实变函数与泛函分析基础 (程其襄著)【khdaw

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A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C). VW SR 1.?\ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). XU x ∈ (A ∪ (B ∪ C)). x ∈ A,9 x ∈ A ∪ B, x ∈ A ∪ C, x ∈ (A ∪ B) ∩ (A ∪ C). x ∈ B ∩ C,9!0 x ∈ A ∪ Bd x ∈ A ∪ C, x ∈ (A ∪ B) ∩ (A ∪ C), V [ x ∈ (A ∪ B) ∩ (A ∪ C). x ∈ A,0 x ∈ A ∪ (B ∩ C). x 6∈ A,/ x ∈ A ∪ Bd x ∈ A ∪ C, x ∈ Bd x ∈ C, x ∈ B ∩ C,!0 x ∈ A ∪ (B ∩ C), (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C). A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). 2.?\ XU (1)A − (A ∩ B) = A ∩ ∁s(A ∩ B) = A ∩ (∁sA ∪ ∁sB) = (A ∩ ∁sA) ∪ (A ∩ ∁sB) = 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); 4.?\ ∁s( Ai, i,x 6∈ Ai, x ∈ ∁sAi, XU x ∈ ∁s( (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. (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. (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 ∩ (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) = Ai),9 x ∈ S, x 6∈ ∞Ti=1 ∁sAi. (A − B) ∩ (A − C) = (A ∩ ∁sB) ∩ (A ∩ ∁sC) = A ∩ ∁sB ∩ ∁sC = A ∩ ∁s(B ∪ C) = A− (B ∪ C). B ∩ ∁sC) = A ∩ (B ∩ ∁sC) = A ∩ (B − C); (A − B) ∪ (A ∩ C); Ai) = ∞Si=1 ∞Si=1 ∞Si=1 1

∞Si=1 nSi=1 (Aα − B). {Bn} (Aα − B); (Aα − B). {An} Aν = (1 ≤ i ≤ n). Ai, ∞Si=1 x ∈ ∞Ti=1 (2) Tα∈Λ Bν, 1 ≤ n ≤ ∞. Bi ∩ Bj ⊂ Ai ∩ (Aj − nSν=1 j−1[n=1 ∁sAi. Ai) = (Aα − B); ∞Si=1 ∞Ti=1 Aα) − B = Sα∈Λ Aα) ∩ ∁sB = Sα∈Λ Aα) ∩ ∁sB = Tα∈Λ Aα) − B = Tα∈Λ An) = Ai ∩ Aj ∩ ∁sA1 ∩ ∁sA2 ∩ ··· ∩ ∁sAi ∩ ··· ∩ ∁sAj−1 = ∅. (2)( Tα∈Λ (Aα ∩ ∁sB) = Sα∈Λ (Aα ∩ ∁sB) = Tα∈Λ n−1Sν=1 ∞Ti=1 ( Sα∈Λ Aα − B = ( Sα∈Λ Aα − B = ( Tα∈Λ ∁sAi. x ∈ ∁sAi,9 i,x ∈ ∁sAi, x ∈ S,x 6∈ Ai, x ∈ S, x 6∈ x ∈ ∁s( Ai). ∁s( 5.?\ (1) XU (1) Sα∈Λ Aν ), n > 1.?\ 6. }U>8 B1 = A1, Bn = An − ( }U: G>d nSν=1 XU i 6= j,. i < j. Bi ⊂ Ai / Bi ⊂ Ai(1 6= i 6= n) nSi=1 Bi. x 6∈ A1,W in}y x ∈ Ain , x ∈ Ai, x ∈ A1,9 x ∈ B1 ⊂ Bi. nSi=1 Ai x ∈ Ain .U >7> T lim x ∈ (0,∞),98 N ,y x < N , n > Nw 0 < x < n, x ∈ A2n, x2 N _D> x2 An, x ∈ lim An,1 lim lim An = ∅,98 N ,y n > N ,0 x ∈ An. 2n − 1 > Nw 0 x ∈ lim x ∈ A2n−1, 0 < x < 1 n .W n → ∞ 0 < x ≤ 0,^ lim 8.?\ lim An,98 N ,y n > N ,x ∈ An, x ∈ XU x ∈ lim Am, m ≥ n,0 lim Am,90 n,y x ∈ Am. x ∈ x ∈ An, x ∈ lim lim An = (0,∞). An = ∅; nSi=1 in−1Si=1 ∞Sn=1 ∞Tm=n n→∞ An ⊂ (0,∞), ∞Sn=1 ∞Tm=n n→∞ An = (0,∞); x 6∈ in−1Si=1 ∞Tm=n+1 An = ∅. n→∞ Bi ⊂ Ai. n→∞ ∞Sn=1 n→∞ Ai = Bin ⊂ An = Am. n→∞ Am ⊂ Am, Ai = Bi. nSi=1 An ⊂ n→∞ nSi=1 {An} n→∞ lim n→∞ n→∞ ∞Tm=n ∞Sn=1 ∞Tm=n ∞Sn=1 An. ∞Tm=n An = Am. n→∞ nSi=1 n→∞ ∞Tm=n 2

, 1 − z 2 )2 = ( 1 y 1 − z ∈ M. ϕ(x, y, z) = x 9.0 (−1, 1)7 (−∞, +∞)<z T ϕ : (−1, 1) → (−∞, +∞). x ∈ (−1, 1), ϕ(x) = tan π 2 x. ϕ} (−1, 1)7 (−∞,∞) 10.?\Fg[k"!93!Æ>87=0a[!Æ>8} XUE"?\g[ S : x2 + y2 + (z − 1 2 )2k" (0, 0, 1)!94 xOya[ M /g= (x, y, z) ∈ S\(0, 0, 1), ? ϕ} S4 M} 11.?\/B℄:GiCÆ> A59 AFÆ> XU G = {△z|△z}B:GiC },8Y △zGj!0 rz, △z4 rzÆ y △z}:G<03}/20 } G40 >ERT3 GF} 12.?\0Æ0 zÆ> XU An} n0 z n = 1, 2,··· ,9 A = An.An/ n + 10 RB6# nz n + 100 bGj 0 0 bj 0 Y0B6R` 0>/ §4# 6,An = a,1/ §4# 4,A = a. 13. A}a[0 ! (}0 )ÆG0 Æ69 A}> XU AG6/0RB6# : (x, y, r).bG (x, y)}6 r}6 x, y1` 0 r` 2 00 }> A = a. 14.?\:5S!E}0 XU f} (−∞,∞):5BS!Æ E,// (1) x ∈ (−∞,∞), f (x + △x) = f (x + 0)? lim f (x + △x) = f (x − 0) 8 (2)x ∈ E/" DÆ f (x + 0) > f (x − 0). (3) x1, x2 ∈ E, x1 < x2,9 f (x1 − 0) < f (x1 + 0) ≤ f (x2 − 0) < f (x2 + 0),Y x ∈ E,2BiC (f (x − 0), f (x + 0)),d/ (3) EG! x

ϕ(0) = r1, eA = ϕ(1) = r2, ϕ(rn) = rn+2, n = 1, 2,··· x ∈ ((0, 1)\R), ϕ(x) = x, TB (0, 1)G0 R = {r1, r2,···},W 9 ϕ} [0,1] (0,1) 16. A}>89 A00>Æ>8  XU A = {x1, x2,···}, A0>Æ eA.An = {x1, x2,··· , xn}, An> Æ eAnG20 2n05 eAn.A ∞Sn=1 eAn, eAFÆ1 AG 05Æ>8} eA} 17.?\ [0, 1] Æ>8b<Æ c. {r1, r2,···}, XUB [0,1] Æ A,[0,1]0 Æ W 9 ϕ} A [0,1]/ [0,1]<Æ c, A<#} c. 18.> AGY05/:R0D# A = {ax1x2x3···} ,Y 0 xij 0<Æ c>9 A<#} c. XU xi ∈ Ai, Ai = c, i = 1, 2,··· .0 Aix> R ϕi.W ϕ} A E∞ ax1x2x3··· ∈ A.ϕ(ax1x2x3···) = (ϕ1(x1), ϕ2(x2), ϕ3(x3),··· ).[?\ ϕ} ϕ(ax1x2x3···) = ϕ(ax′ i)./2 ϕi} xi = ···),9 i, ϕi(xi) = ϕi(x′ i, ax1x2x3··· = ax′ ···. (a1, a2, a3,··· ) ∈ E∞, ai ∈ R, i = 1, 2,··· ,Æ ϕi} 0 xi ∈ Ai,y ϕi(xi) = ai.0 ax1x2x3··· ∈ A,y ϕ(ax1x2···) = (ϕ1(x1), ϕ2(x2),··· ) = (a1, a2,··· ), ϕ} A4 E∞<2 c. An<Æ c,?\8 n0,y An0 <#} c. 19. ∞Sn=1 An = E∞..? An < c, n = 1, 2,··· . Pi XU/2 E∞ = c,Z. ∞Sn=1 Æ E∞ RG# x = (x1, x2,··· , xn,··· ) ∈ E∞,9 Pi(x) = xi.W ,···) ⊂ A ) = rn, n = 1, 2,··· , n = 1, 2,··· √2 n + 1 , 2 B =(√2 √2 ) = 2n √2 2n + 1 ϕ( ϕ( x 6∈ B. √2 3 ,··· , √2 n ϕ(x) = x, x′ 1x′ 2x′ 3 1x′ 2x′ 3 A∗ i = Pi(Ai), i = 1, 2,··· , 4

∞Sn=1 9 A∗ i < Ai < c, i = 1, 2,··· .Y0 i,8 ξi ∈ R\A∗ i ,2} ξ = (ξi, ξ2,··· , ξn,··· ) ∈ E∞. ? ξ 6∈ An.|x ξ ∈ An,98 i,y ∈ Ai,2} ξi = Pi(ξ) ∈ Pi(Ai) = A∗ i ,< An = E∞,<14 ξ ∈ E∞XF8℄0 i0,y 4 ξ ∈ R\A∗ X ξ 6∈ 20.BYjCÆ 0; 1UÆ>8Æ T ,h? T<Æ c. XU T = {{ξ1, ξ2,···} | ξi = 0 or 1, i = 1, 2,···}. T E∞ ϕ : {ξ1, ξ2,···} → {ξ2, ξ3,···},9 ϕ} T E∞> ϕ(T ) A ≤ E∞ = c,A (0,1]iC4 2f>4{Y0 x ∈ (0, 1]Æ x = 0.ξ1ξ2 ··· ,bGY0 ξiÆ 0; 1,W f (x) = {ξ1, ξ2,···},9 f} (0,1] T> f ((0, 1]) T ≥ (0, 1] = c. A = c. Ai0 = c. i ∞Sn=1 ∞Sn=1 5

f E x9 DF E EoBG E ′BC ¯EBA 1.2℄ P0 ∈ E ′|!8 P0V U (P, δ) (Æ" P07)7 =" P0 P1~ E ({x1 P1g2), P0 ∈ Eo/| 8 P0V U (P, δ)(Æ" P07).y U (P, δ) ⊂ E. P0 ∈ E ′,/8 P0V U (P, δ), P07V U (P0) ⊂ U (P, δ), . P1 ∈ E ∩ U (P0) ⊂ E ∩ U (P, δ)d P1 6= P ,D;8 P0V78 P1" P0~ E.38 P0V" P0 P1~ E, P0V U (P0)7 " P0 P1~ E, P0 ∈ E ′. P0 ∈ Eo,/ U (P0) ⊂ E. 3 P0 ∈ U (P, δ) ⊂ E, U (P0) ⊂ U (P, δ) ⊂ E,/ P0 ∈ Eo. 2. E1| [0, 1]7kT h E1. R1_ E ′ 3. E2 = {(x, y)|x2 + y2 < 1}.h E2. R2_ E ′ E ′ 4. E3|9 ,x 6= 0, x = 0 ?C<. R2_X E3 E ′ Eo E ′ 1 , ¯E13|^ 6 5.. R27 23 E ′ y = E ′ 6.2℄ C F C| ¯F = F. F| C/ F ′ ⊂ F,# ¯F = F ∪ F ′ = F. ¯F = F,/ F ′ ⊂ F ∪ F ′ = ¯F = F, # F| C 7.2℄C C>C|C CC>| C G|C F| C/ ∁G| C ∁F|C G − F = G ∩ ∁F| C F − G = F ∩ ∁G| C 8. f (x)| (−∞,∞)x4U9/! a, E = {x|f (x) > a} | C E = {x|f (x) ≥ a} =| C 3 = E3 ∪ {(0, y)| − 1 ≤ y ≤ 1}, Eo ¯E1 = {(x, y)|x2 + y2 ≤ 1}. 2 = {(x, y)|x2 + y2 ≤ 1}, Eo 1 = {(x, y)|x2 + y2 < 1}, 1 = {(x, 0)|0 ≤ x ≤ 1,}, Eo 1 = ∅, ¯E1 = E ′ 1. E ′ 1 = [0, 1], Eo ¯E1 = [0, 1]. 2, Eo 2 , ¯E2. 1, Eo 1 , ¯E1. 1 = ∅, sin 1 x 0, 3 = ∅. 3 3 . 1, Eo 1

n→∞ f (xn) ≥ a, d(x0, y) ≥ 1 n , y∈F 1 n − ǫ = 1 n . d(x, y0) ≤ d(x0, x) + d(x0, y0) < ǫ + δ = ǫ + d(x, y) ≤ d(x, y0) < 1 x0 ∈ E,/ f (x0) > a.# f (x)|U . δ > 0,y! x ∈ (−∞,∞),|x − x0| < δ f (x) > a,D! x ∈ U (x0, δ) x ∈ E, U (x0, δ) ⊂ E,E| C xn ∈ E,d xn → x0(n → ∞)./ f (xn) ≥ a, f (x)U f (x0) = lim D x0 ∈ E,# E| C 9.2℄\2 C|R2CC\2CR zR2 C:C n,Gn|C! x0 ∈ Gn, d(x0, F ) < 1 F| CW Gn =x|d(x, F ) < 1 n , . y0 ∈ F ,y d(x0, y0) = δ < 1 n ,/ d(x0, F ) = inf n . (1/! y ∈ F, d(x0, y) ≥ 1 d(x0, F ) < 1 Z). W ǫ = 1 n − δ > 0,! x ∈ U (x0, ǫ), d(x0, x) < ǫ. | d(x, F ) = inf n , x ∈ Gn.1 U (x0, ǫ) ⊂ Gn,7 Gn|C Gn,! n, x ∈ Gn, d(x, F ) < 1 x ∈ n .W n → ∞, d(x, F ) = 0. F| C x ∈ F (1/ x 6∈ F ,. yn ∈ F,y d(x, yn) → 0, x ∈ F ′ ⊂ F ,Z),D ∞Tn=1 Gn ⊂ F. Gn ⊃ F, n = 1, 2,··· , ∞Tn=1 Gn ⊃ F ,# ∞Tn=1 Gn = F ,F|R2CC G|C/ ∁G| C C Gn,y ∁G = Gn, ∁Gn| C# G|R2 C:C 10.2℄vÆz [0,1]7waÆ:< 7 C [0,1]7Æ< 7| (0.7,0.8). [0,1]7Æ< 7| [0,1]7 nÆ< 7| a7 ai(i = 1, 2,··· , n− 1) 0 9 7b; |jY> 38R` n − 12F1kiG (0.a1a2 ··· an−17, 0.a1a2 ··· an−18), {a1, a2,··· , an−1} (0.07, 0.08) (0.17, 0.18) G = ∁(∁G) = ∁( Gn) = ∁Gn, ··· (0.97, 0.98). y∈F ∞Tn=1 ······ n ∞Tn=1 ∞\n=1 ∞[n=1 An ∞[n=1 2

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