Wyniki wyszukiwania

1

Remarks on measurable Boolean algebras and sequential cardinals

100%

Plebanek G.

Fundamenta Mathematicae

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1993

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tom 143

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nr 1

11-22

EN

The paper offers a generalization of Kalton-Roberts' theorem on uniformly exhaustive Maharam's submeasures to the case of arbitrary sequentially continuous functionals. Applying the result one can reduce the problem of measurability of sequential cardinals to the question whether sequentially continuous functionals are uniformly exhaustive.

2

An example of a non-topologizable algebra

63%

Frankiewicz R., Plebanek G.

Studia Mathematica

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1995

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tom 116

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nr 1

85-87

EN

We present an example of an algebra that is generated by $ω_1$ elements, and cannot be made a topological algebra. This answers a problem posed by W. Żelazko.

3

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

51%

Gabriyelyan S., Kąkol J., Plebanek G.

Studia Mathematica

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2016

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tom 233

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nr 2

119-139

EN

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset 𝒦 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 theory and measure-theoretic properties of ℓ₁, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ₁, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_{ℝ}$-space, (iv) $B_{w}$ is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ₁ iff each closed and convex bounded subset of F is Ascoli in the weak topology. Moreover we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff F is either finite-dimensional or isomorphic to $𝕂^{ℕ}$, where 𝕂 ∈ {ℝ,ℂ}.

Ograniczanie wyników

2 Studia Mathematica

1 Fundamenta Mathematicae

3 Plebanek G.

1 Frankiewicz R.

1 Gabriyelyan S.

1 Kąkol J.

1 2016

1 1995

1 1993

Remarks on measurable Boolean algebras and sequential cardinals

An example of a non-topologizable algebra

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