赋序列范数的矢值Banach序列空间的光滑性与强光滑性

1、Banach(100044)Banachss(E)MR(2000)46B20,46E30Z+x=xkRxkE.EBanach(E)E(E)x=xk(E),(NZ+),suppx=kZ+:xk=0NxN=x1,x2,···,xN,0,0,···=xkekk=1x·A=xkek(AZ+),(xkek=0,0,···,0,xk,0,···,xkkx=xk,y=yk,xkyk,kZ+,1:(E)0,+,(a)(b)mM,kZ+uE,kA|x|= xk E,xy.|x|m u

2、 Einf(uek)sup(uek)M u E;kk(c)(x)=sup(xN).X(E)=x(E):=(x)<,NX(E)E=R,(X(E),)(X(E),)(x)(y).X(E)kZ+,yX(E).X(E)x=xk,y=ykX(E), xk E yk E,kZ+,x=xkX(E),y=yk(E), yk E xk E,x=xk,y=ykX(E), xk E2003-07-19.278yk E,kZ+(|x|y|),X(E)AKBKX(E)Banachpk(x)=xk,x=xkX(E).ssss(E)26j, xj E> yj E,(|x|=|y|),(x)>(y).x=x

3、kX(E),xNxX(E).X(E)pk:X(E)E|x|= xk EssEx=xkss(E)=x(E):|x|= xk Ess,ss(E)Banachss x ss(E)= |x| ss.(2).K¨othess(E) ss=y=yk:|xkyk|<,x=xkss.k=1k=1ss y =sup|xkyk|: x 1.ss(2).ss(E)K¨othess(E)=f=fk:fkE,kZ+,|f|= fk Ess,ss(E) f = |f| ss,ss(E)BK(2),ss,ss,ss(E),ss(E)25BKss(E)19892,ss(E)1223ss1)ss=ss

4、,ssEss(E)AKy=ykss,x,y =2)ss(E)=ss(E), k=1xkyk,x=xkss;f=fkss(E),xk,fk , x,f =33 k=1x=xkss(E);(n)ssAK(n)ss(E)kZ+,xkx(n)=xk,x=xkss(E),Ex(n)xxk|x(n)|x|ss.Banach279kZ+,fk(n)4ssAKfkE|f(n)|f|ss.f(n)=fk,f=fkss(E)(n)2ss(E)1BanachX,xxS(X),fS(X)xXf(x)=1,S(X)fxXX1ssAKssss(E)ssEssss(E)ssEAKssE(ss(E)2,ss=ss,ss(E)

5、=ss(E).).xss(E), x =1.f(1),f(2)ss(E)x1= x,f(i)= k=1(i)xk,fk k=1xk E fk E(i)(i)= |x|,|f(i)| =ksuppxxk E fk E= |x|suppx,|f(i)|suppx x|suppx ss· |f(i)|suppx ss |f(i)|suppx ss |f(i)| ss= f(i) =1,i=1,2(1)|x|,|f(i)| =1(i=1,2), |f(i)| ss= f(i) =1,ss|x|ss|f(1)|,|f(2)|x|f(1)|=|f(2)|,fk E= fk E,suppf(1)=

6、suppf(2)(1)(2)kZ+(2)(3)(1) k=1(i)xk,fk k=1xk E· fk E,(i)xk,fk xk E· fk E,(i)(i)xk,fk xk E· fk E,(i)(i)i=1,2,(4)kB,(3)A=suppxsuppf(i)(i=1,2),B=suppxA.(1)fk=(2)fk=0.kA,(4)(1)(2) x xfkfkkk=1=,xk E f(1) E xk E f(2) Ekk280fk(1)26xk xk Efk E(1)fk E(2)fk(2)xk xk EEEfk(2)ksuppx,(1)fk E(1)=fk(

7、2)fk E(2).fk=fk.xk=0,(1)(1)(2)|f(i)| ss= |f(i)|suppx ss(i=1,2)(5)(i)fk0 Efk(1)=fk,1(2)=0,|f(i)|f(i)|suppx,|f(i)|=|f(i)|suppx,|f(i)| ss> |f(i)|suppx ss,(5)fk(i)=0(i=1,2).k0suppx,xk0=0,ss(ss=ss)kZ+,fk0=0,(i)f(1)=f(2),1<p<+,lpAKss(E)lp(E)Elq13ss(E)2BanachXfS(X),f(n)f2ssssAKxS(X),f(n)S(X),X.fxs

8、s(E)ssx,f(n) 1E),ss(E)ssE(ssEss(E)x=xkss(E), x =1,f(n)=fkss(E)(ss(E)=ss(E), f(n) =1x,f(n) 1,fss(E), f =1 fnf 0.(n)x,f(n)= k=1(n)xk,fk k=1xk E fk E= |x|,|f(n)|(6)(n) |x| ss |f(n)| ss= x · f(n) =1.|x|,|f(n)| 1.f(n) =1,ss|x|ss, |x| ss= x =1|f(n)|ss, |f(n)| ss=|x|gss, g ss=1|f(n)|g ss0.(7)ssBKfk Eg

9、k,(n)kZ+,(8)3Banach281fk E0,(n)gk0,kZ+,k=1g0.1= |x|,g = xk Egk= ksuppx xk E·gk= |x|,g·suppx x · g·suppx ss= g·suppx ss g ss=1|x|,g = |x|,g·suppx =1 g·suppx ss=1.|x|g=g·suppx.ksuppx,gk=0,suppgsuppx.(6) nlim xk,fk(n) = E fk(n) E=1,k=1nlim xkk=1nlim( xk E fk(n) E

10、 xk,fk(n) )=0.k=1xk,fk(n) xk E fk(n) E,(n)nlim( xk E fk E xk,fk(n) )=0,kZ+.(n)(8) xk,fk xk E·gk.ksuppg,suppgsuppx,xkfx,k(n)k Egk1.(8) xkfx(nk(n)k E, fk) 1Exxkk ES(E),f(n)k f(n)k EES(E) xxkk EhkS(E), (n)fkfhk(n) kEE0,fk(n) fk(n) E·hk E0h(1)=hk(1),h(1)=hk,ksuppgk0,ksuppgf=fk,fk=gkhk(1),fkE f

11、k E=gk h(1)k E=gk.|f|= fk E=gk=gss,fss(E) f = |f| ss= g ss=1.(9)282ksuppg,gk=0,26fk=0.(8)fk E= fk(n)(n)fk E0.ksuppg,fk(n)(8)(9)fk E= fk fk= fk(n)(n)(n) fk E·hk+ fk E·hkgkhk E fk E·hk E+ fk E·hkgkhk E fk E·hk E+| fk Egk|· hk E0(n)(n)(n)(n)(n)(n)ss.2kZ+,fkfk4,f(n)f1<p&

12、lt;+,lp(E)lplqAK(n)E.ss(E).|f|=g,ss(E)E(7)|f(n)|f|lp213(1):5361.1996.3ss(E)1999,23(2):7883.4ss(E)2001,25(6):57780.SMOOTHNESSANDSTRONGSMOOTHNESSOFVECTOR-VALUEDBANACHSEQUENCESPACESEQUIPPEDWITHSEQUENTIALNORMSRenLiweiFengGuochen(DepartmentofMathematics,BeijingJiaotongUniversity,Beijing100044)AbstractInthispaper,s