Approximation theorems for compactifications

Kotaro Mine

Displaying similar documents to “Approximation theorems for compactifications”

A first countable supercompact Hausdorff space with a closed $G_{δ}$ nonsupercompact subspace

Murray G. Bell (1980)

Colloquium Mathematicae

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The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh (2012)

Studia Mathematica

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We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s} (X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if...

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

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Let X be a locally compact, separable metric space. We prove that $d i m_{T} X = i n f d i m_{L} X^{'} : X^{'} i s h o m e o m o r p h i c t o X$ , where $d i m_{L} X$ and $d i m_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.

Connectivity functions defined on $I^{n}$

Richard G. Gibson, Fred Roush (1987)

Colloquium Mathematicae

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The degree of approximation by Hausdorff means of a conjugate Fourier series

Sergiusz Kęska (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to analyze the degree of approximation of a function $\bar{f}$ that is a conjugate of a function $f$ belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

When $(E, σ (E, E^{'}))$ is a $D F$ -space?

Dorota Krassowska, Wiesƚaw Śliwa (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let $(E, t)$ be a Hausdorff locally convex space. Either $(E, σ (E, E^{'}))$ or $(E^{'}, σ (E^{'}, E))$ is a $D F$ -space iff $E$ is of finite dimension (THEOREM). This is the most general solution of the problem studied by Iyahen [2] and Radenovič [3].

The polynomial approximation of continuous functions defined on $(- \infty, \infty)$

Leetsch C. Hsu (1959)

Czechoslovak Mathematical Journal

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A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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It is shown that every Polish space X with $d i m_{T} X \geq d$ admits a compact subspace Y such that $d i m_{H} Y \geq d$ where $d i m_{T}$ and $d i m_{H}$ denote the topological and Hausdorff dimensions, respectively.

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

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We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_{γ})$ of compact operators on X such that $s u p_{γ} | | S_{γ} T | | \leq | | T | |$ and $S_{γ} \to I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

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Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x = \sum_{i = 1}^{\infty} c_{i} q^{- i}$ with integer coefficients $c_{i}$ satisfying $0 \leq c_{i} < q$ , i ≥ 1. In this case we say that $(c_{i}) = c ₁ c ₂ . . .$ is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also...

Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

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The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X *, Y^{⊥})$ has the λ-bounded approximation property. Then there exists a net $(S_{α})$ of finite-rank operators on X such that $S_{α} (Y) \subset Y$ and $| | S_{α} | | \leq λ$ for all α, and $(S_{α})$ and $(S *_{α})$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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Univoque sets for real numbers

Fan Lü, Bo Tan, Jun Wu (2014)

Fundamenta Mathematicae

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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_{i} \in 0, 1$ . We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k \geq 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\bar{(x)}$ are nowhere dense.

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

A singular initial value problem for the equation $u^{(n)} (x) = g (u (x))$

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem $u^{(n)} (x) = g (u (x))$ , $u (0) = u^{'} (0) = . . . = u^{(n - 1)} (0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int ₀^{δ} 1 / s {[s / g (s)]}^{1 / n} d s < \infty$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.