实变函数与泛函分析基础-(程其襄-张奠宙-著)-高等教育出版社-课后答案

QVWSR1.?\A∪(B∩C)=(A∪B)∩(A∪C).XUux∈(A∪(B∪C)).px∈A,9x∈A∪B,x∈A∪C,)x∈(A∪B)∩(A∪C).px∈B∩C,9!0x∈A∪Bdx∈A∪C,x∈(A∪B)∩(A∪C),*A∪(B∩C)⊂(A∪B)∩(A∪C).V$-[ux∈(A∪B)∩(A∪C).px∈A,m0x∈A∪(B∩C).px6∈A,/x∈A∪Bdx∈A∪C,Ox∈Bdx∈C,%x∈B∩C,!0x∈A∪(B∩C),*(A∪B)∩(A∪C)⊂A∪(B∩C).%A∪(B∩C)=(A∪B)∩(A∪C).2.?\(1)A−B=A−(A∩B)=(A∪B)−B;(2)A∩(B−C)=(A∩B)−(A∩C);(3)(A−B)−C=A−(B∪C);(4)A−(B−C)=(A−B)∪(A∩C);(5)(A−B)∩(C−D)=(A∩C)−(B∪D);(6)A−(A−B)=A∩B.XU(1)A−(A∩B)=A∩∁s(A∩B)=A∩(∁sA∪∁sB)=(A∩∁sA)∪(A∩∁sB)=A−B;(A∪B)−B=(A∪B)∩∁sB=(A∩∁sB)∪(B∩∁sB)=A−B;(2)(A∩B)−(A∩C)=(A∩B)∩∁s(A∩C)=(A∩B)∩(∁sA∪∁sC)=(A∩B∩∁sA)∪(A∩B∩∁sC)=A∩(B∩∁sC)=A∩(B−C);(3)(A−B)−C=(A∩∁sB)∩∁sC=A∩∁s(B∪C)=A−(B∪C);(4)A−(B−C)=A−(B∩∁sC)=A∩∁s(B∩∁sC)=A∩(∁sB∪C)=(A∩∁sB)∪(A∩C)=(A−B)∪(A∩C);(5)(A−B)∩(C−D)=(A∩∁sB)∩(C∩∁sD)=(A∩C)∩∁s(B∪D)=(A∩C)−(B∪D);(6)A−(A−B)=A∩∁s(A∩∁sB)=A∩(∁sA∪B)=A∩B.3.?\(A∪B)−C=(A−C)∪(B−C);A−(B∪C)=(A−B)∩(A−C).XU(A∪B)−C=(A∪B)∩∁sC=(A∩∁sC)∪(B∩∁sC)=(A−C)∪(B−C);(A−B)∩(A−C)=(A∩∁sB)∩(A∩∁sC)=A∩∁sB∩∁sC=A∩∁s(B∪C)=A−(B∪C).4.?\∁s(∞Si=1Ai)=∞Ti=1∁sAi.XUux∈∁s(∞Si=1Ai),9x∈S,x6∈∞Si=1Ai,*&n(i,x6∈Ai,%x∈∁sAi,*)1x∈∞Ti=1∁sAi.ux∈∞Ti=1∁sAi,9&n(i,x∈∁sAi,x∈S,x6∈Ai,*x∈S,x6∈∞Si=1Ai,x∈∁s(∞Si=1Ai).%∁s(∞Si=1Ai)=∞Ti=1∁sAi.5.?\(1)(Sα∈ΛAα)−B=Sα∈Λ(Aα−B);(2)(Tα∈ΛAα)−B=Tα∈Λ(Aα−B).XU(1)Sα∈ΛAα−B=(Sα∈ΛAα)∩∁sB=Sα∈Λ(Aα∩∁sB)=Sα∈Λ(Aα−B);(2)Tα∈ΛAα−B=(Tα∈ΛAα)∩∁sB=Tα∈Λ(Aα∩∁sB)=Tα∈Λ(Aα−B).6.u{An}}$U8OB1=A1,Bn=An−(n−1Sν=1Aν),n1.?\{Bn}}$U:G)dnSν=1Aν=nSν=1Bν,1≤n≤∞.XUpi6=j,.uij.mBi⊂Ai(1≤i≤n).Bi∩Bj⊂Ai∩(Aj−j−1[n=1An)=Ai∩Aj∩∁sA1∩∁sA2∩···∩∁sAi∩···∩∁sAj−1=∅./Bi⊂Ai(16=i6=n)nSi=1Bi⊂nSi=1Ai.ux∈nSi=1Ai,px∈A1,9x∈B1⊂nSi=1Bi.px6∈A1,Win}MJmyx∈Ain,x6∈in−1Si=1Ai)x∈Ain.!x∈Ain−in−1Si=1Ai=Bin⊂nSi=1Bi.%nSi=1Ai=nSi=1Bi.7.uA2n−1=0,1n
,A2n=(0,n),n=1,2,···,hU{An}r7Tlimn→∞An=(0,∞);ux∈(0,∞),98N,yxN,*nNw0xn,x∈A2n,%x2N$_D)x2(An,x∈limn→∞An,1mlimn→∞An⊂(0,∞),%limn→∞An=(0,∞).limn→∞An=∅;p0x∈limn→∞An=∅,98N,yn(nN,0x∈An.*p2n−1Nwx∈A2n−1,0x1n.Wn→∞0x≤0,O^%limn→∞An=∅.8.?\limn→∞An=∞Sn=1∞Tm=nAm.XUux∈limn→∞An,98N,y$nN,x∈An,%x∈∞Tm=n+1Am⊂∞Sn=1∞Tm=nAm,%limn→∞An⊂∞Sn=1∞Tm=nAm.ux∈∞Sn=1∞Tm=nAm,90n,yx∈∞Tm=nAm,&n(m≥n,0x∈An,%x∈limn→∞An.*limn→∞An=∞Sn=1∞Tm=nAm.29.O$0(−1,1)7(−∞,+∞)$$&+$$&+HzTϕ:(−1,1)→(−∞,+∞).&n(x∈(−1,1),ϕ(x)=tanπ2x.ϕK}(−1,1)7(−∞,∞)$$&+10.?\Fg[k$!%93!Æ87=0a[r!Æ8}&XUE?\g[S:x2+y2+(z−12)2=(12)2k(0,0,1)!94xOya[M&OO/g=,PN&n((x,y,z)∈S\(0,0,1),ϕ(x,y,z)=x1−z,y1−z∈M.&?ϕ}$$-r*S4M}&11.?\/Br℄:GMiCOÆA59AF(ÆOXUuG={△z|△z}Br:GMiC},8Y$△zGnj$!0Qrz,y△z4rz&+*Æ△z}:G*0&+3}$&$/20Q}O)G40QIERT$$&+3%GF(}O12.?\0Æ0Q(zLÆ$OXUuAn}n0Q(zln=1,2,···,9A=∞Sn=0An.An/n+10%RB6L#n(zn+100QbGOj0%$0QbOj$0Q*Y0B6%R‘$0O*/§4#Q6,An=a,1/§4#Q4,A=a.13.uA}a[r%0Q!(P$}0Q)ÆG0QÆJ6l9A}OXUn(AG6/q0%RB6L#:(x,y,r).bG(x,y)}6Pr}6Jx,y1J‘0Q)r‘200Q*)$}O%A=a.14.?\:5S!M(E}O(0XUuf}(−∞,∞)r:5BS!lÆE,//O(1)n(x∈(−∞,∞),lim△x→0+f(x+△x)=f(x+0)?lim△x→0−f(x+△x)=f(x−0)$8(2)x∈E/DÆf(x+0)f(x−0).(3)n(x1,x2∈E,px1x2,9f(x1−0)f(x1+0)≤f(x2−0)f(x2+0),*Y$x∈E,&+2BrMiC(f(x−0),f(x+0)),d/(3)OEG!x&+!MiC}:G*/11OF(}O15.~;y(0,1)7[0,1]AC$$&+$H-*3TB(0,1)G0QlR={r1,r2,···},Wϕ(0)=r1,ϕ(1)=r2,ϕ(rn)=rn+2,n=1,2,···ϕ(x)=x,x∈((0,1)\R),9ϕ}[0,1](0,1)r$$-t16.uA}$O89A00IOÆ8’OXUuA={x1,x2,···},A0IlÆeA.An={x1,x2,···,xn},AnIlÆeAn.&AeAnG202n05)eA=∞Sn=1eAn,*eAF(ÆO1AG$05LÆ8}O*)eA}O17.?\[0,1]rlQOÆ8bÆc.XUB[0,1]rQlÆA,[0,1]r0QlÆ{r1,r2,···},mB=(√22,√23,···,√2n,···)⊂AWϕ(√22n)=√2n+1,n=1,2,···ϕ(√22n+1)=rn,n=1,2,···ϕ(x)=x,x6∈B.9ϕ}A[0,1]$$&+/[0,1]Æc,OA#}c.18.pAGY05/:%RO0D$&$L#A={ax1x2x3···},)Y0xij$0Æc9A#}c.XUuxi∈Ai,Ai=c,i=1,2,···.*)0AixR$$-tϕi.Wϕ}AE∞$-t&n(ax1x2x3···∈A.ϕ(ax1x2x3···)=(ϕ1(x1),ϕ2(x2),ϕ3(x3),···).[?\ϕ}$$-tpϕ(ax1x2x3···)=ϕ(ax′1x′2x′3···),9&n(i,ϕi(xi)=ϕi(x′i)./2ϕi}$&$*xi=x′i,%ax1x2x3···=ax′1x′2x′3···.&n((a1,a2,a3,···)∈E∞,ai∈R,i=1,2,···,*Æϕi}-r0xi∈Ai,yϕi(xi)=ai.%0ax1x2x3···∈A,yϕ(ax1x2···)=(ϕ1(x1),ϕ2(x2),···)=(a1,a2,···),ϕ}$$-t%A4E∞2c.19.p∞Sn=1AnÆc,?\8n0,yAn0#}c.XU/2E∞=c,Z.u∞Sn=1An=E∞..+?*pAnc,n=1,2,···.uPiÆE∞RGo#)-tpx=(x1,x2,···,xn,···)∈E∞,9Pi(x)=xi.WA∗i=Pi(Ai),i=1,2,···,49A∗iAic,i=1,2,···.%&Y0i,8ξi∈R\A∗i,2}ξ=(ξi,ξ2,···,ξn,···)∈E∞.?ξ6∈∞Sn=1An.|xrpξ∈∞Sn=1An,98i,y∈Ai,2}ξi=Pi(ξ)∈Pi(Ai)=A∗i,4ξ∈R\A∗iX’%ξ6∈∞Sn=1An=E∞,14ξ∈E∞X’*Fs8℄0i0,yAi0=c.20.BYjCÆ0;1UlÆ8ÆT,h?TÆc.XUuT={{ξ1,ξ2,···}|ξi=0or1,i=1,2,···}.OTE∞-tϕ:{ξ1,ξ2,···}→{ξ2,ξ3,···},9ϕ}TE∞Iϕ(T)$$-t%A≤E∞=c,+A(0,1]iC42If4{$$&+%Y0x∈(0,1]$O$Æx=0.ξ1ξ2···,bGY0ξiÆ0;1,Wf(x)={ξ1,ξ2,···},9f}(0,1]TIf((0,1])r$$-t*)T≥(0,1]=c.KrA=c.5fu--+EQ-x9DFNEIPEoBMKGE′BMCH¯EBMA1.2℄P0∈E′I|%m!8’P0V,U(P,δ)(ÆP07)7=’)P0P1~)E({xq1P1’g’2),(P0∈EoI/|’8’P0V,U(P,δ)(ÆP07).yU(P,δ)⊂E.OLoP0∈E′,/%m8P0V,U(P,δ),’P07V,U(P0)⊂U(P,δ),.P1∈E∩U(P0)⊂E∩U(P,δ)dP16=P,Dm;8’P0V,78’P1)P0~)E.,3om8’P0V,’)P0P1~)E,l%mP0V,U(P0)7’)P0P1~)E,P0∈E′.oP0∈Eo,/’U(P0)⊂E.,3oP0∈U(P,δ)⊂E,