ʹॱংߏ଄ͱ୯ௐੑʹجͮ͘΋ͷ͕ࡏΔɻ͜͜Ͱ͸ओʹ Amann ͱ Zeidler ʹैͬͯɺॱং

ॱংߏ଄ʹجͮ͘ෆಈ఺ఆཧ

ฏ੒ 28 ೥ 11 ݄ খᖒɹప http://www.ozawa.phys.waseda.ac.jp/index2.html

ෆಈ఺ఆཧͷ୅දతͳྫͱͯ͠ڑ཭ͱ׬උੑʹجͮ͘΋ͷ (όφϋͷෆಈ఺ఆཧ౳) ΍ ίϯύΫτੑʹجͮ͘΋ͷ (ϒϥ΢ϫʔͷෆಈ఺ఆཧ΍γϟ΢μʔͷෆಈ఺ఆཧ౳) ͷ֎

ʹॱংߏ଄ͱ୯ௐੑʹجͮ͘΋ͷ͕ࡏΔɻ͜͜Ͱ͸ओʹ Amann ͱ Zeidler ʹैͬͯɺॱং

ू߹ʹԙ͚Δෆಈ఺ఆཧΛవΊͯஔ͜͏ɻঘɺπΥϧϯͷิ୊ (ٴͼಉ஋ͳҰ࿈ͷ໋୊) ͸

༻͍ͣɺ ʮۭͰͳ͍೚ҙͷશॱং෦෼ू߹্͕ݶΛ࣋ͭॱংू߹ʯͷ࿮૊Ͱٞ࿦ΛਐΊΔɻ

̍ɽॱংू߹

ۭͰͳ͍ू߹ X ͸࣍ͷੑ࣭ (O1)-(O3) Λຬͨ͢ೋ߲ؔ܎ ≤ Λ࣋ͭͱ͖ ( ൒ ) ॱংू߹

((partially) orderd set) Ͱ͋ΔͱҦ͍ɺೋ߲ؔ܎ ≤ Λ X ্ͷॱং (order on X) ͱҦ͏

(O1) ೚ҙͷ x ∈ X ʹର͠ x ≤ x

(O2) x, y ∈ X ͸ x ≤ y ׌ͭ y ≤ x ͳΔͱ͖ x = y (O3) x, y, z ∈ X ͸ x ≤ y ׌ͭ y ≤ z ͳΔͱ͖ x ≤ z

෉ʑ(O1)ɺ(O2)ɺ(O3) ͷੑ࣭Λॱং ≤ ͕൓ࣹత (reflexive)ɺ൓ରশత (antisymmetric)ɺ ਪҠత (transitive) Ͱ͋ΔͱҦ͏ɻ x < y ͱ͸ x ≤ y ׌ͭ x = y Ͱ͋Δࣄͱఆٛ͢Δɻ x ≤ y Λ y ≥ x ͱදͨ͠Γ x < y Λ y > x ͱද͢ࣄ΋͋ΔɻX ͷ೚ҙͷೋͭͷݩ x, y ʹର͠

x ≤ y ຢ͸ y ≤ x ͷԿΕ͔͕ৗʹ੒ཱͭͱ͖ॱং ≤ ͸ X ʹԙ͚Δશॱং (total order) ҃͸

ઢܕॱং (linear order) Ͱ͋ΔͱҦ͍શॱংΛ࣋ͭॱংू߹Λશॱংू߹ (totally ordered set) ҃͸ઢܕॱংू߹ (linearly ordered set) ͱҦ͏ɻ

ఆٛɹॱংू߹ X ͷݩ x ʹର͠

(1) x ͸ X ͷ࠷େݩ (greatest element) ⇔

def. ೚ҙͷ y ∈ X ʹର͠ y ≤ x (2) x ͸ X ͷۃେݩ (maximal element) ⇔

def. x < y ͳΔ y ∈ X ͸ଘࡏ͠ͳ͍

(3) x ͸ X ͷ࠷খݩ (least element) ⇔

def. ೚ҙͷ y ∈ X ʹର͠ x ≤ y (4) x ͸ X ͷۃখݩ (minimal element) ⇔

def. y < x ͳΔ y ∈ X ͸ଘࡏ͠ͳ͍

໋୊ɹॱংू߹ X ʹର͕࣍͠੒ཱͭ

(1) X ͷ࠷େݩ ( ͕ଘࡏ͢Δ৔߹ ) ʹ͸Ұҙతʹఆ·Δɻ

(2) X ͷ࠷খݩ (͕ଘࡏ͢Δ৔߹) ʹ͸Ұҙతʹఆ·Δɻ

(3) X ͷۃେݩ (͕ଘࡏ͢Δ৔߹) ʹ͸Ұҙతʹఆ·Δɻ

(4) X ͷۃখݩ (͕ଘࡏ͢Δ৔߹) ʹ͸Ұҙతʹఆ·Δɻ

(5) X ͷ࠷େݩ (͕ଘࡏ͢Δ৔߹) ͸།ҰͷۃେݩͰ͋Δɻ

(6) X ͷ࠷খݩ ( ͕ଘࡏ͢Δ৔߹ ) ͸།ҰͷۃখݩͰ͋Δɻ (7) x ͕ X ͷۃେݩͰ͋Δҝͷඞཁॆ෼৚݅͸

ʮx ≤ y ͳΔ y ∈ X ͕ଘࡏͨ͠ͱ͢Ε͹ y = xʯͰ͋Δɻ (8) x ͕ X ͷۃখݩͰ͋Δҝͷඞཁॆ෼৚݅͸

ʮy ≤ x ͳΔ y ∈ X ͕ଘࡏͨ͠ͱ͢Ε͹ y = xʯͰ͋Δɻ

(9) X ͕શॱংू߹ͳΒ͹ɺ࠷େݩͱۃେݩͷ֓೦ٴͼ࠷খݩͱۃখݩͷ֓೦͸Ұக

͢Δɻ

(ূ໌) (1) ࠷େݩ͕ೋͭ͋ͬͨͱͯͦ͠ΕΒΛ x ͱ x ͱ͢Δɻx ͸ X ͷ࠷େݩ͔ͩΒ

x ∈ X ʹରͯ͠ x ≤ x ͱͳΔɻಉ༷ʹ x ͸ X ͷ࠷େݩ͔ͩΒ x ∈ X ʹରͯ͠ x ≤ x ͱͳΔɻ൓ରশ཯ (O2) ΑΓ x = x ͕ै͏ɻ

(2) (1) ͱಉ༷Ͱ͋Δɻ

(3) x Λۃେݩͱ͠೚ҙͷ x ∈ X \ { x } ΛऔΔͱ x < x Ͱ͸༗Γಘͳ͍ͷͰ x < x ͱͳ Δɻ͜Ε͸ x ͕ۃେݩͰ͋Δࣄʹ൓͢Δɻނʹ x ͸།ҰͷۃେݩͰ͋Δɻ

(4) (3) ͱಉ༷Ͱ͋Δɻ

(5) x Λ࠷େݩͱ͢Δͱ೚ҙͷ y ∈ X ʹର͠ y ≤ x ͱͳΔ͔Β (ͦ͏Ͱͳ͍)x < y ͳΔ ݩ y ∈ X ͸ଘࡏ͠ͳ͍ɻଈͪ x ͸ۃେݩͰ͋Δɻ

(6) (5) ͱಉ༷Ͱ͋Δɻ (7) x ͸ X ͷۃେݩ

⇐⇒ x < y ͳΔ y ∈ X ͸ଘࡏ͠ͳ͍

⇐⇒ x ≤ y ͳΔ y ∈ X ͸ଘࡏͨ͠ͱ͢Ε͹ x = y Ͱ͋Δɻ (8) (7) ͱಉ༷Ͱ͋Δɻ

ఆٛɹॱংू߹ X ͷۭͰͳ͍෦෼ू߹ A ͱҰ఺ x ∈ X ʹର͠

(1) x ͸ A ͷҰͭͷ্ք (an upper bound for A) ⇔

def. ೚ҙͷ a ∈ A ʹର͠ a ≤ x

(2) x ͸ A ͷҰͭͷԼք (an lower bound for A) ⇔

def. ೚ҙͷ a ∈ A ʹର͠ x ≤ a

(3) A ͸্ʹ༗ք (bound from above) ⇔

def. A ʹର͢Δ্ք͕গͳ͘ͱ΋Ұͭଘࡏ͢Δ (4) A ͸Լʹ༗ք (bound from below) ⇔

def. A ʹର͢ΔԼք͕গͳ͘ͱ΋Ұͭଘࡏ͢Δ (5) x ͸ A ͷ্ݶ (the supremum of A) ⇔

def. x ͸ A ͷ࠷খ্ք (্քͷ੒͢ू߹ͷ࠷

খݩ)

(6) x ͸ A ͷԼݶ (the infimum of A) ⇔

def. x ͸ A ͷ࠷খԼք ( Լքͷ੒͢ू߹ͷ࠷େݩ ) A ͷ্ݶ͕ଘࡏ͢Δͱ͖ɺͦͷݩΛ sup A ͱද͠ɺԼݶ͕ଘࡏ͢Δͱ͖ͦͷݩΛ inf A ͱද͢ɻ

໋୊ɹॱংू߹ X ͷۭͰͳ͍෦෼ू߹ A ͱ x ∈ X ʹର͕࣍͠੒ཱͭɻ (1) x = sup A

⇔

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

(i) ೚ҙͷ a ∈ A ʹର͠ a ≤ x

(ii) x ∈ X ͸೚ҙͷ a ∈ A ʹର͠ a ≤ x Λຬ͍ͨͯ͠Δ΋ͷͱ͢Δͱ x ≤ x

(2) x = inf A

⇔

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

(i) ೚ҙͷ a ∈ A ʹର͠ x ≤ a

(ii) x ∈ X ͸೚ҙͷ a ∈ A ʹର͠ x ≤ a Λຬ͍ͨͯ͠Δ΋ͷͱ͢Δͱ x ≤ x

(ূ໌) (1) (i) ͸ x ͕Ұͭͷ্քͰ͋ΔࣄΛҙຯ͠ (ii) ͸ x ͕શͯͷ্քͷ੒͢ू߹ͷ࠷খ

ݩͰ͋ΔࣄΛҙຯ͢Δɻ (2) (1) ͱಉ༷Ͱ͋Δɻ

ఆٛɹॱংू߹ X ٴͼ x ₀ ∈ X ʹର͠

X ₊ (x ₀ ) = { x ∈ X; x ≥ x ₀ } Λ x ₀ ͷӈ੾ย (the right section at x ₀ )

X ₋ (x ₀ ) = { x ∈ X; x ≤ x ₀ } Λ x ₀ ͷࠨ੾ย (the left section at x ₀ ) ͱҦ͍ a, b ∈ X ʹର͠

[a, b] = X ₊ (a) ∩ X ₋ (b) = { x ∈ X; a ≤ x ≤ b }

Λ a, b Λ୺఺ͱ͢Δॱং۠ؒ (order interval between a and b) ͱҦ͏ɻ

ம . ɹ [a, b] = ∅ ⇐⇒ a ≤ b

ఆٛɹॱংू߹ X ͔Βॱংू߹ Y ΁ͷࣸ૾ f : X −→ Y ͸

x ≤ y ͳΔ೚ҙͷ x, y ∈ X ʹର͠ f (x) ≤ f (y)

Ͱ͋Δͱ͖୯ௐ (monotone) ҃͸୯ௐ૿Ճ (monotone increasing) Ͱ͋ΔͱҦ͏ɻ ம . ɹॱংू߹ X ͔ΒͦΕࣗ਎΁ͷ୯ௐࣸ૾ f : X −→ X ͸

• x ₀ ≤ f(x ₀ ) ͳΔ x ₀ ∈ X ͕ଘࡏ͢Ε͹ f (X ₊ (x ₀ )) ⊂ X ₊ (x ₀ )

• f(x ₀ ) ≤ x ₀ ͳΔ x ₀ ∈ X ͕ଘࡏ͢Ε͹ f (X ₋ (x ₀ )) ⊂ X ₋ (x ₀ ) Λຬͨ͢ɻ࣮ࡍ

• x ₀ ≤ f(x ₀ ), x ∈ X ₊ (x ₀ )

⇐⇒ x ₀ ≤ f (x ₀ ), x ₀ ≤ x

= ⇒ x ₀ ≤ f (x ₀ ), f (x ₀ ) ≤ f(x)

= ⇒ x ₀ ≤ f (x)

⇐⇒ f (x) ∈ X ₊ (x ₀ )

• f(x ₀ ) ≤ x ₀ , x ∈ X ₋ (x ₀ )

⇐⇒ f (x ₀ ) ≤ x ₀ , x ≤ x ₀

= ⇒ f (x ₀ ) ≤ x ₀ , f (x) ≤ f(x ₀ )

= ⇒ f (x) ≤ x ₀

⇐⇒ f (x) ∈ X ₋ (x ₀ )

ैͬͯ a ≤ f (a) ׌ͭ f(b) ≤ b ͳΒ͹ f ([a, b]) ⊂ [a, b] ͕੒ཱͭɻ

ఆٛɹॱংू߹ X ͸೚ҙͷೋ఺ x, y ∈ X ʹର͠ೋ఺ू߹ { x, y } ͷ্ݶ sup { x, y } ٴͼԼ ݶ sup { x, y } ͕ (X ͷݩͱͯ͠) ଘࡏ͢Δͱ͖ଋ (lattice) ͱҦ͏ɻଋ X ͸ͦͷ೚ҙͷۭͰ ͳ͍෦෼ू߹্͕ݶٴͼԼݶΛ (X ͷݩͱͯ͠) ࣋ͭͱ͖׬උ (complete) Ͱ͋ΔͱҦ͏ɻ ம . ɹ׬උଋ͸࠷େݩٴͼ࠷খݩΛ࣋ͭɻ࣮ࡍ X ͷ্ݶٴͼԼݶ͕ X ͷݩͱͯ͠ଘࡏ͠ɺ

ͦΕΒ͕෉ʑ࠷େݩٴͼ࠷খݩͱͳΔɻ

̎ɽϒϧόΩɾΫωʔβʔͷෆಈ఺ఆཧ

ఆཧ ( ϒϧόΩɾΫωʔβʔͷෆಈ఺ఆཧ ) X Λॱংू߹Ͱ࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ

(i)X ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹͸্ݶΛ࣋ͭɻ

X ͔ΒͦΕࣗ਎΁ͷࣸ૾ f ͸࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ (ii) ೚ҙͷ x ∈ X ʹର͠ x ≤ f (x)

͜ͷͱ͖ f ͸ෆಈ఺Λ࣋ͭɻͦͷෆಈ఺͸࣍Ͱ༩͑ΒΕΔɻ x ∈ X ʹର͠ ψ(x) = sup F (x) ͱஔ͘ɻ͜͜ʹ

F (x) = { M ⊂ X;M ͸ x ΛؚΈ f ͰෆมͰ

M ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹͸ M ʹ্ݶΛ࣋ͭ }

ͱ͢Δɻψ(x) ͸ X ͷݩͱͯ͠Ұҙతʹఆ·Γ f ͷෆಈ఺ (f (ψ(x)) = ψ (x)) ͱͳΔɻߋ ʹɺ೚ҙͷ x ∈ X ʹର͠ x ≤ ψ (x) Λຬͨ͢ɻ

(ূ໌) X ͸ۭͰͳ͍͔ΒҰͭͷ఺Λ೚ҙʹऔΓ x ₀ ͱ͢ΔɻX ͷ෦෼ू߹ͷ଒ F (x ₀ ) Λ

্ͷ༷ʹఆٛ͢Δɻ

ଈͪ M ∈ F (x ₀ ) ͱ͸࣍ͷࡾͭͷੑ࣭ (a)(b)(c) Λຬͨ͢΋ͷͰ͋Δɿ (a) x ₀ ∈ M

(b) f(M ) ⊂ M

(c) ∅ = C ⊂ M ͳΔ೚ҙͷશॱং෦෼ू߹ C ʹର͠ sup C ∈ M

ͯ͞ (i) ΑΓ X ∈ F (x ₀ ) Ͱ͋Δ͔Β F (x ₀ ) = ∅ ͕ै͏ɻN =

F (x ₀ ) =

M∈F (x

)

M ͱஔ͘ɻx ₀ ∈ N ނ N = ∅ ͱͳΔɻҎԼɺ؆୯ͷҝ F (x ₀ ) Λ F ͱද͢ɻ

ୈ̍ஈɹ N ∈ F ͳΔࣄΛࣔͦ͏ɻ

(a) ೚ҙͷ M ∈ F ʹର͠ x ₀ ∈ M Ͱ͋Δ͔Β x ₀ ∈ N (b) x ∈ N ⇔ ∀ M ∈ F , x ∈ M

⇒ ∀ M ∈ F , f (x) ⊂ f(M ) ⊂ M

⇒ f(x) ∈ N

(c) ∅ = C ⊂ N ͳΔશॱং෦෼ू߹Λ C ͱ͢Δɻ೚ҙʹ M ∈ F ΛऔΔɻN ⊂ M ނ C

͸ M ͷશॱং෦෼ू߹Ͱ͋Δ͔Β C ͷ্ݶ sup C ͸ M ͷݩͱͯ͠ଘࡏ͢ΔɻM ͸

೚ҙͰ͔͋ͬͨΒ sup C ∈ N ͕ै͏ɻ

ୈ̎ஈɹ N ⊂ X ₊ (x ₀ ) ͳΔࣄΛࣔͦ͏ɻX ₊ (x ₀ ) ∈ F Λࣔͤ͹ॆ෼Ͱ͋Δɻ (a) ఆٛΑΓ x ₀ ∈ X ₊ (x ₀ ) Ͱ͋Δɻ

(b) x ∈ X ₊ (x ₀ ) ⇒ x ₀ ≤ x ≤ f (x) ⇒ f (x) ∈ X ₊ (x ₀ )

(c) ∅ = C ⊂ X ₊ (x ₀ ) ͳΔશॱং෦෼ू߹Λ C ͱ͢ΔɻC ͸ X ͷશॱং෦෼ू߹Ͱ͋

Δ͔Β sup C ∈ X ͕ଘࡏ͢Δɻ೚ҙͷ x ∈ C ʹର͠ x ₀ ≤ x ≤ sup C Ͱ͋Δ͔Β

sup C ∈ X ₊ (x ₀ ) ͕ै͏ɻ

ୈ̏ஈɹ N ͷ෦෼ू߹ O Λ

O = { x ∈ N ; f(N (x) \ { x } ) ⊂ N (x) }

= { x ∈ N ; y ∈ N, y < x ⇒ f (y) ≤ x } ͱ͠ x ∈ O ʹର͠

N _x = N (x) ∪ N ₊ (f(x))

= { y ∈ N ; y ≤ x ຢ͸ f(x) ≤ y }

ͱఆΊΔɻ೚ҙͷ x ∈ O ʹର͠ N _x = N ͳΔࣄΛࣔͦ͏ɻN _x ∈ F Λࣔͤ͹ॆ෼Ͱ͋Δɻ (a) ୈ̍ஈΑΓ x ₀ ∈ N Ͱ͋Δɻx ∈ O ⊂ N Ͱ͋Γୈ̎ஈΑΓ N ⊂ X ₊ (x ₀ ) Ͱ͋ΔͷͰ

x ₀ ≤ x ͕ै͏ɻނʹ x ₀ ∈ N _x ͱͳΔɻ

(b) y ∈ N _x ΛऔΔɻy ∈ N _x ⊂ N ٴͼ f(N ) ⊂ N ΑΓ f(y) ∈ N ͱͳΔɻނʹ f(y) ∈ N _x Λࣔ͢ʹ͸

ʮf (y) ≤ x ຢ͸ f(x) ≤ f (y)ʯ Λࣔͤ͹ྑ͍ɻy ∈ N _x Ͱ͋Δ͔Β y ͸

(i) y = x ɹ (ii) y < x ɹ (iii) f (x) ≤ y ͷԿΕ͔Λຬ͍ͨͯ͠Δɻ

(i) ͳΒ f (x) = f(y) ΑΓ f (x) ≤ f (y) ͕ै͏ɻ (ii) ͳΒ x ∈ O ނ y < x ΑΓ f(y) ≤ x ͕ै͏ɻ

(iii) ͳΒ f ͷԾఆΑΓ y ≤ f (y) ͱͳΔ͔Β f (x) ≤ y ≤ f (y) ͕ै͏ɻ

(c) ∅ = C ⊂ N _x ͳΔશॱং෦෼ू߹Λ C ͱ͢ΔɻC ͸ N ͷશॱং෦෼ू߹Ͱ͋Γ

ୈ̍ஈΑΓ N ∈ F Ͱ͋Δ͔Β C ͷ্ݶ sup C ͸ N ͷݩͱͯ͠ଘࡏ͢Δɿ sup C ∈ N ҎԼ sup C ∈ N _x ͳΔࣄΛࣔͦ͏ɻ೚ҙͷ y ∈ C ⊂ N _x ͸

ʮy ≤ x ຢ͸ f(x) ≤ yʯ Λຬͨ͢ɻͯ࣍͞ͷೋͭͷ৔߹

(i) ∀ y ∈ C, y ≤ x ɹ (ii) ∃ y ∈ C : y > x ͷԿΕ͔Ұํ͕ඞͣ੒ཱ͍ͯ͠Δɻ

(i) ͳΒ sup C ≤ x ΑΓ sup C ∈ N _x ͕ै͏ɻ

(ii) ͳΒ y > x ͳΔ y ∈ C ͸ f(x) ≤ y Λຬͨ͢ࣄͱͳΔɻ͜ͷͱ͖ y ∈ C ނ f(x) ≤ y ≤ sup C ͱͳΓ sup C ∈ N _x ͕ै͏ɻ

Ҏ্͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

ୈ̐ஈɹ O = N ͳΔࣄΛࣔͦ͏ɻO ∈ F Λࣔͤ͹ॆ෼Ͱ͋Δɻ

(a) ୈ̍ஈΑΓ x ₀ ∈ N Ͱ͋Γୈ̎ஈΑΓ೚ҙͷ x ∈ N ʹର͠ x ₀ ≤ x ͱͳΔɻଈͪ

x < x ₀ ͳΔ x ∈ N ͸ଘࡏ͠ͳ͍ͷͰ x ₀ ∈ O ͱͳΔɻ

(b) x ∈ O ΛऔΔɻO ⊂ N ٴͼ f(N ) ⊂ N ΑΓ f (x) ∈ N ΛಘΔɻͯ͞ y < f(x) ͳΔ

೚ҙͷ y ∈ N ΛऔΔɻ͜ͷͱ͖ f(y) ≤ f (x) Λࣔͦ͏ɻ x ∈ O, y ∈ N = N _x ( ୈ̏

ஈ) ΑΓ y ≤ x ຢ͸ f (x) ≤ y ͕੒ཱ͢Δɻy ∈ N ͸ y < f (x) Λຬ͍ͨͯ͠ΔͷͰ f(x) ≤ y ͕੒ཱͤͣ y ≤ x ͕ै͏ɻy = x ͳΒ f (y) = f(x) ≤ f (x) ͱͳΔɻy < x ͳΒ x ∈ O, y ∈ N ނ O ͷఆٛΑΓ f (y) ≤ x ͕ै͏ɻ f ͷੑ࣭ΑΓ x ≤ f (x) ͱͳΔ ͷͰ f (y) ≤ f (x) ΛಘΔɻҎ্ΑΓ y < f (x) ͳΔ೚ҙͷ y ∈ N ʹର͠ f(y) ≤ f(x)

͕ࣔ͞Εͨɻ͜Ε͸ f(x) ∈ O Λҙຯ͢Δɻ

(c) ∅ = C ⊂ O ͳΔશॱং෦෼ू߹Λ C ͱ͢ΔɻC ͸ N ͷશॱং෦෼ू߹Ͱ N ∈ F

(ୈ̍ஈ) Ͱ͋Δ͔Β C ͷ্ݶ͸ N ͷݩͱͯ͠ఆ·Δɿsup C ∈ N

sup C ∈ O ͳΔࣄΛࣔ͢ʹ͸ y < sup C ͳΔ೚ҙͷ y ∈ N ʹର͠ f(y) ≤ sup C Ͱ͋

ΔࣄΛࣔͤ͹ྑ͍ɻͦͷҝɺ࣍ͷओுΛࣔͦ͏ɻ

y ≤ x ₁ ͳΔ x ₁ ∈ C ͷଘࡏɿɹ೚ҙʹ x ∈ C ⊂ O ΛऔΔɻୈ̏ஈΑΓ y ≤ x ຢ͸

f(x) ≤ y ͱͳ͍ͬͯΔɻશͯͷ x ∈ C ʹब͍ͯ f(x) ≤ y Ͱ͋ͬͨͱ͢Δͱ x ≤ f(x) ΑΓ x ≤ y ͕ै͍ɺ͜ΕΑΓ sup C ≤ y ΛಘΔ͕ y < sup C ʹ൓͢Δɻैͬͯશͯ

ͷ x ∈ C ʹब͍ͯ f(x) ≤ y ͱͳΓಘͳ͍ɻଈͪ f(x ₁ ) > y ͳΔ x ₁ ∈ C ͕ଘࡏ͢Δɻ

͜ͷ x ₁ ͸ (f (x ₁ ) ≤ y Λຬͨ͞ͳ͍ͷͰ)y ≤ x ₁ Λຬͨ͢ɻ

ͯ࣍͞ͷೋͭͷ৔߹ʹ෼͚ͯٞ࿦͢Δɻ (i) y = x ₁ ɹ (ii) y < x ₁

(i) y = x ₁ ͷ৔߹ɿઌͣ y < x ₂ ͳΔ x ₂ ∈ C ͕ଘࡏ͢ΔࣄΛࣔͦ͏ɻ΋ͦ͠͏Ͱͳ

͍ͱ͢Δͱ C ͸શॱংू߹Ͱ͋Γ y = x ₁ ∈ C Ͱ͋Δ͔Βʮ೚ҙͷ x ∈ C ʹର͠

x ≤ yʯͰ͋ΔࣄʹͳΔɻ͜ΕΑΓ sup C ≤ y ΛಘΔ͕ y < sup C ʹ൓͢Δɻނʹ y < x ₂ ͳΔ x ₂ ∈ C ͕ଘࡏ͢Δɻx ₂ ∈ C ⊂ O Ͱ y = x ₁ ∈ C ⊂ N Ͱ͋Δ͔Β (O ͷ ఆٛΑΓ)y < x ₂ ΑΓ f(y) ≤ x ₂ ΛಘΔɻҰํ x ₂ ≤ sup C Ͱ͋Δ͔Β f (y) ≤ sup C

͕ै͏ɻ͜Ε͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

(ii) y < x ₁ ͷ৔߹ɿx ₁ ∈ C ⊂ O Ͱ͋Γ y < x ₁ Ͱ͋Δ͔Β (O ͷఆٛΑΓ)f (y) ≤ x ₁ ͱͳΔɻҰํ x ₁ ≤ sup C Ͱ͋Δ͔Β f (y) ≤ sup C ͕ै͏ɻ͜Ε͕ࣔ͢΂͖ࣄͰ

͋ͬͨɻ

ୈ̑ஈɹ N ͸શॱং෦෼ू߹Λ੒͢ࣄΛࣔͦ͏ɻ

N = O Ͱ͋Δ͔Β೚ҙͷ x, y ∈ N ʹର͠ୈ̏ஈΑΓ y ≤ x ຢ͸ f (x) ≤ y ͱͳΔɻ f (x) ≤ y ͳΒ x ≤ f (x) ΑΓ x ≤ y ͕ै͏ɻҎ্ΑΓ೚ҙͷ x, y ∈ N ʹର͠ y ≤ x ຢ͸

x ≥ y ͱͳΔɻ

ୈ̒ஈɹ u := sup N ͕ f ͷෆಈ఺Ͱ͋ΔࣄΛࣔͦ͏ɻ

N ∈ F Ͱશॱং෦෼ू߹Ͱ͋Δ͔Β N ͸্ݶΛ N ͷݩͱͯ࣋ͭ͠ɿ u = sup N ∈ N

͜ͷ͜ͱΛ f(N ) ⊂ N ΑΓ f (u) ∈ N Ͱ͋Δ͔Β f(u) ≤ sup N = u ͱͳΔɻ

Ұํ f ͷੑ࣭ΑΓ u ≤ f(u) Ͱ͋Δɻैͬͯ f (u) = u ͱͳΓ u ͸ f ͷෆಈ఺ͱͳΔɻߋʹ x ₀ ∈ N ނ x ₀ ≤ u ͕ै͏ɻ

̏ɽΞϚϯͷෆಈ఺ఆཧ

ఆཧ ( ΞϚϯͷෆಈ఺ఆཧ ) ɹ X Λॱংू߹Ͱ࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ (i) X ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹͸্ݶΛ࣋ͭɻ

X ͔ΒͦΕࣗ਎΁ͷࣸ૾ f ͸࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ (ii) f ͸୯ௐͰ͋Δ (ଈͪ x ≤ y ͳΒ͹ f(x) ≤ f (y))ɻ (iii) x ₀ ≤ f(x ₀ ) ͳΔ x ₀ ∈ X ͕ଘࡏ͢Δɻ

͜ͷͱ͖ f ͸ X ₊ (x ₀ ) ͷதʹ࠷খͷෆಈ఺Λ࣋ͭɻ

(ূ໌) ୈ̍ஈ (ෆಈ఺ͷଘࡏ)

M = { x ∈ X; x ≤ f(x), x ₀ ≤ x } ͱஔ͘ɻԾఆ (iii) ΑΓ x ₀ ∈ M Ͱ͋Γ M = ∅ ͱͳΔɻ f (M ) ⊂ M ͳΔࣄɿ೚ҙʹ x ∈ M ΛऔΔɻx ≤ f (x) ٴͼ x ₀ ≤ x ͕੒ཱͭɻ

(ii) ΑΓ f(x) ≤ f (f(x)) ٴͼ f(x ₀ ) ≤ f(x) ͕ै͏ɻޙऀͱ (iii) ΑΓ x ₀ ≤ f(x ₀ ) ≤ f(x) Ͱ

͋Δ͔Β f(x) ∈ M ͕੒ཱͭɻ

M ͷۭͰͳ͍શॱং෦෼ू߹͸ M ʹ্ݶΛ࣋ͭࣄɿ

∅ = C ⊂ M ͳΔશॱং෦෼ू߹ C ΛऔΔɻ C ⊂ X ނ (i) ΑΓ u := sup C ∈ X ͕ଘࡏ͢

Δɻ͜ͷͱ͖೚ҙͷ x ∈ C ʹର͠ x ≤ u ͱͳΓ (ii) ΑΓ f (x) ≤ f (u) ͕ै͏ɻx ∈ C ⊂ M ΑΓ x ≤ f(x) ͱͳΔ͔Β x ≤ f(u) ͕ै͏ɻx ͸೚ҙͩͬͨͷͰ u ≤ f(u) ͕ै͏ɻҰํ

C ⊂ M ΑΓ x ₀ ≤ x Ͱ͋Γ x ≤ u Ͱ͋Δ͔Β x ₀ ≤ u ͕ै͏ɻ͜ΕΑΓ u ∈ C ΛಘΔɻ Ҏ্ΑΓ f | M : M −→ M ͸ϒϧόΩɾΫωʔβʔͷෆಈ఺ఆཧͷԾఆΛຬ͍ͨͯ͠

Δɻނʹ f | M ͸ M ͷதʹෆಈ఺Λ࣋ͭɻಛʹ f ͸ෆಈ఺Λ࣋ͭɻ

ୈ̎ஈ (X ₊ (x ₀ ) ͷதͷ࠷খෆಈ఺ͷଘࡏ)

M ^∗ = { x ∈ M ; f (x) = x, x ₀ ≤ x } , N = { y ∈ M ; ∀ x ∈ M ^∗ , y ≤ x } ͱஔ͘ɻୈ̍ஈΑΓ M ^∗ = ∅ Ͱ͋Γ x ₀ ∈ N ނ N = ∅ ͱͳΔɻ

f (N ) ⊂ N ͳΔࣄɿ೚ҙʹ y ∈ N ΛऔΔɻN ⊂ M ނୈ̍ஈΑΓ f(y) ∈ f (N ) ⊂ f (M ) ⊂

M ͕ै͏ɻy ∈ N Ͱ͋Δ͔Β೚ҙͷ x ∈ M ^∗ ʹର͠ y ≤ x Ͱ͋Δɻ(ii) ΑΓ f (y) ≤ f(x)

ͱͳΓ x ∈ M ^∗ ނ f(x) = x ͱͳΔɻଈͪ೚ҙͷ x ∈ M ^∗ ʹର͠ f (y) ≤ x ͕੒ཱͭɻ͜Ε

ΑΓ f (y) ∈ N ͕ै͏ɻ

N ͷۭͰͳ͍શॱং෦෼ू߹͸ N ʹ্ݶΛ࣋ͭࣄɿ

∅ = C ⊂ N ͳΔશॱং෦෼ू߹ C ΛऔΔɻN ⊂ M ނ C ͸ M ͷதʹ্ݶΛ࣋ͭɿ u := sup C ∈ M

೚ҙͷ y ∈ C ΛऔΔɻ C ⊂ N Ͱ͋Δ͔Β೚ҙͷ x ∈ M ^∗ ʹର͠ y ≤ x ͱͳΓɺ͜ΕΑΓ u ≤ x ͕ै͏ɻ͜Ε͸ u ∈ N Λҙຯ͢Δɻ

Ҏ্ΑΓ f | N : N −→ N ͸ϒϧόΩɾΫωʔβʔͷෆಈ఺ఆཧͷԾఆΛຬ͓ͨͯ͠Γ ෆಈ఺ x ^∗ ∈ N Λ࣋ͭɻN ⊂ M ΑΓ x ₀ ≤ x ^∗ ͱͳΔͷͰ x ^∗ ∈ M ^∗ ͕ै͏ɻҰํ x ^∗ ∈ N Ͱ

͋Δ͔Β೚ҙͷ x ∈ M ^∗ ʹର͠ x ^∗ ≤ x ͱͳΔɻ͜Ε͸ x ^∗ ͕ X ₊ (x ₀ ) ͷதͰ࠷খͰ͋Δࣄ

Λҙຯ͢Δɻ

ܥ ( ΞϚϯͷෆಈ఺ఆཧͷܥ ) ɹ X Λॱংू߹Ͱ࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ (i) X ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹͸ԼݶΛ࣋ͭɻ

X ͔ΒͦΕࣗ਎΁ͷࣸ૾͸࣍ͷ৚݅Λຬͨ͢΋ͷͱ͢Δɿ (ii) f ͸୯ௐͰ͋Δ (ଈͪ x ≤ y ͳΒ͹ f(x) ≤ f (y))ɻ (iii) f(x ₀ ) ≤ x ₀ ͳΔ x ₀ ∈ X ͕ଘࡏ͢Δɻ

͜ͷͱ͖ f ͸ X (x ₀ ) ͷதʹ࠷େͷෆಈ఺Λ࣋ͭɻ

̐ɽԠ༻

͜ͷઅͰ͸ΞϚϯͷෆಈ఺ఆཧ͔Βز͔ͭͷෆಈ఺ఆཧΛಋ͜͏ɻ

̐ . ̍ɹλϧεΩͷෆಈ఺ఆཧ ఆཧ ( λϧεΩͷෆಈ఺ఆཧ )

׬උଋ X ͔ΒͦΕࣗ਎΁ͷ୯ௐࣸ૾͸࠷খٴͼ࠷େͷෆಈ఺Λ࣋ͭɻ

( ূ໌ ) ׬උଋ͸࠷খݩٴͼ࠷େݩΛ࣋ͭɻ෉ʑ u = min X = inf X ٴͼ v = max X = sup X ͱ͢Δͱ f (u), f (v) ∈ X ނ u ≤ f(u) ٴͼ f (v ) ≤ v ΛಘΔɻैͬͯΞϚϯͷෆಈ఺ఆཧ ٴͼͦͷܥΑΓ௚ͪʹ݁࿦ΛಘΔɻ

̐ . ̎ɹ௚ܘॖখࣸ૾ʹؔ͢Δෆಈ఺ఆཧ

ఆٛɹ (ۭͰͳ͍) ڑ཭ۭؒ (X, d) ͔ΒͦΕࣗ਎΁ͷࣸ૾ f ͕௚ܘॖখࣸ૾ (diametric

contraction) Ͱ͋Δͱ͸ 0 ≤ k < 1 ͳΔ k ∈ R ͕ଘࡏ͠ f (M ) ⊂ M ͳΔ೚ҙͷ M ⊂ X ʹ ର͠ධՁ

diam(f (M )) ≤ k diam(M )

͕੒ཱͭࣄΛҦ͏ɻ͜͜ʹ diam(M) ͸ M ͷ௚ܘ (diameter of M ) ଈͪ

diam(M ) = sup { d(x, y); x, y ∈ M } Ͱ͋Δɻ

ྫ̍ɽ ɹॖখఆ਺ k ∈ [0, 1) Λ࣋ͭॖখࣸ૾ (contraction) f : X −→ X ଈͪ೚ҙͷ x, y ∈ X ʹର͠ධՁ

d(f (x), f (y)) ≤ kd(x, y) Λຬͨࣸ͢૾͸௚ܘॖখࣸ૾Ͱ͋Δɻ

ྫ̎ɽɹॖখఆ਺ k ∈ [0, 1) Λ࣋ͭ४ॖখࣸ૾ (quasi-contraction) f : X −→ X ଈͪ೚ҙ ͷ x, y ∈ X ʹର͠ධՁ

d(f(x), f (y)) ≤ k max(d(x, y), d(x, f (x)), d(y, f (y)), d(x, f (y)), d(y, f (x))) Λຬͨࣸ͢૾͸௚ܘॖখࣸ૾Ͱ͋Δɻ

ఆཧ ( ௚ܘॖখࣸ૾ʹର͢Δෆಈ఺ఆཧ )

( ۭͰͳ͍ ) ༗քͳ׬උڑ཭্ۭؒͷ௚ܘॖখࣸ૾͸ෆಈ఺Λ།Ұͭ࣋ͭɻ

(ূ໌) f : X −→ X Λ༗քͳ׬උڑ཭ۭؒ X ͔ΒͦΕࣗ਎΁ͷ௚ܘॖখࣸ૾ͱ͠ F _f Λ f ͷෆಈ఺ͷ੒͢෦෼ू߹ͱ͢ΔɿF _f = { x ∈ X; f (x) = x }

Ұҙੑɹ f(F _f ) = F _f Ͱ͋Δɻ࣮ࡍ x ∈ F _f ͳΒ f(f(x)) = f(x) ނ f(x) ∈ F _f ͱͳΓ x ∈ f (F _f ) ͳΒ y ∈ F _f ͕ଘࡏ͠ x = f(y) ͱͳΓ f (x) = f(f (y)) = f(y) = x ނ x ∈ F _f ͱ ͳΔɻैͬͯ

diam(F _f ) = diam(f (F _f )) ≤ k diam(F _f ) ΑΓ

0 ≤ (1 − k)diam(F _f ) ≤ 0

ΛಘΔͷͰ diam(F _f ) = 0 ͕ै͏ɻ F _f ͕ҟͳΔೋ఺ x, y Λ࣋ͬͨͱ͢Δͱ diam(F _f ) ≥ d(x, y) > 0 ͱͳΓໃ६ΛಘΔɻैͬͯ F _f ͸ଟ͘ͱ΋Ұ఺Λ࣋ͭɻ

ଘࡏɹ X ͷ෦෼ू߹͔Β੒Δ଒ X Λ

X = { M ⊂ X; M ͸ۭͰͳ͍ f ෆมͳดू߹ }

= { M ⊂ X; ∅ = M = M ׌ͭ f(M ) ⊂ M } ͱ͠ೋ߲ؔ܎ ≤ Λ

M ≤ N ⇔

def. M = N ·ͨ͸ M ⊂ f(N )

ͱఆٛ͢Δɻ

(X , ≤) ͸ॱংू߹Λ੒͢ࣄɿɹ

൓ࣹ཯ɿ M = N ΑΓ M ≤ M ͕ै͏ɻ

൓ରশ཯ɿM ≤ N ׌ͭ N ≤ M ͱ͢Δɻ࣍ͷ̐ͭͷ৔߹͕ى͜Δɿ (i) M = N ׌ͭ N = M ɹ (ii) M = N ׌ͭ N ⊂ f(M )

(iii) M ⊂ f (N ) ׌ͭ N = M ɹ (iv) M ⊂ f(N ) ׌ͭ N ⊂ f (M) (i)-(iii) ͷ৔߹͸௚ͪʹ M = N ͱͳΓ (iv) ͷ৔߹͸

• M ⊂ f(N ) ٴͼ f (N ) ⊂ N = N ނɹ M ⊂ N

• N ⊂ f(M ) ٴͼ f (M ) ⊂ M = M ނɹ N ⊂ M ͷ૒ํΑΓ M = N ͕ै͏ɻ

ਪҠ཯ɿL ≤ M ׌ͭ M ≤ N ͱ͢Δɻ࣍ͷ̐ͭͷ৔߹͕ى͜Δɿ (i) L = M ׌ͭ M = N ɹ (ii) L = M ׌ͭ M ⊂ f(N )

(iii) L ⊂ f (M ) ׌ͭ M = N ɹ (iv) L ⊂ f(M ) ׌ͭ M ⊂ f (N ) (i) ͷ৔߹͸ L = N ͱͳΔɻ(ii) ͱ (iii) ͷ৔߹͸ L ⊂ f(N ) ͱͳΔɻ (iv) ͷ৔߹͸

f(M ) ⊂ M = M ΑΓ L ⊂ f(M ) ⊂ M ⊂ f (N )ͱͳΔɻ Ҏ্ΑΓ L ≤ N ͕ै͏ɻ

X ͷۭͰͳ͍શॱং෦෼ू߹ʹର͢ΔԼݶͷଘࡏɿɹ

C Λ X ͷۭͰͳ͍શॱং෦෼ू߹ͱ͢Δɻ C ʹ࠷খݩ͕ࡏΔͱ͢Ε͹ԼݶͱͳΔɻͦ͜

Ͱ C ʹ࠷খݩ͕ଘࡏ͠ͳ͍৔߹Λߟ͑Δɻैͬͯ೚ҙͷ C ∈ C ʹର͠ D ∈ C ͕ଘࡏ͠

D < C ͱͳΔɻ C ⊂ X ނ D ⊂ f(C) ͱͳΔɻ͜ΕΑΓ

diam(D) ≤ diamf(C) = diamf(C) ≤ k diam(X)

͕ै͏ɻ D ∈ X ނ ∅ = D ⊂ f(C) ⊂ C ⊂ C ͕੒ཱ͢Δɻؼೲతʹ (C _n ; n ≥ 1) ⊂ C Ͱ C _n = ∅ ,

C ⊃ C ₁ ⊃ C ₂ ⊃ · · · ⊃ C _n , diam(C _n ) ≤ k ⁿ diam(X)

Λຬͨ͢΋ͷ͕ߏ੒͞ΕΔɻؼೲతʹ (x _n ; n ≥ 1) ⊂ X Ͱ x _n ∈ C _n ͳΔྻΛऔΔɻ͜ͷͱ

͖ m > n −→ ∞ ͱ͢Ε͹

d(x _m , x _n ) ≤ diam(C _n ) ≤ k ⁿ diam(X) → 0

ͱͳΔͷͰ X ͷ׬උੑΑΓ (x _n ) ͸ۃݶ x Λ࣋ͭɻ x ͸ x ∈

n≥1

C _n Λຬͨ͢ɻ C ͸ดू߹

Ͱ͋Δ͔Β x ∈ C Ͱ͋Γ C ͸೚ҙͰ͔͋ͬͨΒ x ∈

C ͕ै͏ɻߋʹ΋͏Ұͭ y ∈ C

͕ࡏͬͨͱ͢Δͱ y ∈

n≥1

C _n Ͱ΋͋Δ͔Β

d(x, y) ≤ d(x, x _n ) + d(x _n , y) ≤ 2k ⁿ diam(X) → 0 ͱͳΓ x = y ͕ै͏ɻނʹ

C = { x } ΛಘΔɻ͜ΕΑΓ { f (x) } = f (

C ) ⊂

{ f(C); C ∈ C } ⊂

{ C; C ∈ C } =

C = { x } ٴͼ

f (

C ) =

C ∈ X ΛಘΔɻ͜ͷ

C ͕ C ͷԼݶͰ͋ΔࣄΛࣔͦ͏ɻ೚ҙͷ C ∈ C ΛऔΔɻD ∈ C ͕ଘࡏ

͠ D < C ͱͳΔɻ C ⊂ X ނ D ⊂ f(C) ͱͳΔɻ͜ΕΑΓ C ⊂ D ⊂ f (C)

ଈͪ

C ≤ C ͕ै͏ɻҰํɺ೚ҙͷ C ∈ C ʹର͠ B ≤ C ͳΔ B ∈ X ΛऔΔͱ B = C

·ͨ͸ B ⊂ f (C) Ͱ͋Δɻಛʹ B ⊂ C ∪ f (C) ⊂ C Ͱ C ∈ C ͸೚ҙͰ͔͋ͬͨΒ B ⊂

C = f (

C ) ⊂ f ( C ) ͱͳΓ B ≤

C ͕ै͏ɻଈͪ

C = inf C Ͱ͋Δɻ͜Ε͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

f ͷෆಈ఺ͷଘࡏɿ

M ∈ X ʹର͠ ϕ(M) = f(M ) ͱஔ͘ɻf(M ) ⊂ M ΑΓ f (M) ⊂ M = M ͱͳΔͷͰ f(ϕ(M )) = f (f(M )) ⊂ f(M ) ⊂ f(M ) = ϕ(M)

͕ಋ͔ΕΔɻ·ͨɺ ϕ(M ) ͸ۭͰͳ͍ดू߹Ͱ͋Δ͔Β ϕ(M ) ∈ X ͱͳΔɻଈͪ ϕ ͸ X

͔ΒͦΕࣗ਎΁ͷࣸ૾Ͱ͋Δɻߋʹ

M < N ⇐⇒ M ⊂ f (N )

= ⇒ ϕ(M ) = f (M ) ⊂ f (f (N )) = f (ϕ(N ))

= ⇒ ϕ(M ) ≤ ϕ(N )

Ͱ͋Γ M = N = ⇒ ϕ(M) = ϕ(N ) Ͱ͋Δ͔Β ϕ ͸୯ௐͰ͋Δɻ·ͨ

ϕ(X) = f(X) ⊂ f(X) ΑΓ ϕ(X) ≤ X ΛಘΔɻैͬͯ ϕ : X −→ X ͸ΞϚϯͷෆಈ఺ఆ ཧͷܥΛຬͨ͢ɻނʹ ϕ ͷෆಈ఺ M ^∗ ∈ X ͕ଘࡏ͢Δɿϕ(M ^∗ ) = M ^∗ ଈͪ f (M ^∗ ) = M ^∗

͜ΕΑΓ

diam(M ^∗ ) = diam(f (M ^∗ )) = diam(f(M ^∗ )) ≤ kdiam(M ^∗ )

ͱͳΓ diam(M ^∗ ) = 0 ͕ै͏ɻ͜Ε͸ M ^∗ ͕Ұ఺ू߹ { x ^∗ } Ͱ͋ΔࣄΛҙຯ͠ { x ^∗ } = M ^∗ = f (M ^∗ ) = f( { x ^∗ } ) = { f(x ^∗ ) } = { f(x ^∗ ) } ΑΓ x ^∗ = f (x ^∗ ) ͕ै͏ɻଈͪ x ^∗ ∈ X ͸ f ͷෆಈ

఺Ͱ͋Δɻ

̐ . ̏ɹڽॖࣸ૾ʹର͢Δෆಈ఺ఆཧ

ఆٛɹ (X, d) Λ (ۭͰͳ͍) ڑ཭ۭؒͱ͢ΔɻX ͷۭͰͳ͍༗ք෦෼ू߹ M ʹର͠

κ(M ) = inf { ε > 0; M ͸௚ܘ ε ҎԼͷ༗ݶݸͷ෦෼ू߹Ͱඃ෴͞ΕΔ }

= inf { ε > 0; M ⊂

i∈I

M _i , I < ∞ , diam(M _i ) ≤ ε }

ͱஔ͖ඇίϯύΫτੑ ( Ϋϥτ΢εΩ ) ଌ౓ ((Kuratowski) measure of noncompactness) ͱҦ͏ɻ

໋୊ɹ (X, d) Λ (ۭͰͳ͍) ڑ཭ۭؒͱ͢ΔɻX ͷۭͰͳ͍༗ք෦෼ू߹ʹର͕࣍͠੒

ཱͭɻ

(1) 0 ≤ κ(M ) ≤ diam(M ) (2) κ(M ) = 0 ⇐⇒ M ͸શ༗ք (3) M ⊂ N = ⇒ κ(M ) ≤ κ(N ) (4) κ(M ) = κ(M )

(5) κ(M ∪ N) = κ(M ) ∨ κ(N )

(ূ໌) (1) M ͸ M ͷҰͭͷඃ෴Ͱ͋Δࣄ͔Βै͏ɻ

(2) M ͸શ༗ք

⇐⇒ ∀ ε > 0 ∃ (x _i ; i ∈ I) ⊂ M : I < ∞ , M ⊂

i∈I

B(x _i ; ε)

⇐⇒ κ(M) = 0

(3) N ͷඃ෴ͳΒ͹ M ͷඃ෴Ͱ͋ΓɺԼݶΛऔΔର৅ͱͯ͠͸ M ͷํ͕޿͍ (ڱ͘ͳ

͍) ҝ κ(M ) ≤ κ(N ) ͕ै͏ɻ

(4) ɾM ⊂ M Ͱ͋Δ͔Β (3) ΑΓ κ(M ) ≤ κ(M ) ɾM ⊂

i∈I

M _i , I < ∞ ͳΒ͹ M ⊂

i∈I

M _i , diam(M _i ) = diam(M _i ) ͱͳΔͷͰ κ(M ) ≤ κ(M )

(5) ɾ M, N ⊂ M ∪ N Ͱ͋Δ͔Β (3) ΑΓ κ(M ) ∨ κ(N ) ≤ κ(M ∪ N ) ɾ೚ҙͷ ε > 0 ʹର͠ (M _i ; i ∈ I) ٴͼ (N _j ; j ∈ J) ͕ࡏͬͯ

M ⊂

i∈I

M _i , N ⊂

j∈J

N _j , I < ∞ , J < ∞ , max i∈I diam(M _i )< κ(M ) + ε, max

j∈J diam(N _j ) < κ(N) + ε

ͱͳΔɻ͜ͷͱ͖ M ∪ N ⊂

(i,j)∈I ×J

(M _i ∪ N _j ) Ͱ͋Γ

κ(M ∪ N ) ≤ max

(i,j)∈I ×J (diam(M _i ) ∨ diam(N _j ))

≤ (κ(M ) ∨ κ(N )) + ε

͕ै͏ɻε > 0 ͸೚ҙͰ͔͋ͬͨΒ κ(M ∪ N ) ≤ κ(M ) ∨ κ(N )

໋୊ɹ X ΛϊϧϜۭؒͱ͢ΔɻX ͷۭͰͳ͍༗ք෦෼ू߹ʹର͕࣍͠੒ཱͭɻ (1) κ(M + N ) ≤ κ(M ) + κ(N )

(2) κ(αM ) = | α | κ(M ), α ∈ C (3) κ(M ) = κ(co(M )) = κ(co(M ))

͜͜ʹ co(M ) ٴͼ κ(co(M )) ͸෉ʑM ͷತแٴͼดತแͱ͢Δɻ ( ূ໌ ) (1) M ⊂

i∈I

M _i , N ⊂

j∈J

N _j , I < ∞ , J < ∞ ͱ͢ΔͱͦͷϕΫτϧ࿨ͱͯ͠ఆ·

Δ M + N ͸

M + N ⊂

(i,j)∈I×J

(M _i + N _j ) Λຬͨ͢ɻ͜ͷͱ͖

diam(M _i + N _j ) = sup { x − y ; x, y ∈ M _i + N _j }

= sup { (x + y ) − (y + y ) ; x , y ∈ M _i , x , y ∈ N _j }

≤ sup { x + y ; x , y ∈ M _i } + sup { x − y ; x , y ∈ N _j }

= diam(M _i ) + diam(N _j )

͕ै͏ɻ

(2) ɹ diam(αM _j ) = sup { x − y ; x, y ∈ αM _i }

= sup { αx − αy ; x , y ∈ M _i }

= | α | sup { x − y ; x , y ∈ M _i } = | α | diam(M _i ) ΑΓಋ͔ΕΔɻ

(3) M ⊂ co(M) ٴͼ κ ͷดแʹର͢Δੑ࣭ΑΓ κ(M ) ≤ κ(co(M )) = κ(co(M )) ͕ै

͏ͷͰ κ(co(M)) ≤ κ(M ) Λࣔͤ͹ྑ͍ɻॳΊʹ X ͷ೚ҙͷ༗ք෦෼ू߹ N ʹର͠

diam(N ) = diam(co(N)) ͳΔࣄΛࣔͦ͏ɻN ⊂ co(N ) ΑΓ diam(N ) ≤ diam(co(N )) ͕

ै͏ͷͰ diam(co(N )) ≤ diam(N ) Λࣔͤ͹ྑ͍ɻ΋ͦ͠͏Ͱͳ͍ (ଈͪ δ := diam(N ) <

diam(co(N ))) ͱ͢Δͱ x, y ∈ co(N ) ͕ଘࡏ͠ δ < x − y ͕੒ཱͭࣄͱͳΔɻ͜ͷͱ͖

࣍ͷೋͭͷ৔߹ͷԿΕ͔Ұ͕ͭඞͣ੒ཱͭɿ

(i) N ⊂ B(x; δ) (ii) N ⊂ B (x; δ)

(i) ͷ৔߹ɹ co(N ) ⊂ B(x; δ) ΑΓ y ∈ B(x; δ) ͕ै͏͕ɺ͜Ε͸ x − y ≤ δ Λҙຯ͢Δ ͷͰໃ६Ͱ͋Δɻ

(ii) ͷ৔߹ɹ x ₀ ∈ N ͕ଘࡏ͠ x − x ₀ > δ ͱͳΔɻ x ₀ ∈ N ٴͼ δ ͷఆٛΑΓ N ⊂ B(x ₀ ; δ)

͕ै͍ɺ͜ΕΑΓ co(N ) ⊂ B(x ₀ ; δ) ΛಘΔɻނʹ x ∈ B(x ₀ ; δ) ଈͪ x − x ₀ ≤ δ ΛಘΔ

͕ɺ͜Ε͸ x ₀ ͷ৚݅ʹ൓͢Δɻ

(i)(ii) ԿΕͷ৔߹΋ໃ६ͱͳΔͷͰ diam(co(N )) ≤ diam(N ) ͕੒ཱͭɻ

೚ҙͷ ε > 0 Λ༩͑Δɻ(M _i ; i ∈ I) ͕ࡏͬͯ I < ∞ , M ⊂

i∈I

M _i ,

max i∈I diam(M _i ) < κ(M ) + ε ͕੒ཱͭɻ diam(M _i ) = diam(co(M _i )) ͱͳΔ͔ΒॳΊ͔Β M _i

͸ತͰ͋Δͱͯ͠ྑ͍ɻ

ͯ͞

Λ = { λ = (λ _i ; i ∈ I); λ _i ≥ 0 ∀ i ∈ I,

i∈I

λ _i = 1 } , M (λ) =

i∈I

λ _i M _i ⊂ X, λ ∈ Λ ͱஔ͘ɻ͜ͷͱ͖ (1)(2) ΑΓ

κ(M (λ)) ≤

i∈I

λ _i κ(M _i ) ≤

i∈I

λ _i diam(M _i ) ≤ max

i∈I diam(M _i ) < κ(M) + ε

͕ै͏ɻ࣍ʹ

λ∈Λ

M (λ) ͕ತͰ͋ΔࣄΛࣔͦ͏ɻଈͪ

x, y ∈

λ∈Λ

M (λ), 0 ≤ t ≤ 1 = ⇒ tx + (1 − t)y ∈

λ∈Λ

M (λ) Λࣔͦ͏ɻx ∈ M (λ), y ∈ M (μ) ͳΔ λ, μ ∈ Λ ΛऔΔɻ͜ͷͱ͖

(λ _i ; i ∈ I), (x _i ; i ∈ I), (μ _i ; i ∈ I ), (y _i ; i ∈ I ) ͕ଘࡏ͠

x =

i∈I

λ _i x _i , y =

i∈I

μ _i y _i ͱද͞ΕΔɻ͍· ζ Λ

ζ = tλ + (1 − t)μ ͱఆΊΔͱ

ζ = (ζ _i ; i ∈ I) = (tλ _i + (1 − t)μ _i ; i ∈ I) Ͱ͋Γ tλ _i + (1 − t)μ _i ≥ 0( ∀ i ∈ I) ׌ͭ

i∈I

(tλ _i + (1 − t)μ _i ) = t

i∈I

λ _i + (1 − t)

i∈I

μ _i = t + (1 − t) = 1

ͱͳΔͷͰ ζ ∈ Λ ͱͳΔɻͯ͞ ρ _i Λ

ρ _i = tλ _i /ζ _i , ζ _i > 0 ͷ৔߹

0, ζ _i = 0 ͷ৔߹

ͱఆΊΔͱ tλ _i ≤ tλ _i + (1 − t)μ _i = ζ _i ނ 0 ≤ ρ _i ≤ 1 ͱͳΔɻ͜ͷͱ͖ ζ _i = 0 ͳΔ i ∈ I ʹ ର͠

ζ _i (1 − ρ _i ) = ζ _i (1 − tλ _i ζ _i )

= ζ _i − tλ _i

= (1 − t)μ _i ͱͳΔ͔Β

tx + (1 − t)y =

i∈I

(tλ _i x _i + (1 − t)μ _i y _i )

=

i∈I

ζ _i (ρ _i x _i + (1 − ρ _i )y _i )

͕੒ཱͭɻ֤ M _i ͸ತͱԾఆͨ͠ͷͰ ρ _i x _i + (1 − ρ _i )y _i ∈ M _i ͱͳΓ tx + (1 − t)y ∈ M (ζ) ⊂

λ∈Λ

M (λ)

͕ै͏ɻ͜Ε͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

ͯ͞

M ⊂

i∈I

M _i ⊂

λ∈Λ

M (λ) Ͱ͋Γ࠷ӈล͸ತͰ͋Δ͔Β

co(M ) ⊂

λ∈Λ

M (λ)

͕ै͏ɻ

i∈I

M _i ⊂ B(0; R) ͳΔ R > 0 ΛऔΔɻ೚ҙͷ ε > 0 ͱ λ ∈ Λ ʹର͠

i∈I

| λ _i − μ _i | < ε/R ͳΔ μ ∈ Λ ΛऔΔͱ M(μ) ⊂ M (λ) + B(0; ε) ͱͳΔɻ࣮ࡍ y ∈ M (μ) ʹର͠ y =

i∈I

μ _i y _i (y _i ∈ M _i ) ͱද͠ x =

i∈I

λ _i y _i ͱஔ͘ͱ x ∈ M (λ) Ͱ͋Γ y = x + (y − x) ͱͨ͠ͱ͖ y − x =

i∈I

(μ _i − λ _i )y _i ΑΓ y − x ≤

i∈I

| μ _i − λ _i | y _i ≤

i∈I

| μ _i − λ _i | R < ε ͱ ͳΔ͔ΒͰ͋Δɻނʹ೚ҙͷ ε > 0 ʹର͠ δ > 0 ͕ଘࡏ͠ Λ ⊂

λ∈Λ

B(λ; δ) ׌ͭ | λ − μ | <

δͳΔ೚ҙͷμ ∈ Λʹର͠ M (μ) ⊂ M (λ) + B (0; ε) ͕੒ཱͭɻΛ ͸ίϯύΫτͰ͋Δ͔Β

༗ݶݸͷ (λ ^(j) ; j ∈ J) Λऔͬͯ Λ ⊂

i∈J

B (λ ^(j) ; δ) ͱग़དྷΔɻ͜ͷͱ͖

co(M ) ⊂

j∈J

(M (λ ^(j) ) + B(0; ε))

͕ै͏ɻނʹ໋୊ɹ (3) ٴͼ্ͷ (1) ΑΓ κ(co(M )) ≤ max

j∈J (κ(M (λ ^(j) )) + κ(B(0; ε))

≤ κ(M ) + ε + 2ε

͕ै͏ɻε > 0 ͸೚ҙͰ͔͋ͬͨΒ κ(co(M )) ≤ κ(M ) ΛಘΔɻ

ఆٛɹڑ཭ۭؒ X, d ͔ΒͦΕࣗ਎΁ͷࣸ૾ f : X → X ͸ 0 < k < 1 ͳΔఆ਺ k ͕ࡏͬͯ

೚ҙͷ༗քू߹ M ⊂ X ʹର͠ෆ౳ࣜ

κ(f (M )) ≤ kκ(M )

Λຬͨ͢ͱ͖ू߹ॖখࣸ૾ (set contraction map) ͱҦ͏ɻ·ͨ f : X → X ͸ κ(M ) > 0 ͳΔ೚ҙͷ༗քू߹ M ⊂ X ʹର͠ਅͷෆ౳ࣜ

κ(f (M )) < κ(M ) Λຬͨ͢ͱ͖ڽॖࣸ૾ (condensing map) ͱҦ͏ɻ

ྫ̍ɽɹॖখࣸ૾͸ू߹ॖখࣸ૾Ͱ͋Δɻ༗քू߹ M ٴͼͦͷ༗ݶඃ෴ (M _i ; i ∈ I) Λऔ Δɻf(M ) ⊂ f(

i∈I

M _i ) =

i∈I

f(M _i ) ΑΓ (f (M _i ); i ∈ I) ͸ f (M ) ͷ༗ݶඃ෴Ͱ͋Δɻf ͕

ॖখࣸ૾ͳΒ͹

d(f(x), f (y)) ≤ kd(x, y), x, y ∈ M _i ΑΓ

diam(f (M _i )) ≤ k diam(M _i ) ΛಘΔɻैͬͯ

κ(f (M )) = inf { ε > 0; f(M ) ⊂

j∈J

N _j , J < ∞ , diam(N _j ) ≤ ε }

≤ inf { ε > 0; M ⊂

i∈I

M _i , I < ∞ , diam(f(M _i )) ≤ ε }

≤ inf { ε > 0; M ⊂

i∈I

M _i , I < ∞ , diam(M _i ) ≤ ε/k }

= k inf { ε > 0; M ⊂

i∈I

M _i , I < ∞ , diam(M _i ) ≤ ε }

= kκ(M)

͕੒ཱͭɻ

ྫ̎ɽɹू߹ॖখࣸ૾͸ڽॖࣸ૾Ͱ͋Δɻ

ྫ̏ɽɹϊϧϜۭؒ X ͔ΒͦΕࣗ਎΁ͷࣸ૾ f : X → X ͸ॖখࣸ૾ g ͱίϯύΫτ

ࣸ૾ h(ଈͪ೚ҙͷ༗քू߹ʹରͦ͠ͷ૾ͷดแ͕ίϯύΫτͱͳΔ࿈ଓࣸ૾) ʹґͬͯ

f = g + h ͱද͞Ε͍ͯΔͱ͢Δͱ f ͸ू߹ॖখࣸ૾Ͱ͋Δɻ༗քू߹ M ΛऔΔɻ f (M ) = g(M ) + h(M ) Ͱ͋Γྫ̍ʹڌΓ

κ(g(M )) ≤ kκ(M ) Ͱ໋୊ɹ (2)(4) ΑΓ

κ(h(M )) = κ(h(M )) = 0 ͱͳΔɻै໋ͬͯ୊ɹ (1) ΑΓ

κ(f(M )) ≤ κ(g (M )) + κ(h(M )) ≤ kκ(M )

͕੒ཱͭɻ

ఆཧ ( ڽॖࣸ૾ʹର͢ΔαυϰεΩͷෆಈ఺ఆཧ )

X Λόφϋۭؒͱ͢ΔɻۭͰͳ͍༗քดತ෦෼ू߹͔ΒͦΕࣗ਎΁ͷ࿈ଓڽॖࣸ૾͸ෆ ಈ఺Λ࣋ͭɻ

(ূ໌) M ΛۭͰͳ͍༗քดತू߹Ͱ f : M → M Λ࿈ଓڽॖࣸ૾ͱ͢Δɻγϟ΢μʔͷ ෆಈ఺ఆཧ͕ద༻Ͱ͖Δ༷ʹɺۭͰͳ͍ίϯύΫτू߹ K Λ K ⊂ M ׌ͭ f | K : K → K ͳΔ༷ʹߏ੒͢Δɻ

ୈ̍ஈɹ x ₀ ∈ M ΛҰͭऔΓ

N = { (f ◦ · · · ◦ f

k ݸ

)(x ₀ ) ∈ M ; k ∈ Z _≥0 }

ͱஔ͘ɻ͜ͷͱ͖ N = f (N ) ∪ { x ₀ } ͱͳΔ͔Β

κ(N ) = κ(f (N )) ∨ κ( { x ₀ } ) = κ(f(N )) ∨ 0 = κ(f(N ))

͕ै͏͕ κ(N ) > 0 ͳΒ κ(f (N )) < κ(N ) ͱͳͬͯ͠·͏ҝ κ(N ) = 0 ଈͪ N ͸શ༗քͱ ͳΔɻf(N ) ⊂ N Ͱ f ͸࿈ଓނ f(N ) ⊂ N ͕ै͏ɻ

ୈ̎ஈɹ M ͷ෦෼ू߹ͷ଒ F 0 Λ

F 0 = { F ⊂ M ; ∅ = F = F ⊂ N }

ͱ͠ F ₀ ʹ ⊂ ͰॱংΛಋೖ͢Δɻ N ∈ F ₀ ނ F ₀ = ∅ ͱͳΔɻ N ͸ίϯύΫτͰ͋Δ͔Β

༗ݶަࠥੑʹґΓ F 0 ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹ C ͸Լݶ

C Λ࣋ͭɻ

֤ F ∈ F ₀ ʹର͠ ϕ(F ) = f (F ) ͱஔ͘ɻ͜ͷͱ͖ ϕ ͸ F ₀ ͔ΒͦΕࣗ਎΁ͷࣸ૾Ͱ F ≤ F ⇐⇒ F ⊂ F = ⇒ f(F ) ⊂ f(F )

= ⇒ f(F ) ⊂ f(F ) ⇐⇒ ϕ(F ) ≤ ϕ(F )

ނ ϕ ͸୯ௐͰ͋Δɻ·ͨ ϕ(F ) = f(F ) = f (F ) ⊂ F ͱͳΔ͔Β ϕ(F ) ≥ F Λຬͨ͢ɻ

ैͬͯΞϚϯͷෆಈ఺ఆཧʹڌΓ ϕ ͸ F 0 ʹෆಈ఺ F ₀ Λ࣋ͭɻ͜ͷ F ₀ ͸ ∅ = F ₀ = F ₀ = f (F ₀ ) = f (F ₀ ) ⊂ N Λຬͨ͢ɻ

ୈ̏ஈɹ M ͷ෦෼ू߹ͷ଒ F Λ

F = { F ⊂ M ; F ₀ ⊂ F = F }

ͱ͠ F ʹ ⊂ ͰॱংΛಋೖ͢ΔɻF ₀ ∈ F ނ F = ∅ ͱͳΔɻF ͸ίϯύΫτͰ͋Δ͔Β༗

ݶަࠥੑʹґΓ F ͷ೚ҙͷۭͰͳ͍શॱং෦෼ू߹ C ͸Լݶ

C Λ࣋ͭɻ

֤ F ∈ F ʹର͠ ψ(F ) = co(f(F )) ͱஔ͘ɻ͜ͷͱ͖ ψ ͸ F ͔ΒͦΕࣗ਎΁ͷࣸ૾Ͱ F ≤ F ⇐⇒ F ⊂ F = ⇒ f (F ) ⊂ f (F )

= ⇒ co(f (F )) ⊂ co(f(F )) ⇐⇒ ψ (F ) ≤ ψ(F ) ނ ψ ͸୯ௐͰ͋Δɻߋʹ

ψ(F ₀ ) = co(f (F ₀ )) ≥ f(F ₀ ) = F ₀ = F ₀

͕੒ཱͭɻैͬͯΞϚϯͷෆಈ఺ఆཧʹڌΓ ψ ͸ F ʹෆಈ఺Λ࣋ͭɻͦΕΛ K ͱ͢Δͱ K = ψ(K) = co(f(K)) ΑΓ

κ(K) = κ(co(f (K ))) = κ(f(K))

͕ै͏ɻ κ(K) > 0 ͳΒ κ(f (K )) < κ(K) ͱໃ६͢ΔͷͰ κ(K) = 0 ͕ै͏ɻଈͪ K ͸ί

ϯύΫτू߹ͱͳΔɻf(K) ⊂ co(f(K)) ⊂ K ނ f | K : K → K ͸ίϯύΫτತू߹ K ͔

ΒͦΕࣗ਎΁ͷ࿈ଓࣸ૾ͱͳΔɻैͬͯ f | K ͸ෆಈ఺Λ࣋ͭɻ

̐ . ̐ɹඇ֦େࣸ૾ʹର͢Δෆಈ఺ఆཧ

όφϋۭؒ X ͷۭͰͳ͍෦෼ू߹ M ্Ͱఆٛ͞Εͨࣸ૾ f : M → X ͸೚ҙͷ x, y ∈ M ʹର͠ෆ౳ࣜ

f(x) − f(y) ≤ x − y Λຬͨ͢ͱ͖ඇ֦େࣸ૾ (nonexpansive map) ͱҦ͏ɻ

ෆಈ఺Λ࣋ͨͳ͍ඇ֦େࣸ૾ͷྫɿ X = c ₀ ( Z _≥0 ) ଈͪ 0 ʹऩଋ͢Δ਺ྻ x = (x _n ; n ≥ 0) ͷ

ۭؒͰ

x = sup {| x _n | ; n ≥ 0 }

Ͱ༩͑ΒΕΔϊϧϜΛ࣋ͭόφϋۭؒΛߟ͑ M Λͦͷด୯Ґٿ M = { x ∈ X; x ≤ 1 }

ͱ͢Δɻx ∈ M ʹର͠ y = f (x) Λ

y ₀ = 1 − x , y _n = x _n−1 (n ≥ 1) ͱ͢Δͱ

f (x) = sup {| y _n | ; n ≥ 0 } ≤ (1 − x ) ∨ x ≤ 1, f (x) − f (x ) = sup {| y _n − y _n | ; n ≥ 0 }

≤ x − x ∨ x − x ≤ x − x

ΛಘΔͷͰ f ͸ M ͔ΒͦΕࣗ਎΁ͷඇ֦େࣸ૾Ͱ͋Δɻ΋͠ f ͕ෆಈ఺ x ∈ M Λ࣋ͬͨ

ͱ͢Δͱ x = f (x) ͷ֤੒෼Λൺֱ͠

x ₀ = 1 − x , x _n = x _n−1 (n ≥ 1) ͱͳΓ݁ہશͯͷ n ≥ 0 ʹର͠

x _n = 1 − x

͕੒ཱͭࣄͱͳΓ x _n → 0(n → ∞ ) ΑΓ x = 1 ΛಘΔ͕ɺ͜ΕΑΓશͯͷ n ≥ 0 ʹର͠

x _n = 0 ͱͳΓ x = 0 ͕ै͏ɻ͜Ε͸ໃ६Ͱ͋Δɻ

ඇ֦େࣸ૾͕ෆಈ఺Λ࣋ͭҝʹ͸ఆٛҬʹԿΒ͔ͷߏ଄͕ඞཁʹͳΔࣄΛ্ͷྫ͸ࣔ͠

͍ͯΔɻͦ͜Ͱ࣍ͷ֓೦Λಋೖ͢Δɻ

ఆٛɹόφϋۭؒ X ͸গͳ͘ͱ΋ೋͭͷ఺Λ࣋ͭ೚ҙͷดತ෦෼ू߹ M ͕ ρ(M ) := inf

y∈M sup

x∈M x − y < diam(M ) Λຬͨ͢ͱ͖ਖ਼نߏ଄ (normal structure) Λ࣋ͭͱҦ͏ɻ

໋୊ɹҰ༷ತόφϋۭؒ͸ਖ਼نߏ଄Λ࣋ͭɻ

( ূ໌ ) M Λ d := diam(M) > 0 ͳΔดತ෦෼ू߹ͱ͢Δɻ x, y ∈ M Λ x = y ͳΔ΋ͷͱ

͠ ε = x − y /d ͱ͢Δɻ͜ͷͱ͖ η ₀ = (x + y)/2 ͱ͓͘ͱ η ₀ ∈ M Ͱ͋Γ೚ҙͷ ξ ∈ M ʹର͠

ξ − η ₀ d = 1

2

ξ − x

d + ξ − y d

ͱͳΔɻͯ͞

ξ − x

d ≤ 1, ξ − y

d ≤ 1, ξ − x

d − ξ − y

d = x − y d = ε ͱͳΔ͔ΒҰ༷ತੑΑΓ δ(ε) > 0 ͕ଘࡏ͠

1

d ξ − η ₀ = 1 2

ξ − x

d + ξ − y d

≤ 1 − δ(ε)

͕ै͏ɻ͜ΕΑΓ

ξ − η ₀ ≤ d(1 − δ(ε)) ͱͳΓ ξ ∈ M ͸೚ҙͰ͔͋ͬͨΒ

η∈M inf sup

ξ∈M ξ − η ≤ sup

ξ∈M ξ − η ₀ ≤ d(1 − δ(ε)) < d ΛಘΔɻ

ఆཧ ( ඇ֦େࣸ૾ʹର͢ΔΧʔΫͷෆಈ఺ఆཧ ) ɹ

ਖ਼نߏ଄Λ࣋ͭόφϋۭؒͷۭͰͳ͍ऑίϯύΫτತ෦෼ू߹͔ΒͦΕࣗ਎΁ͷඇ֦େ

ࣸ૾͸ෆಈ఺Λ࣋ͭɻ

(ূ໌) ਖ਼نߏ଄Λ࣋ͭόφϋۭؒͷۭͰͳ͍ऑίϯύΫτತू߹Λ X ͱ͠ f : X → X Λ ඇ֦େࣸ૾ͱ͢Δɻ

ୈ̍ஈɹ M Λ M ⊂ X ͳΔۭͰͳ͍ดತू߹ͱ͠ M ͷνΣϏγΣϑத৺ Ceb(M ) Λ Ceb(M ) := { x ∈ M ; sup

y∈M x − y = ρ(M) } ͱఆٛ͢Δɻ Ceb(M ) ͕ۭͰͳ͍ࣄΛࣔͦ͏ɻ M ͷྻ (x _n ; n ≥ 1) Λ

sup

x∈M x − x _n < ρ(M ) + 1 n

ͱͳΔ༷ʹऔΔɻM ͸ดತू߹Ͱ͋Δ͔Βऑดू߹Ͱ͋ΔɻM ͸ऑίϯύΫτू߹ X ͷ ෦෼ू߹Ͱ͋Δ͔ΒऑίϯύΫτͰ͋ΔɻΤϕϧϥΠϯͷఆཧΑΓ M ͸ऑ఺ྻίϯύΫ τͰ͋Δ͔Β෦෼ྻ (x _n

; j ≥ 1) ٴͼ x ^∗ ∈ M ͕ଘࡏ͠ (x _n

) ͸ x ^∗ ʹऑऩଋ͢Δɻ֤ x ∈ M ʹର͠ (x − x _n

; j ≥ 1) ͸ x − x ^∗ ʹऑऩଋ͢Δ͔Β

x − x ^∗ ≤ lim inf

j→∞ x − x _n

≤ lim inf

j→∞ (ρ(M ) + 1

n _j ) = ρ(M)