关于Hellinger-Toeplitz拓扑的闭图定理  

CLOSED GRAPH THEOREMS ON HELLINGER-TOEPLITZ TOPOLOGIES

作　　者：丘京辉[1] 

机构地区：[1]苏州大学数学系

出　　处：《苏州大学学报（自然科学版）》1991年第3期311-313,共3页Journal of Soochow University(Natural Science Edition)

摘　　要：本文讨论的Hellinger-Toeplitz拓扑α对于任何对偶双(X,Y)有定义。一个可容拓扑α称为Hellinger-Toeplitz拓扑若对于任何两个对偶双(X_1,Y_1)、(X_2,Y_2),只要线性映照t:(X_1,σ(X_1,Y_1))→(X_2,σ(X_2,Y_2))为连续,必t:(X_1,α(X_1,Y_1))→(X_2,α(X_2,Y_2))也连续(见[1],11—1)。称Hellinger-Toeplitz拓扑α具有关于完备化的承继性,若对于任何对偶双(X,Y),(X,α(X,Y))的完备化(X,Y)恰为Y。相似地可定义α关于拟完备化和序列式完备化的承继性。Let α be a Hellinger-Toeplitz topology defined for all dual pairs (see [1], 11-1), We call α to have the hereditary property inherited by completion if for any dual pair (X, Y),where is the completion of(X,α(X,Y)) and α(X,Y) is the topology on induced by α(X,Y). Similarly, the hereditary property inherited by quasi-completion or sequential completion is also defined, σ, τ and β are all the Hellinger-Toeplitz topologies which have the hereditary property inherited by completion, quasi-completion and sequential completion.We prove the following:Theorem 1 Let (F, η) be a locally convex Hausdorff topological vector space and α be a Hellinger-Toeplitz topology which has the hereditary property inherited by sequential completion (resp. quasi-completion or completion), then the following conditions are equivalent: (ⅰ) For any Mackey space (E,ξ) whose topological dual E' is α(E',E)-sequentially complete (reap. quasi-complete or complete), every closed graph linear map t: (E, ξ)→(F, η) is continuous.(ⅱ) For any dense vector subspace H of (F',α(F',F)), the sequential completion (resp. guasi-completion or completion) of (H, α(H,F)) contains F'.(ⅲ) For any vector subspace M of F and any dual pair (M,N), where N M^3, if (N,α(N,M)) is sequentially complete (resp. quasi-complete or complete) and the set {f∈F':f|M∈N} separates points in F, then for any f∈F', f|M∈N.Particularly, α may be α, τ or β.Corollary Let (E, ξ) be a Mackey space whose Mackey dual (E', τ(E',E)) is sequentially complete and (F,η) have a countable fundamental System of absolutely convex (weakly) compact sets, then every closed graph linear map t: (E, ξ)→(F,η) is continuous.

关 键 词：H-T拓扑 团图定理 弱拓扑σ 

分 类 号：O189.1[理学—数学]