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Saak Gabriyelyan

Publications and source records attributed to Saak Gabriyelyan.

At least 19 recordsLinked to original sources
arXiv2026

Baire-type properties of topological vector spaces

Burzyk, Kliś and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. Kcakol and Sánchez Ruiz proved that every sequentially complete Fréchet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $κ$-Fréchet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $κ$-Fréchet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $κ$-Fréchet--Urysohn. Although a $κ$-Fréchet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given.

math.FA↗
arXiv2026

On $κ$-Frechet-Urysohn topological groups

We characterize $κ$-Fréchet--Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $κ$-Fréchet--Urysohn iff it is locally compact, and (2) if $F$ is a closed metrizable subspace of a topological vector space (tvs) $E$ such that the quotient $E/F$ is a $κ$-Fréchet--Urysohn space, then also $E$ is a $κ$-Fréchet--Urysohn space. Consequently, the product of a $κ$-Fréchet--Urysohn tvs and a metrizable tvs is a $κ$-Fréchet--Urysohn space. Under Martin's Axiom, we construct a countable Boolean $κ$-Fréchet--Urysohn group which is not a $k_{\mathbb R}$-space.

math.GN↗
arXiv2026

Completeness and reflexivity type properties of $B_1(X)$

For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given.

math.GN↗
arXiv2025

New classes of compact-type spaces

Being motivated by the notions of $κ$-Fréchet--Urysohn spaces and $k'$-spaces introduced by Arhangel'skii, the notion of sequential spaces and the study of Ascoli spaces, we introduce three new classes of compact-type spaces. They are defined by the possibility to attain each or some of boundary points $x$ of an open set $U$ by a sequence in $U$ converging to $x$ or by a relatively compact subset $A\subseteq U$ such that $x\in \overline{A}$. Relationships of the introduced classes with the classical classes (as, for example, the classes of $κ$-Fréchet--Urysohn spaces, (sequentially) Ascoli spaces, $k_{\mathbb R}$-spaces, $s_{\mathbb R}$-spaces etc.) are given. We characterize these new classes of spaces and study them with respect to taking products, subspaces and quotients. In particular, we give new characterizations of $κ$-Fréchet--Urysohn spaces and show that each feathered topological group is $κ$-Fréchet--Urysohn. We describe locally compact abelian groups which endowed with the Bohr topology belong to one of the aforementioned classes. Numerous examples are given.

math.GN↗
arXiv2025

$κ$-spaces

We say that a Tychonoff space $X$ is a $κ$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $κ$-spaces is strictly between the class of Dieudonné complete spaces and the class of $μ$-spaces. We show that the class of $κ$-spaces has nice stability properties, that allows us to define the $κ$-completion $κX$ of $X$ as the smallest $κ$-space in the Stone--Čech compactification $βX$ of $X$ containing $X$. For a point $z\inβX$, we show that (1) if $z\in\upsilon X$, then the Dirac measure $δ_z$ at $z$ is bounded on each compact subset of $C_p(X)$, (2) $z\in κX$ iff $δ_z$ is continuous on each compact subset of $C_p(X)$ iff $δ_z$ is continuous on each compact subset of $C_p^b(X)$, (3) $z\in\upsilon X$ iff $δ_z$ is bounded on each compact subset of $C_p^b(X)$. It is proved that $κX$ is the largest subspace $Y$ of $βX$ containing $X$ for which $C_p(Y)$ and $C_p(X)$ have the same compact subsets, this result essentially generalizes a known result of R.~Haydon.

math.GN↗
arXiv2025

Functions on products $X \times Y$ with applications to Ascoli spaces, $k_{\mathbb{R}}$-spaces and $s_{\mathbb{R}}$-spaces

We prove that a Tychonoff space $X$ is (sequentially) Ascoli iff for every compact space $K$ (resp., for a convergent sequence $\mathbf{s}$), each separately continuous $k$-continuous function $Φ:X\times K\to \mathbb{R}$ is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the $μ$-completion and the Dieudonné completion of a (sequentially) Ascoli space are (sequentially) Ascoli. We give also cover-type characterizations of Ascoli spaces and suggest an easy method of construction of pseudocompact Ascoli spaces which are not $k_\mathbb{R}$-spaces and show that each space $X$ can be closely embedded into such a space. Using a different method we prove Hušek's theorem: a Tychonoff space $Y$ is a locally pseudocompact $k_\mathbb{R}$-space iff $X\times Y$ is a $k_\mathbb{R}$-space for each $k_\mathbb{R}$-space $X$. It is proved that $X$ is an $s_\mathbb{R}$-space iff for every locally compact sequential space $K$, each $s$-continuous function $f:X\times K\to\mathbb{R}$ is continuous.

math.GN↗
arXiv2025

On $k_\mathbb{R}$-spaces and $s_\mathbb{R}$-spaces

We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or an $s_\mathbb{R}$-space. It is proved that $C_p(X)$ is a $k_\mathbb{R}$-space for any space $X$ with one non-isolated point; if, in addition, $|X|$ is not sequential, then $C_p(X)$ is even an $s_\mathbb{R}$-space. Under $(CH)$, it is shown that there exists a separable metrizable space $X$ such that $C_p(X)$ is an Ascoli space but not a $k_\mathbb{R}$-space.

math.GN↗
arXiv2024

Gelfand--Phillips type properties of locally convex spaces

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of the Gelfand--Phillips property and the (coarse) Gelfand--Phillips property of order $p$ of Banach spaces, we introduce and study different types of the Gelfand--Phillips property of order $(p,q)$ (the $GP_{(p,q)}$ property) and the coarse Gelfand--Phillips property of order $p$ in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the $GP_{(p,q)}$ property. Characterizations of locally convex spaces with the introduced Gelfand--Phillips type properties are given.

math.FA↗
arXiv2024

Weakly and weak$^\ast$ $p$-convergent operators

Let $p\in[1,\infty]$. Being motivated by weakly $p$-convergent and weak$^\ast$ $p$-convergent operators between Banach spaces introduced by Fourie and Zeekoei, we introduce and study the classes of weakly $p$-convergent and weak$^\ast$ $p$-convergent operators between arbitrary locally convex spaces. Relationships between these classes of operators are given, and we show that they have ideal properties. Numerous characterizations of weakly $p$-convergent and weak$^\ast$ $p$-convergent operators are given.

math.FA↗
arXiv2024

Limited type subsets of locally convex spaces

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse $p$-limited subsets of an arbitrary locally convex space $E$. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact and relatively weakly sequentially (pre)compact sets. If $E$ is a Banach space, we show that the class of coarse $1$-limited subsets of $E$ coincides with the class of $(1,\infty)$-limited sets, and if $1<p<\infty$, then the class of coarse $p$-limited sets in $E$ coincides with the class of $p$-$(V^\ast)$ sets of Pełczyński. We also generalize a known theorem of Grothendieck.

math.FA↗
arXiv2024

Dunford--Pettis type properties of locally convex spaces

In 1953, Grothendieck introduced and studied the Dunford--Pettis property (the $DP$ property) and the strict Dunford--Pettis property (the strict $DP$ property). The $DP$ property of order $p\in[1,\infty]$ for Banach spaces was introduced by Castillo and Sanchez in 1993. Being motivated by these notions, for $p,q\in[1,\infty]$, we define the strict Dunford--Pettis property of order $p$ (the strict $DP_p$ property) and the sequential Dunford--Pettis property of order $(p,q)$ (the sequential $DP_{(p,q)}$ property). We show that a locally convex space (lcs) $E$ has the $DP$ property iff the space $E$ endowed with the Grothendieck topology $τ_{Σ'}$ has the weak Glicksberg property, and $E$ has the strict $DP_p$ property iff the space $(E,τ_{Σ'}) $ has the $p$-Schur property. We also characterize lcs with the sequential $DP_{(p,q)}$ property. Some permanent properties and relationships between Dunford--Pettis type properties are studied. Numerous (counter)examples are given. In particular, we give the first example of an lcs with the strict $DP$ property but without the $DP$ property and show that the completion of even normed spaces with the $DP$ property may not have the $DP$ property.

math.FA↗
arXiv2024

Pełczyński's type sets and Pełczyński's geometrical properties of locally convex spaces

For $1\leq p\leq q\leq\infty$ and a locally convex space $E$, we introduce and study the $(V^\ast)$ subsets of order $(p,q)$ of $E$ and the $(V)$ subsets of order $(p,q)$ of the topological dual $E'$ of $E$. Using these sets we define and study the (sequential) Pełczyński's property $V^\ast$ of order $(p,q)$, the (sequential) Pełczyński's property $V$ of order $(p,q)$, and the Pełczyński's property $(u)$ of order $p$ in the class of all locally convex spaces. To this end, we also introduce and study several new completeness type properties, weak barrelledness conditions, Schur type properties, the Gantmacher property for locally convex spaces, and $(q,p)$-summing operators between locally convex spaces. Applications to some classical function spaces are given.

math.FA↗
arXiv2023

A Banach space characterization of (sequentially) Ascoli spaces

We prove that a Tychonoff space $X$ is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space $E$, every $k$-continuous and almost $k$-compact (resp., almost $k$-sequential) map $T$ form $X$ into the Banach dual $E'$ of $E$ is continuous.

math.FA↗
arXiv2022

Compatible group topologies on a locally quasi-convex abelian group and the Mackey group problem

For a locally quasi-convex (lqc) abelian group $G$, we give the first description of all compatible group topologies on $G$ and apply this result to the Mackey group problem for lqc groups. We characterize lqc abelian groups which are Mackey groups or admit a Mackey group topology and provide a characterization of two Mackey groups whose product is Mackey. We obtain the first characterization of locally convex spaces which are Mackey groups.

math.GR↗
arXiv2021

The Josefson--Nissenzweig property for locally convex spaces

We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E',σ(E',E))\to ( E',β^\ast(E',E))$ is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space $X$, the function space $C_p(X)$ has the JNP iff there is a weak$^\ast$ null-sequence $(μ_n)_{n\inω}$ of finitely supported sign-measures on $X$ with unit norm. However, for every Tychonoff space $X$, neither the space $B_1(X)$ of Baire-1 functions on $X$ nor the free locally convex space $L(X)$ over $X$ has the JNP.

math.FA↗
arXiv2021

The Gelfand-Phillips property for locally convex spaces

We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space $E$ to be Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phillips property by standard operations over locally convex spaces is considered. Also we explore the Gelfand-Phillips property in spaces $C(X)$ of continuous functions on a Tychonoff space $X$. If $τ$ and $\mathcal T$ are two locally convex topologies on $C(X)$ such that $\mathcal T_p\subseteq τ\subseteq \mathcal T\subseteq \mathcal T_k$, where $\mathcal T_p$ is the topology of pointwise convergence and $\mathcal T_k$ is the compact-open topology on $C(X)$, then the Gelfand--Phillips property of the function space $(C(X),τ)$ implies the Gelfand--Phillips property of $(C(X),\mathcal T)$. If additionally $X$ is metrizable, then the function space $\big(C(X),\mathcal T\big)$ is Gelfand--Phillips.

math.FA↗
arXiv2021

Locally convex spaces with the strong Gelfand-Phillips property

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of finite scattered height such that for every functionally bounded set $B\subseteq X$ the complement $B\setminus K$ is finite, (b) the subspace $C^b_{\mathcal T}(X)$ of $C_{\mathcal T}(X)$ consisting of all bounded functions on $X$ has the strong Gelfand-Phillips property iff $X$ is a compact countable space of finite scattered height.

math.FA↗