The Experts below are selected from a list of 7776 Experts worldwide ranked by ideXlab platform
Saak Gabriyelyan - One of the best experts on this subject based on the ideXlab platform.
-
on reflexivity and the ascoli property for free Locally Convex Spaces
Journal of Pure and Applied Algebra, 2020Co-Authors: Saak GabriyelyanAbstract:Abstract Let L ( X ) be the free Locally Convex Space over a Tychonoff Space X. If X is Dieudonne complete (for example, metrizable), then L ( X ) is a reflexive group if and only if X is discrete. We prove also that L ( X ) is an Ascoli Space if and only if X is a countable discrete Space.
-
Locally Convex properties of free Locally Convex Spaces
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Saak GabriyelyanAbstract:Abstract Let L ( X ) be the free Locally Convex Space over a Tychonoff Space X. We show that the following assertions are equivalent: (i) L ( X ) is l ∞ -barrelled, (ii) L ( X ) is l ∞ -quasibarrelled, (iii) L ( X ) is c 0 -barrelled, (iv) L ( X ) is ℵ 0 -quasibarrelled, and (v) X is a P-Space. If X is a non-discrete metrizable Space, then L ( X ) is c 0 -quasibarrelled but it is neither c 0 -barrelled nor l ∞ -quasibarrelled. We prove that L ( X ) is a ( D F ) -Space iff X is a countable discrete Space. We show that there is a countable Tychonoff Space X such that L ( X ) is a quasi- ( D F ) -Space but is not a c 0 -quasibarrelled Space. For each non-metrizable compact Space K, the Space L ( K ) is a ( d f ) -Space but is not a quasi- ( D F ) -Space. If X is a μ-Space, then L ( X ) has the Grothendieck property iff every compact subset of X is finite. We show that L ( X ) has the Dunford–Pettis property for every Tychonoff Space X. If X is a sequential Space and a μ-Space (for example, metrizable), then L ( X ) has the sequential Dunford–Pettis property iff X is discrete.
-
a topological group observation on the banach mazur separable quotient problem
Topology and its Applications, 2019Co-Authors: Saak Gabriyelyan, Sidney A. MorrisAbstract:Abstract The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach Space has an infinite-dimensional separable quotient Banach Space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach Spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach Space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach Space has as a quotient group the separable metrizable infinite-dimensional topological group, T ω , where T denotes the compact unit circle group. Indeed it is shown that every Locally Convex Space, which has a subSpace which is an infinite-dimensional Frechet Space, has T ω as a quotient group.
-
Locally Convex properties of free Locally Convex Spaces
arXiv: Functional Analysis, 2018Co-Authors: Saak GabriyelyanAbstract:Let $L(X)$ be the free Locally Convex Space over a Tychonoff Space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-Space. If $X$ is a non-discrete metrizable Space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\ell_\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-Space iff $X$ is a countable discrete Space. We show that there is a countable Tychonoff Space $X$ such that $L(X)$ is a quasi-$(DF)$-Space but is not a $c_0$-quasibarrelled Space. For each non-metrizable compact Space $K$, the Space $L(K)$ is a $(df)$-Space but is not a quasi-$(DF)$-Space. If $X$ is a $\mu$-Space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford--Pettis property for every Tychonoff Space $X$. If $X$ is a sequential $\mu$-Space (for example, metrizable), then $L(X)$ has the sequential Dunford--Pettis property iff $X$ is discrete.
-
on reflexivity and the ascoli property for free Locally Convex Spaces
arXiv: General Topology, 2018Co-Authors: Saak GabriyelyanAbstract:Let $L(X)$ be the free Locally Convex Space over a Tychonoff Space $X$. If $X$ is Dieudonn\'{e} complete (for example, metrizable), then $L(X)$ is a reflexive group if and only if $X$ is discrete. Answering a question posed in [9] we prove also that $L(X)$ is an Ascoli Space if and only if $X$ is a countable discrete Space.
Sidney A. Morris - One of the best experts on this subject based on the ideXlab platform.
-
a topological group observation on the banach mazur separable quotient problem
Topology and its Applications, 2019Co-Authors: Saak Gabriyelyan, Sidney A. MorrisAbstract:Abstract The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach Space has an infinite-dimensional separable quotient Banach Space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach Spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach Space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach Space has as a quotient group the separable metrizable infinite-dimensional topological group, T ω , where T denotes the compact unit circle group. Indeed it is shown that every Locally Convex Space, which has a subSpace which is an infinite-dimensional Frechet Space, has T ω as a quotient group.
-
a topological group observation on the banach mazur separable quotient problem
arXiv: General Topology, 2018Co-Authors: Saak Gabriyelyan, Sidney A. MorrisAbstract:The Banach-Mazur problem, which asks if every infinite-dimensional Banach Space has an infinite-dimensional separable quotient Space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach Spaces. It is also known that every infinite-dimensional non-normable Fr\'{e}chet Space has an infinite-dimensional separable quotient Space, namely $\mathbb{R}^\omega $. It is proved in this paper that every infinite-dimensional Fr\'{e}chet Space (including every infinite-dimensional Banach Space), indeed every Locally Convex Space which has a subSpace which is an infinite-dimensional Fr\'{e}chet Space, has an infinite-dimensional (in the topological sense) separable metrizable quotient group, namely $\mathbb{T}^\omega$, where $\mathbb{T}$ denotes the compact unit circle group.
-
free topological vector Spaces
Topology and its Applications, 2017Co-Authors: Saak Gabriyelyan, Sidney A. MorrisAbstract:Abstract In this paper the free topological vector Space V ( X ) over a Tychonoff Space X is defined and studied. It is proved that V ( X ) is a k ω -Space if and only if X is a k ω -Space. If X is infinite, then V ( X ) contains a closed vector subSpace which is topologically isomorphic to V ( N ) . It is proved that for X a k-Space, the free topological vector Space V ( X ) is Locally Convex if and only if X is discrete and countable. The free topological vector Space V ( X ) is shown to be metrizable if and only if X is finite if and only if V ( X ) is Locally compact. Further, V ( X ) is a cosmic Space if and only if X is a cosmic Space if and only if the free Locally Convex Space L ( X ) on X is a cosmic Space. If a sequential (for example, metrizable) Space Y is such that the free Locally Convex Space L ( Y ) embeds as a subSpace of V ( X ) , then Y is a discrete Space. It is proved that V ( X ) is a barreled topological vector Space if and only if X is discrete. This result is applied to free Locally Convex Spaces L ( X ) over a Tychonoff Space X by showing that: (1) L ( X ) is quasibarreled if and only if L ( X ) is barreled if and only if X is discrete, and (2) L ( X ) is a Baire Space if and only if X is finite.
Manuel Lopezpellicer - One of the best experts on this subject based on the ideXlab platform.
-
a quantitative approach to weak compactness in frechet Spaces and Spaces c x
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Carlos Angosto, J Kakol, Manuel LopezpellicerAbstract:Let EE be a Frechet Space, i.e. a metrizable and complete Locally Convex Space (lcs), E′′E′′ its strong second dual with a defining sequence of seminorms ‖⋅‖n‖⋅‖n induced by a decreasing basis of absolutely Convex neighbourhoods of zero UnUn, and let H⊂EH⊂E be a bounded set. Let ck(H):=sup{d(clustE′′(φ),E):φ∈HN}ck(H):=sup{d(clustE′′(φ),E):φ∈HN} be the “worst” distance of the set of weak ∗∗-cluster points in E″E″ of sequences in HH to EE, and k(H):=sup{d(h,E):h∈H¯} the worst distance of H¯ the weak ∗∗-closure in the bidual of HH to EE, where dd means the natural metric of E′′E′′. Let γn(H):=sup{|limplimmup(hm)−limmlimpup(hm)|:(up)⊂Un0,(hm)⊂H}, provided the involved limits exist. We extend a recent result of Angosto–Cascales to Frechet Spaces by showing that: If x∗∗∈H¯, there is a sequence (xp)p(xp)p in HH such that dn(x∗∗,y∗∗)≤γn(H)dn(x∗∗,y∗∗)≤γn(H) for each σ(E′′,E′)σ(E′′,E′)-cluster point y∗∗y∗∗ of (xp)p(xp)p and n∈Nn∈N. Moreover, k(H)=0k(H)=0 iff ck(H)=0ck(H)=0. This provides a quantitative version of the weak angelicity in a Frechet Space. Also we show that ck(H)≤dˆ(H¯,C(X,Z))≤17ck(H), where H⊂ZXH⊂ZX is relatively compact and C(X,Z)C(X,Z) is the Space of ZZ-valued continuous functions for a web-compact Space XX and a separable metric Space ZZ, being now ck(H)ck(H) the “worst” distance of the set of cluster points in ZXZX of sequences in HH to C(X,Z)C(X,Z), respect to the standard supremum metric dd, and dˆ(H¯,C(X,Z)):=sup{f,C(X,Z),f∈H¯}. This yields a quantitative version of Orihuela’s angelic theorem. If XX is strongly web-compact then ck(H)≤dˆ(H¯,C(X,Z))≤5ck(H); this happens if X=(E′,σ(E′,E))X=(E′,σ(E′,E)) for E∈GE∈G (for instance, if EE is a (DF)-Space or an (LF)-Space). In the particular case that EE is a separable metrizable Locally Convex Space then dˆ(H¯,C(X,Z))=ck(H) for each bounded H⊂RXH⊂RX.
-
a quantitative approach to weak compactness in frechet Spaces and Spaces c x
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Carlos Angosto, J Kakol, Manuel LopezpellicerAbstract:Let EE be a Frechet Space, i.e. a metrizable and complete Locally Convex Space (lcs), E′′E′′ its strong second dual with a defining sequence of seminorms ‖⋅‖n‖⋅‖n induced by a decreasing basis of absolutely Convex neighbourhoods of zero UnUn, and let H⊂EH⊂E be a bounded set. Let ck(H):=sup{d(clustE′′(φ),E):φ∈HN}ck(H):=sup{d(clustE′′(φ),E):φ∈HN} be the “worst” distance of the set of weak ∗∗-cluster points in E″E″ of sequences in HH to EE, and k(H):=sup{d(h,E):h∈H¯} the worst distance of H¯ the weak ∗∗-closure in the bidual of HH to EE, where dd means the natural metric of E′′E′′. Let γn(H):=sup{|limplimmup(hm)−limmlimpup(hm)|:(up)⊂Un0,(hm)⊂H}, provided the involved limits exist. We extend a recent result of Angosto–Cascales to Frechet Spaces by showing that: If x∗∗∈H¯, there is a sequence (xp)p(xp)p in HH such that dn(x∗∗,y∗∗)≤γn(H)dn(x∗∗,y∗∗)≤γn(H) for each σ(E′′,E′)σ(E′′,E′)-cluster point y∗∗y∗∗ of (xp)p(xp)p and n∈Nn∈N. Moreover, k(H)=0k(H)=0 iff ck(H)=0ck(H)=0. This provides a quantitative version of the weak angelicity in a Frechet Space. Also we show that ck(H)≤dˆ(H¯,C(X,Z))≤17ck(H), where H⊂ZXH⊂ZX is relatively compact and C(X,Z)C(X,Z) is the Space of ZZ-valued continuous functions for a web-compact Space XX and a separable metric Space ZZ, being now ck(H)ck(H) the “worst” distance of the set of cluster points in ZXZX of sequences in HH to C(X,Z)C(X,Z), respect to the standard supremum metric dd, and dˆ(H¯,C(X,Z)):=sup{f,C(X,Z),f∈H¯}. This yields a quantitative version of Orihuela’s angelic theorem. If XX is strongly web-compact then ck(H)≤dˆ(H¯,C(X,Z))≤5ck(H); this happens if X=(E′,σ(E′,E))X=(E′,σ(E′,E)) for E∈GE∈G (for instance, if EE is a (DF)-Space or an (LF)-Space). In the particular case that EE is a separable metrizable Locally Convex Space then dˆ(H¯,C(X,Z))=ck(H) for each bounded H⊂RXH⊂RX.
Jorge Vielma - One of the best experts on this subject based on the ideXlab platform.
-
weak sequential convergence and weak compactness in Spaces of vector valued continuous functions
Journal of Mathematical Analysis and Applications, 1995Co-Authors: Surjit Singh Khurana, Jorge VielmaAbstract:Abstract Let X be completely regular Hausdorff Space, E a Hausdorff Locally Convex Space, C(X, E) (Cb(X, E)) the Space of all E-valued (all E-valued bounded) continuous functions on X, and βz a strict topology on Cb(X, E). It is proved that a sequence {ƒn} in (Cb(X, E), βz) converges weakly to 0 if and only if {ƒn} is uniformly bounded and ƒn,(x) → 0, pointwise on X with weak topology on E. Similar results are proved for (C(X, E), β∞c). Some characterizations of relatively weakly compact subsets of (Cb(X, E), βz) and (C(X, E), β∞c) are also given.
Rasoul Nasrisfahani - One of the best experts on this subject based on the ideXlab platform.
-
more on the Locally Convex Space m x beta x of a Locally compact hausdorff Space x
Bulletin of The Belgian Mathematical Society-simon Stevin, 2016Co-Authors: Hossein Javanshiri, Rasoul NasrisfahaniAbstract:In the previous paper [12] we introduced the definition of the strict topology $\beta(X)$ on the measure Space $M(X)$ for a Locally compact Hausdorff Space $X$. In this paper, we consider on $M(X)$ the topology $\beta(X)$ and we show that $\beta(X)$ is the weak topology under all left multipliers induced by a function Space on $M(X)$. We then show that $\beta(X)$ can be considered as a mixed topology. This result is not only of interest in its own right, but also it paves the way to prove that $(M(X),\beta(X))$ is a Mazur Space and the Locally Convex Space $(M(S),\beta(S))$, equipped with the convolution multiplication is a complete semitopological algebra, for a wide class of Locally compact semigroups $S$.
-
the strict topology for the Space of radon measures on a Locally compact hausdorff Space
Topology and its Applications, 2013Co-Authors: Hossein Javanshiri, Rasoul NasrisfahaniAbstract:Abstract In this paper, we deal with the dual of the measure Space M ( X ) for a Locally compact Hausdorff Space X under certain Locally Convex topologies. We first introduce and study a strict topology β ( X ) on M ( X ) under which its strong dual can be identified with L 0 ∞ ( M ( X ) ) , the Banach Space of all F ∈ L ∞ ( M ( X ) ) that vanishes at infinity. Under this duality, we investigate some attributes of ( M ( X ) , β ( X ) ) as a Locally Convex Space.
-
the strict topology on the discrete lebesgue Spaces
Bulletin of The Australian Mathematical Society, 2010Co-Authors: Saeid Maghsoudi, Rasoul NasrisfahaniAbstract:Let Σ be a set and σ be a positive function on Σ. We introduce and study a Locally Convex topology β 1 (Σ, σ ) on the Space l 1 (Σ, σ ) such that the strong dual of ( l 1 (Σ, σ ), β 1 (Σ, σ )) can be identified with the Banach Space . We also show that, except for the case where Σ is finite, there are infinitely many such Locally Convex topologies on l 1 (Σ, σ ). Finally, we investigate some other properties of the Locally Convex Space ( l 1 (Σ, σ ), β 1 (Σ, σ )) , and as an application, we answer partially a question raised by A. I. Singh [‘ L ∞ 0 ( G ) * as the second dual of the group algebra L 1 ( G ) with a Locally Convex topology’, Michigan Math. J. 46 (1999), 143–150].
