arXiv subjects

Jerzy Kakol

Publications and source records attributed to Jerzy Kakol.

11 recordsLinked to original sources

arXiv2026

On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections

We show that for infinite Tychonoff spaces X and Y the weak*-dual of Ck(X x Y) contains a basic sequence; moreover, the weak*-bidual of Ck(X) contains such a sequence as well. When X and Y are infinite compact spaces, we single out a concrete sequence ({\mu}n) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of ({\mu}n) admits a further subsequence which is strongly normal and forms a weak*-basic sequence in the dual C(X x Y)* of the Banach space C(X x Y). We also study the weak*-basic sequence problem for Frechet locally convex spaces in the class of quojections, and prove that for every quojection E the bidual E** admits a weak*-basic sequence, while a long-standing open problem asks whether the dual of every infinite-dimensional Banach space admits a basic sequence in the weak*-topology. Several examples and open questions are included, in particular for spaces C(X) and for inductive limits of Frechet spaces.

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arXiv2025

An Elementary Proof Of The Josefson-Nissenzweig Theorem For Banach Spaces C(KxL)

In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(\mu_n)$ of normalized signed measures on $K\times L$ with finite supports which converges to $0$ with respect to the weak topology of the dual Banach space $C(K\times L).$ In this paper, we return to this construction, limiting ourselves only to elementary combinatorial calculus. The main efects of this construction are additional information about the measures $\mu_n$, this is particularly clearly seen (among the others) in the resulting inequalities $$\frac{1}{2\sqrt{\pi}}\frac{1}{\sqrt{n}} <\sup_{A\times B\subset X\times Y} |\mu_n(A\times B)|<\frac{2}{\sqrt{\pi}}\frac{1}{\sqrt{n}},$$ $n\in\mathbb{N}$, with $\mu_n(f) \to_n 0$ for every $f\in C(X \times Y);$ where X and Y are arbitrary Tychonoff spaces containing infinite compact subsets, respectively. As an application we explicitly describe for Banach spaces $C(X\times Y)$ some complemented subspaces isomorphic to $c_0$. This result generalizes the classical theorem of Cembranos and Freniche, which states that for every infinite compact spaces K and L, the Banach space $C(K\times L)$ contains a complemented copy of the Banach space $c_0.$

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arXiv2024

On the product of Weak Asplund locally convex spaces

For locally convex spaces, we systematize several known equivalent definitions of Fréchet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product $\prod_{α\in A} E_α$ of any family of separable Fréchet spaces; then the product $E \times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_α)$ is an Asplund locally convex space if and only if each $E_α$ is Asplund. Analogues of both results are valid under the same assumptions, if $Y$ is the $Σ$-product of any family $(E_α)$.

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arXiv2022

When is a locally convex space Eberlein-Grothendieck?

In this paper we undertake a systematic study of those locally convex spaces $E$ such that $(E, w)$ is (linearly) Eberlein-Grothendieck, where $w$ is the weak topology of $E$. Let $C_{k}(X)$ be the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology. The main results of our paper are: (1) For a first-countable space $X$ (in particular, for a metrizable $X$) the locally convex space $(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $σ$-compact and locally compact; (2) $(C_{k}(X), w)$ is linearly Eberlein-Grothendieck if and only if $X$ is compact. We characterize $E$ such that $(E, w)$ is linearly Eberlein-Grothendieck for several other important classes of locally convex spaces $E$. Also, we show that the class of $E$ for which $(E, w)$ is linearly Eberlein-Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.

math.FA↗

arXiv2021

On linear continuous operators between distinguished spaces $C_p(X)$

As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Δ$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) \to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,α])$, where $α$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,α]) \to C_p(Y)$ is given. We also observe that for every countable ordinal $α$ all closed linear subspaces of $C_p([1,α])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Δ$-spaces we prove that a linear continuous surjection $T:C_p(X) \to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X \subset \mathbb{R}^n$).

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arXiv2021

Basic properties of $X$ for which spaces $C_p(X)$ are distinguished

In our paper [18] we showed that a Tychonoff space $X$ is a $Δ$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $Δ$ of $Δ$-spaces is invariant under the basic topological operations. We prove that if $X \in Δ$ and $φ:X \to Y$ is a continuous surjection such that $φ(F)$ is an $F_σ$-set in $Y$ for every closed set $F \subset X$, then also $Y\in Δ$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\in Δ$, then also $X\in Δ$. In particular, $σ$-product of any family of scattered Eberlein compact spaces is a $Δ$-space and the product of a $Δ$-space with a countable space is a $Δ$-space. Our results give answers to several open problems posed in \cite{KL}. Let $T:C_p(X) \longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a $Δ$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making use of obtained results, we provide a very short proof for the claim that every compact $Δ$-space has countable tightness. As a consequence, under Proper Forcing Axiom (PFA) every compact $Δ$-space is sequential. In the article we pose a dozen open questions.

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arXiv2020

A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Δ$-space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a Čech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrówka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,ω_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Δ$-spaces is invariant under basic topological operations.

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arXiv2016

Topological properties of function spaces over ordinals

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of $C_p(κ)$ and $C_k(κ)$ over ordinals $κ$. We prove that $C_p(κ)$ is always an Ascoli space, while $C_p(κ)$ is a $k_R$-space iff the cofinality of $κ$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_R$-space, namely $C_p(ω_1)$. We show that $C_k(κ)$ is Ascoli iff $cf(κ)$ is countable iff $C_k(κ)$ is metrizable.

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arXiv2016

The Ascoli property for function spaces

The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $κ$-Fréchet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $κ$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stratifiable, then $C_p(X)$ is an Ascoli space. Consequently: (a) If $X$ is a complete metrizable space, then $C_p(X)$ is Ascoli iff $X$ is scattered. (b) If $X$ is a Čech-complete Lindelöf space, then $C_p(X)$ is Ascoli iff $X$ is scattered iff $C_p(X)$ is Fréchet-Urysohn. Moreover, we prove that for a paracompact space $X$ of point-countable type the following conditions are equivalent: (i) $X$ is locally compact. (ii) $C_k(X)$ is a $k_{\mathbb R}$-space. (iii) $C_k(X)$ is an Ascoli space. The Asoli spaces $C_k(X,[0,1])$ are also studied.

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arXiv2016

Free locally convex spaces with a small base

The paper studies the free locally convex space $L(X)$ over a Tychonoff space $X$. Since for infinite $X$ the space $L(X)$ is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for $L(X)$ is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a $\mathfrak{G}$-base. A space $X$ has a {\em $\mathfrak{G}$-base} if for every $x\in X$ there is a base $\{ U_α: α\in\mathbb{N}^\mathbb{N}\}$ of neighborhoods at $x$ such that $U_β\subseteq U_α$ whenever $α\leqβ$ for all $α,β\in\mathbb{N}^\mathbb{N}$, where $α=(α(n))_{n\in\mathbb{N}}\leq β=(β(n))_{n\in\mathbb{N}}$ if $α(n)\leqβ(n)$ for all $n\in\mathbb{N}$. We show that if $X$ is an Ascoli $σ$-compact space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ admits an Ascoli uniformity $\mathcal{U}$ with a $\mathfrak{G}$-base. We prove that if $X$ is a $σ$-compact Ascoli space of $\mathbb{N}^\mathbb{N}$-uniformly compact type, then $L(X)$ has a $\mathfrak{G}$-base. As an application we show: (1) if $X$ is a metrizable space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ is $σ$-compact, and (2) if $X$ is a countable Ascoli space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ has a $\mathfrak{G}$-base.

math.FA↗

arXiv2015

On topological properties of the weak topology of a Banach space

Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space $E$ and its closed unit ball $B$, both endowed with the weak topology of $E$. We show that $B$ has the Pytkeev property if and only if $E$ in the norm topology contains no isomorphic copy of $\ell_1$, while $E$ has the Pytkeev property if and only if it is finite-dimensional. We extend Schlüchtermann and Wheeler's result by showing that $B$ is a (separable) metrizable space if and only if it has countable $cs^\ast$-character and is a $k$-space. As a corollary we obtain that $B$ is Polish if and only if it has countable $cs^\ast$-character and is Čech-complete, that supplements a result of Edgar and Wheeler.

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