工学のための関数解析 - 2章前半

4a45da7dd02c934534ed38cc44cab89b?s=47 Daiki Tanaka
November 06, 2019

工学のための関数解析 - 2章前半

4a45da7dd02c934534ed38cc44cab89b?s=128

Daiki Tanaka

November 06, 2019
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  1. ޻ֶͷͨΊͷؔ਺ղੳ 1 ڑ཭ۭؒ ׬උڑ཭ۭؒ ։ू߹ͱดू߹

  2. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒ w ڑ཭ۭؒɾ఺ɾڑ཭ͷఆٛ w ڑ཭ۭؒͷ։ٿɾू߹ͷ༗քੑɾ఺ྻͱ෦෼ྻͷఆٛ w ڑ཭ۭؒʹ͓͚Δ఺ྻͷఆٛ w ڑ཭ۭؒʹ͓͚Δ఺ྻɾίʔγʔྻͷੑ࣭

    !2
  3. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒɾ఺ɾڑ཭ͷఆٛ w ʮڑ཭ۭؒʯ͸ ೚ҙͷ ू߹ʹଐ͢Δ఺ؒͷۙ͞΍ɺ఺ྻͷۃݶΛ࿦͡ΔͨΊͷ΋ͷ w ఆٛ ڑ཭ۭؒɾ఺ɾڑ཭ 

    w ू߹9ͷ೚ҙͷݩY Zʹରͯ͠ɺ࣮਺E Y Z ͕ରԠ͠ɺҎԼͷ৚݅Λຬͨ࣌͢ɺ9͸ ڑ཭ۭؒͰ͋Δͱ͍͏ɻ9ͷݩΛ఺ QPJOU ɺE Y Z Λ఺Y Zͷڑ཭ͱݺͿɻ %  ͨͩ͠ɺ ਖ਼஋ੑ %  ରশੑ %  ࡾ֯ෆ౳ࣜ w ڑ཭ۭؒ͸ ͷΑ͏ʹɺू߹ͱڑ཭ؔ਺ͷϖΞͰදݱͰ͖Δɻ d(x, y) ≥ 0 d(x, y) = 0 ⇔ x = y d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z) (X, d) !3
  4. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒͷྫ ࣮਺ू߹ʹର͢Δڑ཭ۭؒ w ࣮਺શମͷू߹ ʹ͓͍ͯɺ఺ ʹରͯ͠ڑ཭Λ  ͱఆٛ͢Δͱɺ ͸ڑ཭ۭؒͱͳΔ

    R x, y ∈ R d(x, y) = |x − y| (R, d) !4
  5. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒͷྫ ࣮ϕΫτϧू߹ʹର͢Δڑ཭ۭؒ w /ݸͷ࣮਺Λॱ൪ʹฒ΂ͯͰ͖ΔશͯͷϕΫτϧ͔ΒͳΔू߹ ʹ͓͍ͯɺ఺ ʹରͯ͠ڑ཭Λ  ͱఆٛ͢Δͱɺ ͸ڑ཭ۭؒͱͳΔɻ

    w 1SPPG w %  % ͸໌Β͔ w ೚ҙͷ ʹରͯ͠ɺࡾ֯ෆ౳ࣜͱϛϯίϑεΩʔͷෆ౳ࣜ  Q Λ༻͍Δͱ  ͕੒Γ ཱͪɺ % ΋੒ཱ͢Δɻ RN x, y ∈ RN dp (x, y) = p N ∑ i=1 |xi − yi |p lp := (RN, dp ) z = (z1 , z2 , . . . , zn ) ∈ RN p N ∑ i=1 (|xi | + |yi |)p ≤ p N ∑ i=1 |xi |p + p N ∑ i=1 |yi |p p N ∑ i=1 |xi − yi |p ≤ p N ∑ i=1 (|xi − zi | + |yi − zi |)p ≤ p N ∑ i=1 |xi − zi |p + p N ∑ i=1 |yi − zi |p = dp (x, z) + dp (y, z) !5
  6. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒͷྫ ࣮਺ྻू߹ʹର͢Δڑ཭ۭؒ w ৚݅ Λຬͨ͢શͯͷ࣮਺ྻ ͔ΒͳΔू߹Λ ͱه͢ɻ  ͷ೚ҙͷ఺

     ʹରͯ͠ɺڑ཭Λ ͱఆٛ͢Δͱɺ ͸ڑ཭ۭؒͱͳΔɻ w ೚ҙͷ࣮਺ྻ   ʹରͯ͠ϛϯίϑεΩʔͷ ఆཧ౳Λ࢖͏ͱ  ͕೚ҙͷO Ͱ੒ΓཱͭͷͰɺ ൪ࠨͷ ࠨล͸Oʹ͍ͭͯ୯ௐ͔ͭ༗քͳ࣮਺ྻͰ͋Δɻ ∞ ∑ i=1 |xi |p < ∞ x := (xi )∞ i=1 lp lp x := (xi )∞ i=1 y := (yi )∞ i=1 dp (x, y) := ( ∞ ∑ i=1 |xi − yi |p ) 1 p (lp, dp ) x := (xi )∞ i=1 y := (yi )∞ i=1 z := (zi )∞ i=1 ∈ lp ( n ∑ i=1 |xi − yi |p ) 1 p ≤ ( n ∑ i=1 |xi |p + |yi |p ) 1 p ≤ ( n ∑ i=1 |xi |p ) 1 p + ( n ∑ i=1 |yi |p ) 1 p ≤ ( ∞ ∑ i=1 |xi |p ) 1 p + ( ∞ ∑ i=1 |yi |p ) 1 p < ∞ !6
  7. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒͷྫ ࣮਺ྻू߹ʹର͢Δڑ཭ۭؒ w ୯ௐ͔ͭ༗քͳ࣮਺ྻͰ͋ΔͷͰɺ Ͱऩଋ͠ɺ  ͱͳΓɺ͜Ε͸  ͳΒ͹

     ্ͷෆ౳͕ࣜ৚݅ Λ͍ࣔͯ͠Δ͔Β ͱ % % ͷ੒ཱΛอূ͢Δɻ w % ʹ͍ͭͯ͸ϛϯίϑεΩʔͷෆ౳ࣜΛ༻͍ͯ೚ҙͷਖ਼੔਺Oʹ͍ͭͯ  ͱͳ Δɻࠨลͷ֤਺ྻ͸୯ௐ͔ͭ༗քͳ࣮਺ྻͱͳΔͷͰ Ͱऩଋ͢Δɿ  ͱͳΓɺ % ΋੒ཱ͢Δɻ n → ∞ dp (x, y) = ( ∞ ∑ i=1 |xi − yi |p ) 1 p = lim n→∞ ( n ∑ i=1 |xi − yi |p ) 1 p ≤ ( ∞ ∑ i=1 |xi |p ) 1 p + ( ∞ ∑ i=1 |yi |p ) 1 p < ∞ x, y ∈ lp x − y := (xi − yi )∞ i=1 ∈ lp ∞ ∑ i∈N |xi |p < ∞ ( n ∑ i=1 |xi − yi |p ) 1 p ≤ [ n ∑ i=1 (|xi − zi | + |zi − yi |) p ] 1 p ≤ ( n ∑ i=1 |xi − zi |p ) 1 p + ( n ∑ i=1 |zi − yi |p ) 1 p ≤ dp (x, z) + dp (z, y) n → ∞ dp (x, y) ≤ lim n→∞ [ n ∑ i=1 (|xi − zi | + |zi − yi |) p ] 1 p ≤ dp (x, z) + dp (z, y) !7
  8. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒͷྫ ࣮਺஋࿈ଓؔ਺ʹର͢Δڑ཭ۭؒ w ։۠ؒ  ͨͩ͠BC Ͱఆٛ͞ΕΔ࣮਺஋࿈ଓؔ਺શମͷ ू߹Λ ͱ͠ɺ೚ҙͷ

    ʹରͯ͠  Λఆٛ͢Δͱ͖ɺ ͸ ڑ཭ۭؒͱͳΔ [a, b] ∈ R C[a, b] f, g ∈ C[a, b] dp ( f, g) := (∫ b a | f(x) − g(x)|p dx ) 1 p (C[a, b], dp ) !8
  9. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒఆٛ ڑ཭ۭؒͷ։ٿɾू߹ͷ༗քੑɾ఺ྻͱ෦෼ྻ   ։ٿPQFOCBMM  w ڑ཭ۭؒ9ͷ఺ ͱਖ਼ͷ਺SΛ༻͍ͯఆٛ͞ΕΔू߹

    ΛɺYͱத৺͢Δ։ٿ PQFO CBMM ͱ͍͏ɻ  ༗քͳू߹CPVOEFETFU  w ڑ཭ۭؒ9ͷ෦෼ू߹4ʹରͯ͠ɺ͋Δ఺ ͱ༗ݶͳਖ਼਺S͕ଘࡏ͠ɺ ͱͰ͖Δ࣌ɺ4͸ʮ༗ք CPVOEFE Ͱ͋Δʯͱ͍͏ɻ  ఺ྻɾ෦෼ྻ  w ࣗવ਺ͷ֤ʑʹɺ9ͷݩY͕ͭͣͭରԠ͢Δͱ͖ɺ ͸ʮ9ͷ఺ྻͰ͋Δʯͱ͍͍͏ɻ w 9ͷ఺ྻ ͕༩͑ΒΕΔͱ͖ɺ͜ΕΛؒҾ͍ͯ ཁૉΛൈ͍ͯ ఆٛ͞ΕΔ৽͍͠఺ྻΛɺ ͷ෦෼ྻ ͱ͍͏ɻ w ෦෼ྻ͸ɺਖ਼੔਺ ͔Β৽ͨͳਖ਼੔਺ ΛରԠ͚ͮΔڱٛ୯ௐ૿Ճؔ਺ Λ༻͍ͯɺ ͱද ͤΔɻ x ∈ X BX := {y ∈ X|d(x, y) < r} x ∈ X S ⊂ B(X, r) {xn }∞ n=1 {xn }∞ n=1 {xn }∞ n=1 k ∈ N m(k) ∈ N m {xm(k) }∞ k=1 !9
  10. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒڑ཭ۭؒʹ͓͚Δ఺ྻͷऩଋͷఆٛ w ఆٛ ڑ཭ۭؒ9ͷ఺ྻͷऩଋ   ఺ྻͷऩଋ w ڑ཭ۭؒ

    ͷ఺ྻ ʹରͯ͠ɺ͋Δ఺ ͕ଘࡏ͠ɺ ͱͳΔ ࣌ɺ఺ྻ ͸ʮۃݶYʹऩଋ͢Δʯͱ͍͏ɻ w  ࿦๏తʹ͸ɿʮ೚ҙͷਖ਼਺ʹରͯ͠ɺ͋Δࣗવ਺ ͕ଘࡏ͠ɺ ͳΒ͹  ʯ  ίʔγʔྻ w ڑ཭ۭؒ ͷ఺ྻ ͕ ͱͳΔ࣌ɺ ͸ʮ9ͷίʔγʔ ྻͰ͋Δʯͱ͍͏ɻ (X, d) {xn }∞ n=1 x ∈ X lim n→∞ d(xn , x) = 0 {xn }∞ n=1 ϵ − δ ϵ N N < n d(xn , x) < ϵ (X, d) {xn }∞ n=1 d(xn , xm ) → 0(n, m → ∞) {xn }∞ n=1 !10
  11. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒੑ࣭ڑ཭ۭؒͷऩଋ఺ྻɺίʔγʔྻͷجຊੑ࣭ " ऩଋઌͷҰҙੑ ڑ཭ۭؒ 9 E ͷ఺ྻ ͕ऩଋ͢ΔͳΒ ͹ɺऩଋઌ͸Ұҙʹఆ·Δ

    # ڑ཭ۭؒ9ͷ఺ྻ ͕͋Δ఺ ʹऩଋ͢ΔͳΒ͹ɺ ͸ίʔγʔྻͱͳΔ ஫ҙٯ͸੒Γཱͨͳ͍  $ ڑ཭ۭؒ9ͷίʔγʔྻ ͸༗քͰ͋Δ (xn )∞ n=1 (xn )∞ n=1 x ∈ X (xn )∞ n=1 (xn )∞ n=1 !11
  12. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒऩଋઌͷҰҙੑʹ͍ͭͯͷূ໌  ऩଋઌͷҰҙੑ ڑ཭ۭؒ 9 E ͷ఺ྻ ͕ऩଋ͢ΔͳΒ͹ɺऩଋઌ͸Ұҙʹఆ·Δ <ํ਑>ҟͳΔͭͷ఺

    ʹ͍ͭͯɺ ͱ ͕ಉ࣌ʹ੒ཱ͢Δ͜ͱΛԾఆ͠ɺໃ ६Λಋ͘ɻ ڑ཭ͷఆٛ  ɺͨͩ͠ɺ ͔Βɺ  ·ͨԾఆ͔Βɺ͋Δࣗવ਺ ͱ ͕ଘࡏͯ͠ɺʮ ͳΒ͹ ʯ ͱʮ ͳΒ͹ ʯ͕੒ཱ͢Δɻ (xn )∞ n=1 x, y ∈ R lim n→∞ xn = x lim n→∞ xn = y d(x, y) ≥ 0 d(x, y) = 0 ⇔ x = y d(x, y) > 0 → d(x, y) 3 > 0 N1 N2 N1 < n d(xn , x) < d(x, y) 3 N2 < n d(xn , y) < d(x, y) 3 !12
  13. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒऩଋઌͷҰҙੑʹ͍ͭͯͷূ໌  ऩଋઌͷҰҙੑ ڑ཭ۭؒ 9 E ͷ఺ྻ ͕ऩଋ͢ΔͳΒ͹ɺऩ ଋઌ͸Ұҙʹఆ·Δ

    ͕ͨͬͯ͠ɺ ͳΒ͹ ͱͳΓɺ͜Ε͸ໃ६ɻ ূ໌ऴΘΓ  ࠨͷෆ౳ࣜ͸ࡾ֯ෆ౳ࣜɺӈͷෆ౳ࣜ͸ઌ΄Ͳͷͭͷෆ౳ࣜΛ଍ ͨ͠΋ͷɻ  (xn )∞ n=1 max{N1 , N2 } < n d(x, y) ≤ d(x, xn ) + d(xn , y) < 2 3 d(x, y) !13
  14. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒऩଋͱίʔγʔྻͷؔ܎ʹ͍ͭͯͷূ໌ ڑ཭ۭؒ9ͷ఺ྻ ͕͋Δ఺ ʹऩଋ͢ΔͳΒ͹ɺ ͸ίʔγʔྻ ͱͳΔ ڑ཭ͷఆ͔ٛΒɺ  

    ͸఺Yʹऩଋ͢Δ͜ͱ͔Βɺ ɺ Ͱ͋Δͷ ͰɺڬΈܸͪͷݪཧͰɺ  ূ໌ऴΘΓ  (xn )∞ n=1 x ∈ X (xn )∞ n=1 0 ≤ d(xn , xm ) ≤ d(xn , x) + d(x, xm ) = d(xn , x) + d(xm , x) (xn )∞ n=1 lim n→∞ d(xn , x) = 0 lim n→∞ d(xm , x) = 0 lim n→∞ d(xn , xm ) = 0 !14
  15. ,ZPUP6OJWFSTJUZ ڑ཭ۭؒίʔγʔྻ͕༗քͰ͋Δ͜ͱͷূ໌ ڑ཭ۭؒ9ͷίʔγʔྻ ͸༗քͰ͋Δ w  ͸ίʔγʔྻͳͷͰɺ༗ݶͳࣗવ਺ ͕ଘࡏ͠ɺ ͳΒ͹ ͱͳΔɻ

    ίʔγʔྻͷఆٛΛ؇࿨ͨ͠  w ͕ͨͬͯ͠ɺ͋Δ ʹରͯ͠ɺ ͳΒ͹   ࠨͷෆ౳ࣜ͸ڑ཭ͷఆٛࡾ֯ෆ ౳ࣜ  w ͞Βʹɺ ͕༗ݶ֬ఆ஋ ࣮਺ͷ༗ݶݸͷ ू߹ͷ࠷େ஋͸༗ݶ֬ఆ஋ ʹͳΓɺ݁ہ೚ҙͷࣗવ਺Oʹରͯ͠ɺ  ͱͳΓɺ ͸༗քͱͳΔɻ ূ໌ऴΘΓ (xn )∞ n=1 (xn )∞ n=1 N N < n d(xN+1 , xn ) < 1 y ∈ X N < n d(y, xn ) ≤ d(y, xN+1 ) + d(xN+1 , xn ) < d(y, xN+1 ) + 1 A := max {d(y, x1 ), d(y, x2 ), . . . , d(y, xN )} < ∞ d(y, xn ) ≤ max {A, d(y, xN+1 )} < ∞ (xn )∞ n=1 !15
  16. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒ w ׬උڑ཭ۭؒͷఆٛ w ࣮਺ۭؒ ͷ׬උੑ RN !16

  17. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒԿ͕ศརͳͷ͔ʁ w ڑ཭ۭؒ9ͷ఺ྻ͕ɺ͋Δ஋ʹऩଋ͢Δ͜ͱΛ൑ఆ͢Δ͜ͱ͸Ͱ͖Δɻ w ͔͠͠ɺʮ఺ྻͷۃݶ͕9ͷதʹଘࡏ͢Δ͔ʁʯͱ͍͏໰͍ʹ౴͑Δ͜ͱ͸ ೉͍͠ɻ FH͋Δ఺ྻͷۃݶ͕9ͷதʹଘࡏ͢Ε͹ͦͷ఺ྻ͸ίʔγʔྻ ͕ͩɺίʔγʔྻ͔ͩΒͱ͍ͬͯۃݶ͕9ͷதʹଘࡏ͢Δ͔͸Θ͔Βͳ͍ 

    w 9ͷཁૉYʹͭͣͭʹ͋ͨΓΛ͚ͭɺͦͷ౎౓൑ఆΛߦΘͳ͚Ε͹ͳΒ ͳ͍ɻ w ఺ྻͷ৘ใͷΈ͔ΒۃݶͷଘࡏੑΛ൑ఆ͍ͨ͠  w ʮ׬උڑ཭ۭؒʯ͸ͦͷΑ͏ͳੑ࣭Λຬͨ͢ڑ཭ۭؒ → !17
  18. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒ׬උڑ཭ۭؒͷఆٛ w ఆٛ ׬උڑ཭ۭؒ  w ڑ཭ۭؒ 9 E

    ͷ೚ҙͷίʔγʔྻʹରͯ͠ɺͦͷ఺ྻͷۃݶ ͕ɺ9ʹॴଐ͢Δ఺ͱͯ͠ଘࡏ͢Δ͜ͱ͕อূ͞ΕΔ࣌ɺ9͸ ʮ׬උ DPNQMFUF Ͱ͋Δʯͱ͍͏ɻ w ׬උͳڑ཭ۭؒΛ׬උڑ཭ۭؒ DPNQMFUFNFUSJDTQBDF ͱ ͍͏ !18
  19. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒ׬උͰ͸ͳ͍ڑ཭ۭؒͷྫ w 3Λ࣮਺શମͷू߹ɺ2Λ༗ཧ਺શମͷू߹ͱ͢Δ w ࣮਺ྻ ʮ  ʜ Λখ਺఺ҎԼୈOܻͰଧͪ੾ͬͨ༗ཧ਺ʯ

    ͸ɺڑ཭ۭؒ 3 E ͨͩ͠ Ͱ͸ ʹऩଋ͢ΔͷͰ Ҏ ֎ͷ࣮਺ʹ͸ऩଋ͠ͳ͍ ऩଋઌͷҰҙੑ ɻ w ·ͨɺ ͸ ʹऩଋ͢Δ͜ͱ͔Βɺڑ཭ۭؒ 3 E ͷίʔγʔྻ Ͱ͋Δ ੑ࣭ ɻ͕ͨͬͯ͠ɺ 2 E ͷίʔγʔྻͰ΋͋Δɻ w ͔͠͠ɺ ͷۃݶ͸2಺ʹ͸ଘࡏ͠ͳ͍ͨΊɺ 2 E ͸׬උڑ཭ۭؒͰ ͸ͳ͍ɻ xn := 2 ∈ Q d(x, y) = |x − y| 2 ∈ R\Q 2 (xn )∞ n=1 ⊂ Q 2 ∈ R (xn )∞ n=1 ⊂ Q !19
  20. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒ׬උͰͳ͍ڑ཭ۭؒΛ׬උʹ͢Δ͜ͱ͕Ͱ͖Δ w ׬උͰͳ͍ڑ཭ۭؒ 9 E ͕༩͑ΒΕΔ࣌ɺ9ʹ৽ͨͳ఺Λ௥Ճ͠ɺ ৽ͨͳू߹ Λߏ੒͠ɺ w

    ͞Βʹɺ ʹ Λຬͨ͢৽͍͠ڑ཭ Λಋೖ ͢Δ͜ͱʹΑΓɺ׬උͳڑ཭ۭؒ Λߏ੒͢Δ͜ͱ͕Ͱ͖ Δɻ w ͜ͷϓϩηεΛ׬උԽ DPNQMFUJPO ͱ͍͏ɻ͜Ε͸ষͰৄ͘͠ ৮ΕΒΕΔɻ ̂ X ⊃ X ̂ X ̂ d(x, y) = d(x, y), ∀x, y ∈ X ̂ d ( ̂ X, ̂ d) !20
  21. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒڑ཭ۭؒ ͷ׬උੑ  RN w ఆཧڑ཭ۭؒ ͷ׬උੑ  

    w ࣮਺શମͷू߹ ͸ɺڑ཭ ͷ΋ͱͰ׬උڑ཭ۭ ؒʹͳΔɻ w ࣮ࡍɺڑ཭ۭؒ ͷ೚ҙͷίʔγʔྻʹରͯ͠ɺ͋Δ ͕ඞͣଘࡏ͠ɺ  R R d(x, y) = |x − y| (R, d) α ∈ R α = lim n→∞ an = lim n→∞ sup an = lim n→∞ inf an ∈ R !21
  22. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒڑ཭ۭؒ ͷ׬උੑ  ఆཧQSPPG  RN w  ͕ίʔγʔྻͰ͋Δͱ͖ɺ͜Ε͕

    ͷͲ͔͜ʹऩଋ͢Δ ͜ͱΛࣔͤ͹Α͍ɻ w ڑ཭ۭؒͷίʔγʔྻ͸༗քͰ͋Δ͔Βɺ  ΋༗քͳ୯ௐݮগ਺ྻͰ͋Γɺ ༗ݶ֬ఆ஋ʹऩଋ͢Δ ੑ࣭ ɻͦͷۃݶ஋Λ  ͱ͢Δɻ w ͋ͱ͸ɺ ʹͳΔ͜ͱΛࣔͤ͹Α͍ɻ (an )∞ n=1 R ̂ ap := sup{an |n ≥ p}, (p = 1,2,...) α := lim p→∞ ̂ ap = lim n→∞ sup an ∈ R α = lim n→∞ an !22
  23. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒڑ཭ۭؒ ͷ׬උੑ  ఆཧQSPPG  RN w  Λࣔ͢ɻ

    w  ͸ίʔγʔྻͰ͋ΔͷͰɺ೚ҙͷਖ਼਺ʹରͯ͠ɺ͋Δࣗવ਺ ͕ଘࡏͯ͠ɺ ͳΒ ͹ɺ Ͱ͋ΔͷͰɺ ͱͳΔɻ͜Ε͸ ͕ ͷ ্քͷͭͰ͋Δ͜ͱΛද͢ɻ w ಛʹɺ Ͱɺ ͱ͢Δͱɺ ͳΒ͹  w ্ݶͷఆ͔ٛΒɺ ΑΓɺ ͱ͢Ε͹ɺ ͳΒ͹   w Αͬͯɺ ͳΒ͹ ͱͳΓɺ ͕ࣔ͞Εͨɻ α = lim n→∞ an (an )∞ n=1 ϵ N1 N1 < n, m d(an , am ) = |an − am | < ϵ am − ϵ < an < am + ϵ am + ϵ {ak |k ≥ n} am − ϵ < an ≤ ̂ an n → ∞ N1 < m am − ϵ ≤ lim n→∞ ̂ an = lim n→∞ sup{ak |k ≥ n} = α ̂ an = sup{ak |k ≥ n} ≤ am + ϵ n → ∞ N1 < n lim n→∞ ̂ an = α ≤ am + ϵ N1 < n |am − α| ≤ ϵ α = lim n→∞ an !23
  24. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒڑ཭ۭؒ ͷ׬උੑ  RN w ఆཧڑ཭ۭؒ ͷ׬උੑ  

    w ڑ཭  Q    ͷ΋ͱͰɺڑ཭ۭؒ  ͸׬උڑ཭ۭؒʹͳΔɻ RN dp (x, y) = p N ∑ i=1 |xi − yi |p ∞ RN !24
  25. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒڑ཭ۭؒ ͷ׬උੑ  ఆཧQSPPG RN w ڑ཭  Q

       ͷ΋ͱͰɺڑ཭ۭؒ ͸׬උڑ཭ۭؒʹͳΔɻ w  ͨͩ͠ɺ ͕ ͷίʔγʔྻͰ͋Ε͹ɺ೚ҙͷਖ਼਺ʹରͯ͠ɺ͋Δࣗ વ਺ ͕ଘࡏ͠ɺ ͳΒ͹  ίʔγʔྻͷఆ͔ٛΒ  w ͞Βʹɺ͢΂ͯͷL L  / ʹ͍ͭͯ  Q৐͢Ε͹໌Β͔  w Αͬͯɺ ͸ڑ཭ۭؒ3ͰͷίʔγʔྻͰɺఆཧ  ͔Β ͸͋Δ࣮਺ ʹऩଋ͢ Δɻ w ໰͔Βɺ ͕͋Δ࣮਺ ʹऩଋ͢Δͱ͖ ͸ ʹऩଋ͢Δɻ ূ໌ऴΘΓ dp (x, y) = p N ∑ i=1 |xi − yi |p ∞ RN (am )∞ m=1 am := (a(m) 1 , a(m) 2 , …, a(m) N ) RN ϵ N1 N1 < m1 , m2 d(am1 , am2 ) < ϵ |a(m1 ) k − a(m2 ) k | < d(am1 , am2 ) < ϵ (a(m) k )∞ m=1 (a(m) k )∞ m=1 αk ∈ R (a(m) k )∞ m=1 αk ∈ R (am )∞ m=1 α ∈ RN !25
  26. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒྫ୊  ͷ׬උੑ   lp w ྫ୊ڑ཭ۭؒ 

    ͸׬උͰ͋Δɻ w  ͔ΒͳΔ఺ྻ ͕ ͷίʔγʔྻͰ͋Δͱ͖ɺ͜Ε͕  ͷ͋Δ఺ʹऩଋ͢Δ͜ͱΛࣔͤ͹Α͍ɻ೚ҙͷ ʹରͯ͋͠Δਖ਼੔਺ ͕ଘ ࡏ͠ɺ ͳΒ͹ɿ ͱͳΔɻ͜ΕΑΓɺશ ͯͷK  ʹରͯ͠ɺ ͳΒ͹ɿ Ͱ͋ΔɻKΛ ݻఆ͠ɺ Λఆٛ͢Ε͹͜Ε͸3ͷίʔγʔྻʹͳ͍ͬͯΔɻ3͸׬උͰ͋ ΔͷͰ ͸ऩଋ͠ɺ ͕ఆٛͰ͖Δɻ (lp, dp ) (1 ≤ p ≤ ∞) xn := (ξ(n) 1 , ξ(n) 2 , . . . ) ∈ lp (xn )∞ n=1 lp lp ϵ > 0 N N < n, m dp (xm , xn ) = ∞ ∑ j=1 |ξ(m) j − ξ(n) j |p 1 p < ϵ N < n, m |ξ(m) j − ξ(n) j | ≤ dp (xm , xn ) < ϵ (ξ(n) j )∞ n=1 (ξ(n) j )∞ n=1 ξj = lim n→∞ (ξ(n) j ) ∈ R !26
  27. ,ZPUP6OJWFSTJUZ ׬උڑ཭ۭؒྫ୊  ͷ׬උੑ   lp w ྫ୊ڑ཭ۭؒ 

    ͸׬උͰ͋Δɻ w  ͸఺ྻ ͷऩଋઌͷ༗ྗͳީิͰ͋Δɻ࣮ࡍ ͱͳΓɺ ͱͳΔ͜ͱΛࣔ ͢ɻ ͔Βɺ೚ҙͷࣗવ਺Lʹରͯ͠/N OͳΒ͹ ͱͳΔ ͷͰɺ/NͳΒ͹ ͕੒ཱ͢Δɻ͜ͷ͜ͱ͔Β఺ྻ ͸Lʹ্͍ͭͯʹ༗քͳ୯ௐ૿Ճ࣮਺ྻͰ͋ΔͷͰɺ ͱͨ͠ͱ͖ۃݶ͕ఆ·Δɻ͕ͨͬͯ͠N/ͳΒ ͹ɺ ΋ಘΔͷͰ ͕อূ͞ΕΔɻ͞Βʹɺ ʹ ରͯ͠ Ͱ͋Δ ࣜ ͷͰɺ ͱͳΔɻΑͬͯN /ͳΒ͹ ͕੒ཱ͠ɺ ͱͳΔɻ ূ໌ऴΘΓ (lp, dp ) (1 ≤ p ≤ ∞) x := (ξ1 , ξ2 , . . . ) ∈ R (xn )∞ n=1 x ∈ lp lim n→∞ xn = x dp (xm , xn ) = ∞ ∑ j=1 |ξ(m) j − ξ(n) j |p 1 p < ϵ k ∑ j=1 |ξ(m) j − ξ(n) j |p < ϵp k ∑ j=1 |ξ(m) j − ξj |p = lim n→∞ k ∑ j=1 |ξ(m) j − ξ(n) j |p ≤ ϵp k ∑ j=1 |ξ(m) j − ξj |p ∞ k=1 k → ∞ ∞ ∑ j=1 |ξ(m) j − ξj |p = lim k→∞ k ∑ j=1 |ξ(m) j − ξj |p ≤ ϵp (ξ(m) j − ξj) ∞ j=1 ∈ lp ∀x, y ∈ lp x − y := (xi − yi )∞ i=1 ∈ lp x = (ξj )∞ ( j=1) = (ξ(m) j )∞ ( j=1) − (ξ(m) j − ξj) ∞ ( j=1) ∈ lp dp (xm , x) ≤ ϵ lim n→∞ xn = x !27
  28. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ w ։ू߹ɾดू߹ͷఆٛ w ։ू߹ɾดू߹ͷྫ w ։ू߹ɾดू߹ͷੑ࣭ w ಺఺ɾूੵ఺ͷఆٛ

    w ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ !28
  29. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹Կ͕ศརʁ w ࣮਺௚ઢ্ͷด۠ؒͱ։۠ؒ͸ڑ཭ۭؒͷดू߹ͱ։ू߹ͷ֓೦ ʹҰൠԽ͞ΕΔ w ղੳֶͷୈҰา఺Λগ͚ͩ͠ಈ͔ͯ͠ΈΔɺ͜ͱ w ։ू߹Ͱ͸఺Λ౓Ͳͷํ޲ʹಈ͔ͯ͠΋ू߹͔Βඈͼग़͢ ͜ͱ͕ͳ͍͜ͱ͕อূ͞ΕΔ

    ಈ͔͢ํ޲ʹྫ֎Λઃఆͯ͠ ٞ࿦͢Δඞཁ͕ͳ͘ͳΔ → !29
  30. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ఆٛ ։ू߹ͱดू߹ w ڑ཭ۭؒ 9 E ͷ෦෼ू߹ ʹ͍ͭͯ w

    ೚ҙͷ ʹରͯ͠ɺΛத৺ͱ͢Δਖ਼ͷ൒ܘSΛ࣋ͭ։ٿ ͕ͭͰ ΋ଘࡏ͠ɺ ͱͰ͖Δͱ͖ɺ4͸ʮ։ू߹Ͱ͋Δʯͱ͍͏ɻશମू߹9ͱۭू߹ ೚ҙͷू߹ ͷ෦෼ू߹ ΋։ू߹Ͱ͋Δɻ w ։ू߹ͷ఺͸Ͳͷํ޲ʹ গ͚ͩ͠ ಈ͔ͯ͠΋ू߹಺ʹͱͲ·Δ w 4ͷิू߹ ͕։ू߹ͱͳΔͱ͖ɺ4͸ʮดू߹ʯͰ͋Δͱ͍͏ɻશମू߹9ͱۭू߹͸ดू߹Ͱ ΋͋Δɻ w ू߹4ΛؚΉ࠷খͷดू߹Λ4ͷดแ DMPTVSF ͱ͍͍ɺ Ͱද͢ɻ 4ΛؚΉશͯͷดू߹ͷڞ௨෦෼ ͕ ͱͳΔɻ  w  ͕4ͷ৮఺Ͱ͋Δɺͱ͸೚ҙͷਖ਼਺Sʹରͯ͠ ͱͳΔ͜ͱͰ͋Δ ͲΜͳʹؤ ுͬͯ΋ۙ๣͕4ʹೖͬͯ͠·͏ ɻ4ͷดแ͸4ͷ৮఺શମͷू߹Ͱ͋Δɻ S ⊂ X x ∈ S x B(x, r) := {y ∈ X|d(x, y) < r} B(x, r) ⊂ S X \S ¯ S ¯ S x ∈ X B(x, r) ∩ S ≠ ∅ !30
  31. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ྫ୊ ։ू߹ͱดू߹ͷྫ w ڑ཭ۭؒ   ͨͩ͠ ʹରͯ͠ɺ։ٿ ͷਖ਼ମ͸։۠ؒ

     ͱͳΔɻ೚ҙͷ։۠ؒ΍ͦͷ೚ҙݸͷ߹ซ΍༗ݶݸͷڞ௨෦෼ ू߹͸3ͷ։ू߹ͱͳΔɻҰํͰɺด۠ؒ΍ɺ೚ҙͷ఺ ͔ΒͳΔू߹  ΍͜ΕΒͷ༗ݶݸͷ߹ซɺ೚ҙݸͷڞ௨෦෼ू߹͸ดू߹ʹͳΔɻ w 1SPPG3ͷதͷ։۠ؒΛ ͱ͢Δͱɺͦͷ։۠ؒͷ೚ҙͷ఺ ʹରͯ͠ɺ Λຬͨ͢Α͏ʹਖ਼਺SΛબΜͰ͜Εͯɺ։ٿ  ͸ ͷ෦෼ू߹ʹͳ͍ͬͯΔͷͰɺ ͸3ͷ։ू߹ͱ ͳ͍ͬͯΔɻ w ։۠ؒ B C ͷดแ͸ด۠ؒ<B C>Ͱ͋Δ R, d d(x, y) = |x − y| B(x, r) (x − r, x + r) x ∈ X {x} ⊂ X (a, b) x ∈ (a, b) a < x − r, x + r < b (x − r, x + r) (a, b) (a, b) !31
  32. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ྫ୊ ։ू߹ͱดू߹ͷྫ w  Λ༻͍ͯఆٛ͞ΕΔ ͷ෦෼ू߹   ͸ڑ཭ۭؒ

    ͷ։ू߹ͱͳ Δɻ w 1SPPG4ͷ೚ҙͷཁૉͷۙ๣ZΛௐ΂ͯɺۙ๣΋4ͷཁૉͰ͋Δ͜ͱΛࣔ ͢ɻ w ೚ҙͷ ʹରͯ͠ɺ  ͕อূ͞ΕΔɻ·ͨ೚ҙͷ ʹରͯ͠ɺ  ͕͢΂ͯͷJʹ͍ͭͯ੒ཱ͢ΔɻΑͬͯ  ͕੒Γཱͪɺ ͱͳΓɺ4͸։ू߹ͱͳΔɻ t := (t1 , t2 , . . . , tN ) ∈ RN RN S := {x = (x1 , x2 , . . . , xN )|xi < ti } (∀i = 1,2,...,N) (RN, dp ) ˜ x = (˜ x1 , ˜ x2 , . . . , ˜ xN ) ∈ S r := min{t1 − ˜ x1 , . . . , tN − ˜ xN } > 0 y = (y1 , y2 , . . . , yN ) ∈ B(˜ x, r) |yi − ˜ xi | ≤ dp (y, ˜ x) < r ≤ ti − ˜ xi yi < ti y ∈ S !32
  33. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ w B ༗ݶݸͷ։ू߹  J   O

    ʹରͯ͠ɺڞ௨෦෦෼ू߹  ͸։ू߹ͱͳΔɻͨͩ͠Մࢉແݶݸͷ։ू߹ͷڞ௨෦෼ू ߹ ͸։ू߹ʹͳΔͱ͸ݶΒͳ͍ɻ w C ༗ݶݸͷดू߹   ͸։ू߹ ͷ߹ซ ͸ดू߹ ͱͳΔɻͨͩ͠Մࢉແݶ͜ͷดू߹ͷ߹ซ ͸ดू߹ʹͳΔ ͱ͸ݶΒͳ͍ɻ·ͨɺ೚ҙݸͷดू߹ͷڞ௨෦෼ू߹͸ดू߹ͱ ͳΔɻ Ui ∩n i=1 Ui ∩∞ i=1 Ui Si := X\Ui Ui ∪n i=1 Si ∪∞ i=1 Si !33
  34. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ B ূ໌ w B ༗ݶݸͷ։ू߹  J 

     O ʹରͯ͠ɺڞ௨෦෦෼ू߹ ͸։ू߹ͱͳΔɻ ͨͩ͠Մࢉແݶݸͷ։ू߹ͷڞ௨෦෼ू߹ ͸։ू߹ʹͳΔͱ͸ݶΒͳ͍ɻ w ڞ௨෦෼ू߹͕ۭू߹ͳΒ͹ɺ੒Γཱͭɻ ۭू߹͸։ू߹  w ۭͰͳ͚Ε͹ɺఆ͔ٛΒ ʹ͍ͭͯ  J   O ͱͳΔ։ٿ͕ ଘࡏ͢Δɻ Λ൒ܘͱ͢Δ։ٿ ͕ Λຬ ͨ͢ɻ w ҰํͰ Λத৺ͱ͠ɺ  J   ͕ଘࡏ͢Δ͕ɺ ͱ ͳΔՄೳੑ͕͋ΓɺYΛத৺ͱ͢Δਖ਼ͷ൒ܘΛ࣋ͭ։ٿΛ ʹऔΕΔͱ͸ݶ Βͳ͍ɻ Ui ∩n i=1 Ui ∩∞ i=1 Ui x ∈ ∩n i=1 Ui B(x, ri ) ⊂ Ui rmin := min{ri }n i=1 > 0 B(x, rmin ) B(x, rmin ) ⊂ ∩n i=1 Ui x ∈ ∩∞ i=1 Ui B(x, ri ) ⊂ Ui inf{ri }∞ i=1 = 0 ∩∞ i=1 Ui !34
  35. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ C ͷূ໌ w C ༗ݶݸͷดू߹   ͸։ू߹

    ͷ߹ซ ͸ดू߹ ͱͳΔɻ w υɾϞϧΨϯͷ๏ଇ͔Βɺ  ͱදͤΔɻ ͸։ू ߹ͱͳΔ ੑ࣭ B ͷͰɺ ͸ดू߹ͱͳΔɻ Si := X\Ui Ui ∪n i=1 Si ∪n i=1 Si = [( ∪n i=1 Si) C ] C = [( ∩n i=1 Ui)] C ∩n i=1 Ui ∪n i=1 Si !35
  36. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ఆٛ ಺఺ͱूੵ఺  ڑ཭ۭؒͷดू߹ͱ։ू߹͸఺ྻͷۃݶͷ֓೦Ͱಛ௃͚ͮΒΕΔ w ڑ཭ۭؒ9ͷ෦෼ू߹4ͱ఺ ʹରͯ͠ɺ͋Δਖ਼਺SΛ͕ଘࡏ͠ɺ։ٿ  ʹରͯ͠ɺ

    ͱͰ͖Δͱ͖ɺY͸4ͷ಺఺Ͱ͋ Δͱ͍͏ɻ4ͷ಺఺ΛશͯूΊͯͰ͖Δू߹Λ4ͷ಺෦ͱ͍͍ɺ Ͱද͢ 4ʹؚ ·ΕΔ࠷େͷ։ू߹ɾ4ʹؚ·ΕΔ։ू߹ͷ߹ซ ɻ w ʮ4͕։ू߹ʯ ʮ ʯ w  Λ4ͷڥքͱ͍͏ɻ ʮY͕4ͷ৮఺Ͱ͋Δʯ ʮY͸4ͷ಺෦͔ڥքʯ  w ڑ཭ۭؒ9ͷ෦෼ू߹4ͱ఺ ͕༩͑ΒΕɺYʹऩଋ͢Δ఺ྻ ͕ఆٛ Ͱ͖Δͱ͖ɺY͸4ͷूੵ఺Ͱ͋Δͱ͍͏ɻ4ͷूੵ఺ΛશͯूΊͯͰ͖Δू߹Λ 4ͷಋू߹ͱ͍͍ɺ Ͱද͢ɻ x ∈ S B(x, r) := {y ∈ X|d(x, y) < r} B(x, r) ⊂ S SO ⇔ S = So ∂S := ¯ S\So ⇔ x ∈ X xn ∈ S\{x} Sd !36
  37. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ w ڑ཭ۭؒ9ͷ෦෼ू߹4ʹ͍ͭͯɺ w B  ɺͨͩ͠ ͸4ͷดแ

    4ΛؚΉ࠷খͷดू߹  w C ʮ4͸ดू߹ʯ ʮ ʯ w D ʮ4͸ดू߹ʯ ʮ ʹऩଋ͢Δ఺ྻ  O   ͕ଘࡏ͢ΔͳΒ ͱͳΔʯ ¯ S = S ∪ Sd ¯ S ⇔ Sd ⊂ S ⇔ x ∈ X xn ∈ S x ∈ S !37
  38. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ ͷূ໌ w B ڑ཭ۭؒ9ͷ෦෼ू߹4ʹ͍ͭͯɺ ɺͨͩ͠ ͸4ͷดแ 4ΛؚΉ࠷খͷดू߹

       ͸ดแͳͷͰɺ ͕ดू߹ʹͳΔ͜ͱΛࣔͤ͹Α͍ɻͭ·Γ ͕։ू߹Ͱ ͋Δ͜ͱΛࣔͤ͹Α͍ɻ೚ҙͷ ͸4ͷूੵ఺Ͱ͸ͳ͍ͷͰ  ͸ूੵ఺ͷू߹ͳͷͰ ɺ   O   ͱͳΔΑ͏ͳ఺ྻ ͸ଘࡏ͠ͳ͍ɻ ଘࡏ͢ΔͱY͕ूੵ఺ʹͳΔɻ ͕ͨͬͯࣗ͠વ਺/͕ଘࡏ͠೚ҙͷ ʹରͯ͠ ͱͳΔɻ·ͨɺશͯͷूੵ఺  ʹରͯ͠ ͱͳΔ఺ྻ ͕ଘࡏ͠ɺࡾ֯ෆ౳͔ࣜΒ೚ҙͷࣗવ਺Oʹରͯ͠  ͕੒ཱ͠ɺ ͱͳΓ  ɺΑͬͯ ͱͳΓɺ ͸։ू߹ͱͳΔɻ ূ໌ऴΘ Γ ¯ S = S ∪ Sd ¯ S ¯ S ⊂ S ∪ Sd ¯ S S ∪ Sd (S ∪ Sd) C x ∈ (S ∪ Sd) C Sd xn ∈ B (x, 1 n) ∩ S (xn )∞ n=1 ϵ ∈ (0, 1 N ) B (x, ϵ) ∩ S = ∅ y ∈ Sd lim n→∞ yn = y (yn )∞ n=1 d(x, y) ≥ d(x, yn ) − d(y, yn ) ≥ ϵ − d(y, yn ) d(x, y) ≥ lim n→∞ ϵ − d(y, yn ) = ϵ B (x, ϵ 2) ∩ (S ∪ Sd) = ∅ B (x, ϵ 2) ⊂ (S ∪ Sd) C (S ∪ Sd) C !38
  39. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ ͷূ໌ w B ڑ཭ۭؒ9ͷ෦෼ू߹4ʹ͍ͭͯɺ ɺͨͩ͠ ͸4ͷ ดแ

    4ΛؚΉ࠷খͷดू߹    ͷఆ͔ٛΒɺ ͸໌Β͔ͳͷͰ Λࣔͤ͹ े෼ɻ ͱͳΔY͕ଘࡏ͢ΔͱԾఆ͠ໃ६Λಋ͘ɻ Ͱɺ ͸։ू߹ͳͷͰ͋Δਖ਼਺͕ଘࡏ͠ɺ ͱͰ ͖Δɻ͜Ε͔Βɺ ΛಘΔ͕ɺ͜Ε ͸ ʹໃ६͢Δɻ ¯ S = S ∪ Sd ¯ S ¯ S ⊃ S ∪ Sd ¯ S ¯ S ⊃ S ¯ S ⊃ Sd x ∈ Sd\ ¯ S x ∈ ( ¯ S) C ( ¯ S) C ϵ B(x, ϵ) ⊂ ( ¯ S) C (B(x, ϵ) ∩ S) ⊂ (B(x, ϵ) ∩ ¯ S) = ∅ x ∈ Sd !39
  40. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ w C ڑ཭ۭؒ9ͷ෦෼ू߹4ʹ͍ͭͯɺʮ4͸ดू߹ʯ ʮ ʯ w 

    4͕ดू߹ͳΒ4ࣗ਎͕ʮ4ΛؚΉ࠷খͷดू߹ʯͱͳΔ ͷͰɺ ͱͳΔɻ·ͨੑ࣭ B  ͔Βɺ ͱ ͳΔɻ w   ͳΒ͹ɺੑ࣭ B  ͔Β ͔ͭɺ ΑΓ ɻ͕ͨͬͯ͠4͸ดू߹ 4ࣗ਎͕4ͷด แ ͱͳΔɻ ⇔ Sd ⊂ S ⇒ ¯ S = S ¯ S = S ∪ Sd S ⊃ Sd ⇐ Sd ⊂ S ¯ S = S ∪ Sd S ⊃ (S ∪ Sd) = ¯ S ¯ S ⊃ S S = ¯ S !40
  41. ,ZPUP6OJWFSTJUZ ։ू߹ͱดू߹ੑ࣭ ఺ྻʹΑΔดू߹ͷಛ௃͚ͮ w D ڑ཭ۭؒ9ͷ෦෼ू߹4ʹ͍ͭͯɺʮ4͸ดू߹ʯ ʮ ʹऩଋ ͢Δ఺ྻ 

    O   ͕ଘࡏ͢ΔͳΒ͹ ͱͳΔʯ w  4͕ดू߹ͳΒ͹ɺੑ࣭ C ΑΓɺ ͕อূ͞ΕΔɻ఺ ྻ ͕ ͔ͭ ͳΒ͹ɺY͸4ͷूੵ఺ͱͳΔ͕ɺ ͜Ε͸ ʹໃ६ ೚ҙͷूੵ఺͸4ͷதʹ͋Δ  w  ʮ ʯΛ͔֬ΊΕ͹Α͍ɻ ੑ࣭ C  ͕ଘࡏ͢ Δͱ͢Δͱɺूੵ఺ͷఆ͔ٛΒYʹऩଋ͢Δ఺ྻ ͕ଘࡏ͠ɺ͜ Ε͸৚݅ʹໃ६ ⇔ x ∈ X xn ∈ S x ∈ S ⇒ Sd ⊂ S xn ∈ S lim n→∞ xn = x ∈ X x ∉ S Sd ⊂ S ⇐ Sd ⊂ S x ∈ Sd\S xn ∈ S !41
  42. ,ZPUP6OJWFSTJUZ w IUUQTXJJTJOGPNBUISFBMOVNCFSUPQPMPHZBEIFSFOUQPJOU w IUUQXXXNBUIUJUFDIBDKQdLPUBSPDMBTTTFUQEG !42