Strict topologies and Banach-Steinhaus type theorems

Marian Nowak

Displaying similar documents to “Strict topologies and Banach-Steinhaus type theorems”

On Meager Additive and Null Additive Sets in the Cantor Space $2^{ω}$ and in ℝ

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let T be the standard Cantor-Lebesgue function that maps the Cantor space $2^{ω}$ onto the unit interval ⟨0,1⟩. We prove within ZFC that for every $X \subseteq 2^{ω}$ , X is meager additive in $2^{ω}$ iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in $2^{ω}$ and ℝ.

Addendum to “On Meager Additive and Null Additive Sets in the Cantor space $2^{ω}$ and in ℝ” (Bull. Polish Acad. Sci. Math. 57 (2009), 91-99)

Tomasz Weiss (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove in ZFC that there is a set $A \subseteq 2^{ω}$ and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), $H^{- 1} (X)$ is null additive in $2^{ω}$ . This settles in the affirmative a question of T. Bartoszyński.

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $(x_{k (n)}) ₙ$ such that $(T x_{k (n)}) ₙ$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

On Vitali-Hahn-Saks-Nikodym type theorems

Barbara T. Faires (1976)

Annales de l'institut Fourier

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A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $(a_{n})$ , $(b_{m})$ in $𝒜$ with $a_{n} \leq b_{m}$ for all $n, m$ , there exists an element $b$ in $𝒜$ such that $a_{n} \leq b \leq b_{n}$ for all $n$ . Let $𝒜$ denote an algebra with the property (I). It is shown that if $(μ_{n} : 𝒜 \to X)$ ( $X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_{n} μ_{n} (a)$ exists for each $a \in 𝒜$ , then $μ (a) = {lim}_{n} μ_{n} (a)$ defines a strongly additive map from $𝒜$ to $X$ the $μ_{n}^{'} s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive...

Weakly precompact operators on $C_{b} (X, E)$ with the strict topology

Juliusz Stochmal (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T : C_{b} (X, E) \to F$ . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T : C_{b} (X, E) \to F$ is weakly precompact.

Strong measure zero and meager-additive sets through the prism of fractal measures

Ondřej Zindulka (2019)

Commentationes Mathematicae Universitatis Carolinae

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We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{ω}$ is meager-additive if and only if it is $ℰ$ -additive; if $f : 2^{ω} \to 2^{ω}$ is continuous and $X$ is meager-additive, then so is $f (X)$ .

Topological properties of some spaces of continuous operators

Marian Nowak (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{β} (C_{b} (X, E), F)$ of all $(β, | | \cdot | |_{F})$ -continuous linear operators from $C_{b} (X, E)$ to F, equipped with the topology $τ_{s}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_{s}$ -compact subsets of $L_{β} (C_{b} (X, E), F)$ in terms of properties of the corresponding sets of the representing...

A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Fateme Kouchakinejad, Alexandra Šipošová (2017)

Kybernetika

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For an aggregation function $A$ we know that it is bounded by $A^{*}$ and $A_{*}$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^{*}$ is directionally convex, then $A = A^{*}$ and $A_{*}$ is linear; similarly, if $A_{*}$ is directionally concave, then $A = A_{*}$ and $A^{*}$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively. ...

On vector valued measure spaces of bounded $Φ$ -variation containing copies of $ℓ_{\infty}$

María J. Rivera (2001)

Czechoslovak Mathematical Journal

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Given a Young function $Φ$ , we study the existence of copies of $c_{0}$ and $ℓ_{\infty}$ in ${c a b v}_{Φ} (μ, X)$ and in ${c a b s v}_{Φ} (μ, X)$ , the countably additive, $μ$ -continuous, and $X$ -valued measure spaces of bounded $Φ$ -variation and bounded $Φ$ -semivariation, respectively.