新的泛函分析基本原理_英文_李容录

文章编号:1004-4353(2004)03-0157-04New basic principles of functional analysisL I Rong -lu 1,ZHON G Shu -hui 1,CU I Cheng -ri 2(1.Dep t.M ath.,H ar bin I ns titute of T echnology ,H ar bin 150001,China ;2.D ep t.M ath.,College of Science and Engineer ing,Yanbian Univer sity ,Yanj i 133002,China )Abstract:New basic principles of functional analysis such as equicontinuity theorem,uniform boundedness principle,open mapping theorem and closed graph theorem are established by smash -ing the bonds of linearity of concerned m appings.Key words:Second category;Open mapping;Closed graphC LC number:O177.3 Document code:ALet X ,Y be topological vector spaces and N (X )the family of neighborhoods of 0I X.As usual,L (X ,Y)is t he space of continuous linear operators.1 New equicontinuity theoremLet C (0)={U I C C :lim t y 0U (t )=U (0)=0;|U (t )|\|t|,P |t|[1}.Definition 1.1 A mapping f :X y Y is said to be dissecting if there exist U I C (0)and U I N (X )such that f (x +tu )=rf (x )+sf (u)w henever x I X ,u I U and |t|[1,w here r and s are determined by x ,u and t ,|r -1|[|U (t)|,|s|[|U (t)|.Let F U ,U (X ,Y )be the family of dissecting mappings w hich are related to U I C(0)and U I N (X ).It is trivial that F U ,U (X ,Y )includes all linear operators.How ever,many nonlinear mappings are also dissecting.In fact,for U (t )=et and U =(-1,1),F U ,U (R ,R )includes many nonlinear functions such as |x |,sin x ,e |x|-1,etc.Example 1.1 Let (X ,+#+)be a seminormed space and U ={x I X :+x +<12},U (t)=e 2t,P t I C.Then every linear operator T :X y Y yields various nonlinear dissecting mappings in F U ,U (X ,Y)such as f T (x )=11++x +T (x ),g T (x )=(1+12sin +x +)T (x ),h T (x )=e -+x +T (x ),etc.Let F <Y X ,x I X.F is said to be equicontinuous at x if P V I N (Y )v W I N (X )such that f (x +W )<f (x )+V ,P f I F [1].F is equicontinuous on X if F is equicontinuous at each x I X .Theorem 1.1(Equicont inuity T heorem) Suppose that X is of second category in itself,and F <F U ,U (X ,Y)such that each f I F is continuous.If F is pointwise bounded on X ,第30卷 第3期2004年9月 延边大学学报(自然科学版)Journal of Yanbian U niversity (Natural Science) Vol.30No.3 Sep.2004收稿日期:2003-12-20 基金项目:国家自然科学基金资助项目(10361005)作者简介:李容录(1943-),男(朝鲜族),吉林龙井人,哈尔滨工业大学数学系教授.158延边大学学报(自然科学版)第30卷then F is equicont inuous on X.Since L(X,Y)<F U,U(X,Y),the usual equicontinuity theorem[2]is a special case of T heorem1.1.T he proof of T heorem1.1,see[3],[4].2New uniform boundedness principleF(<Y X)is said to be uniformly bounded on each bounded subset of X if{f(x):f I F, x I B}is bounded for each bounded B<X.Theorem2.1(U niform Boundedness Principle)Suppose that X is of second category in itself,and F<F U,U(X,Y)such that each f I F is continuous.If F is pointw ise bounded on X,then F is uniformly bounded on each bounded subset of X.The usual resonance theorem[5,6]is a special case of Theorem2.1.T he proof of Theorem 2.1,see[3],[4].Another consequence of T heorem1.1is the follow ing:Theorem2.2(New Banach-Steinhaus Closure T heorem)Suppose that X is of second category in itself,and{f n}<F U,U(X,Y)such that each f n is continuous.If lim n f n(x)= f(x)exists at each x I X,t hen f is also cont inuous and f I F U,U(X,Y).3New open mapping theorem+#+:X y[0,+])is called a paranorm if+0+=0,+-x+=+x+,+x+u+ [+x+++u+and+t n x n-tx+y0w henever|t n-t|y0and+x n-x+y0.(X,+#+)is a Fr chet space if it is a complete separated paranormed space.Definition3.1Let(X,+#+)be a Fr chet space.A mapping f:X y Y is said to be conditionally additive if f(0)=0and for every x,u I X t here exist A,B I C,|A|[1,|B|[ 1,such that f(x)+f(u)=f(A x+B u).M oreover,f:X y Y is said to be conditionally countably additive if f(0)=0and for every{x i}<X w ith E]i=1+x i+<+]there exists {u i}<X such that+u i+[+x i+for all i and E]i=1f(x i)=f(E]i=1u i).If f:X y Y is linear and continuous,then f is conditionally countably additive.In fact, continuous linear operator T:X y Y satisfies a more strong condit ion:E]x i converges]E]i=1T(x i)=T(E]i=1x i).i=1How ever,many nonlinear mappings are also conditionally countably additive.To see this, letA(R)={U I R R:U is continuous,U(0)=0;P t,s I R v A,B I[-1,1]such that U(t)+U(s)=U(A t+B s)}.A(R)includes U(t)=2t.If U I R R such that U(0)=0,U c(t)>0for t X0and U c(t)is increasing in(0,+])but decreasing in(-],0),then U I A(R),e.g.,U(t)=e t -1,t \0,t , t <0,W (t )=t 2, t \0,t 3, t <0,etc.Example 3.1 Let U I A (R ), e.g.,U (t )=e t -1,t \0,t , t <0.For every continuous linear T :c 0y Y define f :c 0y Y by f (E ]j =1t j e j )=T (E ]j =1U (t j )e j ), E ]j =1t j e jI c 0.T hen f is conditionally countably additive but f is not linear whenever T X 0.Definition 3.2 A mapping f :X y Y is said to be slightly transitive if for every x I X and U I N (X )there is a V I N (X )such that f (x )+f (V )<f (x +U ).It is trivial that every linear operator is slightly transitive,and many nonlinear mappings are also slightly transitive, e.g.,f :c 0y Y in Example 3.1.Definition 3.3 Let (X ,+#+)be a paranormed space and U a ={x I X :+x +<a},P a >0.A mapping f :X y Y is said to be slightly homogeneous if(i)P D >0and n I N v m I N such that f (n U D )<mf (U D ),(ii)P D >0v G (D )>0such that -f (U D )<f (U G (D ))and lim D y 0G (D )=0.Trivially,linear operators are slightly homogeneous,and many nonlinear mappings are also slightly homogeneous,e.g.,f :c 0y Y in Example 3.1.Thus,if U (t)=e t -1,t \0,t, t <0,then each T I L (c 0,Y)yields a nonlinear,conditionally countably additive,slightly transitive and slightly homogeneous f :c 0y Y:f (E ]j =1t j e j )=T (E ]j =1U (t j )e j ), E ]j =1t j e j I c 0.Theorem 3.1 Let X ,Y be Fr chet spaces.If f :X y Y is conditionally countably additive,slightly transitive,slightly homogeneous and onto,i.e.,f (X )=Y ,Then f is open,i.e.,f (G )is open for each open G A X .If,in addition,f is continuous and one to one,then f is a homeomorphism of X onto Y .The usual open mapping theorem[7]is a special case of Theorem 3.1.T he proof ofT heorem 3.1,see [8].4 New closed graph theoremDefinition 4.1 A mapping f :X y Y is said to be harmonious at 0ifx n y x in Xlim n,m y +]f (x n -x m )=0]lim nf (x n -x )=0.If f (0)=0and f is cont inuous at 0,then f is harmonious at 0.The converse is not true.Example 4.1 Let f :R y R such that f (0)=0,f (t )=1,t X 0.Then f is harmonious at 0but f is not continuous at 0.Moreover,the graph {(t ,1):t X 0}G {(0,0)}is not closed in R @R .159 第3期李容录,等:新的泛函分析基本原理We have a trivial fact as follows.Lemma4.1Let Y=(Y,+#+)be a paranormed space.If f:X y Y is linear,then(1)f(0)=0;(2)+f(-x)+=+f(x)+,P x I X;(3)+f(x+u)+[+f(x)+++f(u)+,P x,u I X;(4)+f(t n x n-tx)+y0,if|t n-t|y0and+f(x n-x)+y0;(5)v M\L>0such thatL+f(x-u)+[+f(x)-f(u)+[M+f(x-u)+,P x,u I X.M any nonlinear mappings also satisfy the conditions(1)-(5).Example4.2LetU(t)=32n+(t-2n),2n[t[2n+1,n=0,1,2,,,32n+1+12(t-2n-1),2n+1[t[2n+2,n=0,1,2,,, -U(-t),t<0.Let+E]j=1t j e j+=sup j|t j|for E]j=1t j e j I c0<R N andf(E]j=1t j e j)=E]j=1U(t j)e j,E]j=1t j e j I c0.T hen f is not linear but f satisfies the condit ions(1)-(5).Lemma4.2Let Y be a Fr chet space.If f:X y Y satisfies the condition(5)and,in addition,f has closed graph,then f is harmonious at0.We are interested in the follow ing(5c)w hich is strictly w eaker than(5).(5c)limn f(x n-x)=0]limnf(x n)=f(x).Example4.3Let f(x)=e|x|-1,x I R.Then f sat isfies(5c)but(5)fails for f.Theorem4.1Let X,Y be Fr chet spaces.If a mapping f:X y Y satisfies t he conditions(1)-(4),(5c)and,in addition,f is harmonious at0,then f is continuous.C orollary4.1Let X,Y be Fr chet spaces.If a mapping f:X y Y satisfies t he conditions(1)-(5)and,in addition,f has closed graph,then f is continuous.The usual closed graph t heorem is a special case of Corollary4.1.T he proof of Theorem 4.1,see[9].References:[1]Wilansky A.T opolog y for Analysis[M].Jo hn Wiley,1970.129.[2]KÊthe G.T o pological V ector Spaces I[M].Spr ing er-Verlag,1969.169.[3]L i Rong lu,Zhong Shuhui,Cui Chengr i.A new uniform boundedness principle[J].to appear.[4]L i Rong lu.A n Intro duction to Funct ional Analysi s[M].Harbin:HIT Press,2003.[5]L i R onglu,Swartz C.Spaces for w hich t he unifor m boundedness pr inciple holds[J].Studia Sci M ath Hungar,1992,(27):379-384.[6]Swartz C.Infinite M atrices and the Gliding Hump[M].Singa-pore-N ew Jersey-L ondon-Hong Kong:World Sci.,1996.[7]Wilansky A.M odern M ethods in T opological V ector Spaces[M].M cGr aw-H ill,1978.58.(下转第189页) 160延边大学学报(自然科学版)第30卷3 结果与分析从数值仿真结果可以得到如下结论:3.1 一般情况下,驱动力矩的频率小于简支梁第一阶固有频率时,高阶频率的影响很小,所以只取前二阶模态就可以满足工程精度要求.3.2 由图4可知,驱动力矩的频率与简支梁的某个固有频率相同时发生/共振0.3.3 若驱动力矩的频率与简支梁某高阶固有频率相近时,计算模态应取到比该高阶模态多一、二个模态,这样才能提高计算精度,满足高精度的工程要求.参考文献:[1] 陆佑方.弹性多体系统动力学[M ].北京:高等教育出版社,1996.[2] 冯冠民,王彬,陆佑方.转动规范理论及其在柔体系统动力学中的应用[A].多体系统动力学)))理论、计算方法和应用[C].上海:上海交通大学出版社,1992.Influence of rotational flexibility machine arm high level moldshape to the power responseJIN Za -i quan(Dep ar tment of M echanical Engineer ing ,College of Science and Engineer ing ,Yanbian Univ ersity ,Y anj i 133002,China )Abstract:T his paper,basing on the rotational flexibility machine arm Benoull-i Euier beam kinetics pattern w hich established from the rotational specification theory ,discusses the influence of high level mode shape on the pow er response,and explains how to mode shape cuts off in the imitate the ture system of the rotational flex ibility machine arm Benoull-i Euier beam.Key words:Flex ibility machine arm;Mode shape;T he power response(上接第160页)[8] L i Rong lu,Zhong Shuhui.A new open mapping theorem[J].to appear.[9] L i Rong lu,Zhong Shuhui.A new closed g raph theorem[J].to appear.新的泛函分析基本原理李容录1, 钟书慧1, 崔成日2( 1.哈尔滨工业大学数学系,黑龙江哈尔滨150001; 2.延边大学数学系,吉林延吉133002)摘要:突破了线性分析的框架,对包含所有线性算子在内的更广泛的一类算子族建立了新的等度连续原理、一致有界原理、开映射原理和闭图像原理等泛函分析基本原理.关键词:第二纲;开映射;闭图像189 第3期金在权:转动弹性机械臂高阶模态对动力响应的影响。