of $C (X)$ is given and using this, it turns out that $q (X)$ with the $m$ -topology is connected if and only if $X$ is a pseudocompact almost $P$ -space, if and only if $C (X)$ with $r$ -topology is connected. We also...

A hit-and-miss topology for $2^{X}$ , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

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A hit-and-miss topology ( $τ_{H M}$ ) is defined for the hyperspaces $2^{X}$ , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between $τ_{H M}$ and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

On Pólya's Theorem in several complex variables

Ozan Günyüz, Vyacheslav Zakharyuta (2015)

Banach Center Publications

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Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{- k - 1}$ be its Taylor expansion at ∞, and $H_{s} (f) = d e t {(a_{k + l})}_{k, l = 0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $l i m s u p_{s \to \infty} {| H_{s} (f) |}^{1 / s ²} \leq d (K)$ , where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

Nicoleta Ularu, Nicoleta Breaz (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

A pure smoothness condition for Radó’s theorem for $α$ -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

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Let $Ω \subset ℂ^{n}$ be a bounded, simply connected $ℂ$ -convex domain. Let $α \in ℤ_{+}^{n}$ and let $f$ be a function on $Ω$ which is separately $C^{2 α_{j} - 1}$ -smooth with respect to $z_{j}$ (by which we mean jointly $C^{2 α_{j} - 1}$ -smooth with respect to $Re z_{j}$ , $Im z_{j}$ ). If $f$ is $α$ -analytic on $Ω ∖ f^{- 1} (0)$ , then $f$ is $α$ -analytic on $Ω$ . The result is well-known for the case $α_{i} = 1$ , $1 \leq i \leq n$ , even when $f$ a priori is only known to be continuous.

Displaying similar documents to “The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ( R r )”

Displaying similar documents to “The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ( $R^{r}$ )”