Some Qualities Of Operator Function In The Topology Of Type I~,Convergence On The Field Of Operator

In this paper , the author mainly researches the qualities of the operator function which is valued on the field of operators Q,like strong continuity,strong uniform continuity,strong uniform convergence and so on.J.Mikusinski introduced the concept of continuity,continuity,differentiability on the field of operators Q,and also the concept of type I convergence,but we can't give the topology of type I convergence on the field of operators Q,actually the topology in which the type of the column convergence equal it that base on operator isn't exist on the field of operators Q ,because the convergengce doesn't satisfy the so-called Urysohn condition about column convergence.So,some scholars later extended to the concept of the type I ' convergence ,actually by way of add the Urysohn condition on the first concept of the type I convengence,thus got a topology T and the topological space (Q , T).In the recent years,other scholars researched and proved that topological space (Q , T) is a semilinear topology and it can't be a topology algebra,obviously it limited the extension of the concept of continuity, differentiability of operator function ,so they conformationed a new topology (?) by way of the strict inductive limit again and proved that the topological space (Q ,(?)) was a complete,Hausdorff linear topological space.In this paper,author extends a series of concepts and some important conclusions in the frechet space to the topological space (Q ,(?)) by way of the strict inductive limit and tends to get some profound conclusions.In the chapter 1,author mainly introduces a series of concepts and correlative nature of continuity,differentiability,integrability,strong continuity,strong uniform continuity and so on of operator function which is valued on the subspace Q0 (Frechet space) in the type I ' convergent topology (the topology is limited to be a quasi norm in Q0 ), and gives an outline of correlative concepts and nature topology linear space that there isn't partial convex condition) and some conformation of topological space (Q ,(?)) to provide the underlying basis for subsequent article.In the second chapter,first of all,author compares the topological spaces which is conform by the type I ' convergent topology and the topology(?) of strict inductive limit and gives their relation.Then gets the important conclusion :the type of column convergence in (Q ,(?)) equal it in the type I ' convergent,thus introduces the concept of continuity,bounded variation and differentiability of operator function in the topology (?) and discusses the nature of I 'continuity, I 'differentiability, (?) -continuity and(?) -differentiability.In chapter 3,the main study is making further effort on discussing the concept of convergence,uniform continuity,R-S integration and useful conclusion,and gets more profound progeny.