In this paper, by msss_mappings, the relations between metric spaces and spaces with σ _locally countable cs_networks or spaces with σ _locally countable weak bases are established. These are some answers to A...In this paper, by msss_mappings, the relations between metric spaces and spaces with σ _locally countable cs_networks or spaces with σ _locally countable weak bases are established. These are some answers to Alexandroff’s problems.展开更多
In this paper, we introduce the signed weak gliding hump property in a dual pair with the structure of a system of sections and show that if a dual pair [E, F] has the signed weak gliding hump property, then the β-du...In this paper, we introduce the signed weak gliding hump property in a dual pair with the structure of a system of sections and show that if a dual pair [E, F] has the signed weak gliding hump property, then the β-dual space of E is a weak sequentially complete space if and only if for every n ∈N,(F[n] ,σ(F[n] ,E[n] )) is sequentially complete. Furthermore, we also prove that if [E,F] has the signed weak gliding hump property, then (E,τ(E,E<β> )) is an AK- space.展开更多
Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequ...Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequentially) compactly regular but need not be weakly (sequentially) compactly regular. For countable inductive limits of weakly sequentially complete Frechet spaces, Retakh's condition (M0) implies weakly (sequentially) compact regularity. Particularly for Kothe (LF)-sequence spaces Ep(1 ≤ p < ∞), Retakh's condition (M0) is equivalent to weakly (sequentially) compact regularity. For those spaces, the characterizations of weakly (sequentially) compact regularity are given by using the related results of Vogt.展开更多
Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of ...Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.展开更多
文摘In this paper, by msss_mappings, the relations between metric spaces and spaces with σ _locally countable cs_networks or spaces with σ _locally countable weak bases are established. These are some answers to Alexandroff’s problems.
基金Supported by Research Fund of Kumoh National Institute of Technology,Korea
文摘In this paper, we introduce the signed weak gliding hump property in a dual pair with the structure of a system of sections and show that if a dual pair [E, F] has the signed weak gliding hump property, then the β-dual space of E is a weak sequentially complete space if and only if for every n ∈N,(F[n] ,σ(F[n] ,E[n] )) is sequentially complete. Furthermore, we also prove that if [E,F] has the signed weak gliding hump property, then (E,τ(E,E<β> )) is an AK- space.
基金Supported by the Natural Science Foundation of the Education Committee of Jiangsu Province (Q1107107)
文摘Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequentially) compactly regular but need not be weakly (sequentially) compactly regular. For countable inductive limits of weakly sequentially complete Frechet spaces, Retakh's condition (M0) implies weakly (sequentially) compact regularity. Particularly for Kothe (LF)-sequence spaces Ep(1 ≤ p < ∞), Retakh's condition (M0) is equivalent to weakly (sequentially) compact regularity. For those spaces, the characterizations of weakly (sequentially) compact regularity are given by using the related results of Vogt.
基金supported by the Fundamental Research Funds for the Central UniversitiesProgram for Changjiang Scholars and Innovative Research Team in UniversityNational Natural Science Foundation of China(Grant Nos.11301063 and 11171057)
文摘Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.
