Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík; Jiří Spurný

Displaying similar documents to “Descriptive properties of elements of biduals of Banach spaces”

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ ⁰_{η}$ sets which can be interesting in its own right.

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

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Let E be a Banach space and let $ℬ ₁ (B_{E *})$ and $₁ (B_{E *})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E *}$ , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $d i s t (f, ℬ ₁ (B_{E *}))$ and $d i s t (f, ₁ (B_{E *}))$ , where f is an affine function on $B_{E *}$ . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with...

On the $k$ -Baire property

Alessandro Fedeli (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this note we show the following theorem: “Let $X$ be an almost $k$ -discrete space, where $k$ is a regular cardinal. Then $X$ is $k^{+}$ -Baire iff it is a $k$ -Baire space and every point- $k$ open cover $𝒰$ of $X$ such that $card (𝒰) \leq k$ is locally- $k$ at a dense set of points.” For $k = ℵ_{0}$ we obtain a well-known characterization of Baire spaces. The case $k = ℵ_{1}$ is also discussed.

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{δ}$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{σ δ}$ sets.

Non-separable Banach spaces with non-meager Hamel basis

Taras Banakh, Mirna Džamonja, Lorenz Halbeisen (2008)

Studia Mathematica

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We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ ⁺; ∙ a dense non-meager Hamel basis if $| X | = κ^{ω} = 2^{κ}$ for some cardinal κ.

On the Dirichlet problem for functions of the first Baire class

Jiří Spurný (2001)

Commentationes Mathematicae Universitatis Carolinae

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Let $ℋ$ be a simplicial function space on a metric compact space $X$ . Then the Choquet boundary $Ch X$ of $ℋ$ is an $F_{σ}$ -set if and only if given any bounded Baire-one function $f$ on $Ch X$ there is an $ℋ$ -affine bounded Baire-one function $h$ on $X$ such that $h = f$ on $Ch X$ . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$ .

Normal P-spaces and the $G_{δ}$ -topology

R. Levy, M. D. Rice (1981)

Colloquium Mathematicae

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Uniqueness of measure extensions in Banach spaces

J. Rodríguez, G. Vera (2006)

Studia Mathematica

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Let X be a Banach space, $B \subset B_{X *}$ a norming set and (X,B) the topology on X of pointwise convergence on B. We study the following question: given two (non-negative, countably additive and finite) measures μ₁ and μ₂ on Baire(X,w) which coincide on Baire(X,(X,B)), does it follow that μ₁ = μ₂? It turns out that this is not true in general, although the answer is affirmative provided that both μ₁ and μ₂ are convexly τ-additive (e.g. when X has the Pettis Integral Property). For a Banach space Y...

The Covering Principle for Darboux Baire 1 functions

Piotr Szuca (2007)

Fundamenta Mathematicae

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We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected $G_{δ}$ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

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We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on $ℝ^{d}$ . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

Weak Baire measurability of the balls in a Banach space

José Rodríguez (2008)

Studia Mathematica

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Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of $ℓ^{\infty} (ℕ)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised...

Baire classes of complex $L_{1}$ -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

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Let $X$ be a complex $L_{1}$ -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ of the extreme points of the dual unit ball $B_{X^{*}}$ to the whole unit ball $B_{X^{*}}$ . As a corollary we show that, given $α \in [1, ω_{1})$ , the intrinsic $α$ -th Baire class of $X$ can be identified with the space of bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ when $ext B_{X^{*}}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...