Các phiên bản của định lý hahn banach

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ P❍❆◆ ❚❍➚ ◆●❹◆ ❈⑩❈ P❍■➊◆ ❇❷◆ ❈Õ❆ ✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ P❍❆◆ ❚❍➚ ◆●❹◆ ❈⑩❈ P❍■➊◆ ❇❷◆ ❈Õ❆ ✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P t ữớ ữợ ❤å❝ ❚❤❙✳ ◆●❯❨➍◆ ❱❿◆ ❚❯❨➊◆ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ▲❮■ ữủ ỷ ỡ tợ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❚✉②➯♥ ◆❣✉②➵♥ ❱➠♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ❝❤➳✳ ❑➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ P❤❛♥ ❚❤à ◆❣➙♥ ữợ sỹ ữợ t õ ữủ t ổ trũ ợ t ✤➲ t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ P❤❛♥ ❚❤à ◆❣➙♥ ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ỡ ỵ rở ỵ ❍❛❤♥ ✲ ❇❛♥❛❝❤ ✈➔ ❝→❝ ù♥❣ ❞ö♥❣ ✷✵ ✷✳✶✳ ❉↕♥❣ rở ỵ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳ ❈→❝ ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✶✳ Ù♥❣ ❞ö♥❣ tr♦♥❣ ❣✐↔✐ t ỵ tr t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✸✳ ▼ët →♣ ❞ư♥❣ ❝❤♦ t♦→♥ tû ✤❛ trà ✤ì♥ ✤✐➺✉ ✳ ỵ tỗ t↕✐ ❦❤ỉ♥❣ ❝â ❜✐➯♥ ÷✉ t✐➯♥ ✳ ✳ ✳ ✳ ✸✷ ❑➳t ❧✉➟♥ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✽ ✐✐✐ ỵ ữủ t ữ ởt ỵ tú õ ự t tr t ỵ ữủ ♠ët ❞↕♥❣ ❣✐↔✐ t➼❝❤ ❝õ❛ ❚✐➯♥ ✤➲ ❝❤å♥ ✈➔ ❧➔ ♠ët ✈✐➯♥ ♥❣å❝ ❝õ❛ ●✐↔✐ t➼❝❤ ❤➔♠✳ ❈→❝ ❞↕♥❣ q✉❛♥ trồ ỵ ỵ rở trở ởt ỵ t P sỡ ỵ ởt ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X tr R M p ởt ữợ t t ữợ t t t ❞÷ì♥❣✮ ①→❝ ✤à♥❤ tr➯♥ ✈➔ ✤÷đ❝ ❧➔♠ trë✐ ❜ð✐ t✉②➳♥ t➼♥❤ F ❝õ❛ f p✳ X ✈➔ ❝❤♦ f ❧➔ ởt t t tr M ỵ r tỗ t ởt rở tr t X s❛♦ ❝❤♦ F ✈➝♥ ✤÷đ❝ ❧➔♠ trë✐ ❜ð✐ p tr➯♥ t ổ s ỵ q trồ ỵ ởt ỵ tỗ t rt ❜✐➺t ❝õ❛ ♥â ❧➔ ❝→❝ ❝→❝❤ t✐➳♣ ❝➟♥ ✤è✐ ✈ỵ✐ t t t ởt số ữợ t ỗ ỵ tt ố ỵ tt t➼❝❤ ♣❤➙♥ ❈❛✉❝❤② ❝❤♦ ❤➔♠ ❣✐↔✐ t➼❝❤ ❣✐→ trà ✈❡❝t♦r x : D → X, X ♣❤➥♥❣ ♣❤ù❝ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ D ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ ♠➦t C✳ • ❚✐➯✉ ❝❤✉➞♥ ❍❡❧❧② ❝❤♦ ❤➺ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ♣❤↔♥ ①↕ ự sỹ tỗ t r tr ỵ tt tr q ỗ tr ỵ tt trỏ ỡ tr t ỹ ợ ỵ ự q trồ tr ữủ sỹ ữợ t ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✏ ❝õ❛ ỵ õ tốt õ ỗ ữỡ ỡ ỵ ữỡ rở ỵ ✲ ❇❛♥❛❝❤ ✈➔ ❝→❝ ù♥❣ ❞ư♥❣ ✷ ❈❤÷ì♥❣ ✶ ❈→❝ ỡ ỵ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ K ❧➔ tr÷í♥❣ R ❤♦➦❝ C✱ ✈➔ X ❧➔ ♠ët K✲❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r✳ ❆✳ ▼ët ❤➔♠ sè ✭✐✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët tü❛✲♥û❛ ❝❤✉➞♥ ♥➳✉ q(x + y) ≤ q(x) + q(y) ∀x, y ∈ X ❀ q(tx) = tq(x) ∀x ∈ X, ∀t ∈ R, t ≥ 0✳ ✭✐✐✮ ❇✳ ▼ët ❤➔♠ sè ✭✐✮ q:X→R q:X→R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ ♥➳✉ q(x + y) ≤ q(x) + q(y) ∀x, y ∈ X ❀ ✭✐✐✬✮ q(λx) = |λ|q(x) ∀x ∈ X, ∀λ ∈ K✳ ❉➵ t❤➜② r➡♥❣ ♥➳✉ q ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ t❤➻ q(x) ≥ ✈ỵ✐ ♠å✐ x ∈ X✳ ❙❛✉ ✤➙②✱ t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè ♣❤✐➯♥ ❜↔♥ q✉❡♥ t❤✉ë❝ ❝õ❛ ỵ ỵ ỵ ♣❤✐➯♥ ❜↔♥ t❤ü❝✮ ✸ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r tr➯♥ tr÷í♥❣ sè t❤ü❝✳ ●✐↔ sû q : X → R ❧➔ ♠ët tü❛✲ ♥û❛ ❝❤✉➞♥✳ ●✐↔ sû Y ⊂ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ❝♦♥ ❝õ❛ X ✈➔ φ : Y → R ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ä❛ ♠➣♥✿ φ(y) ≤ q(y) ∀y ∈ Y õ tỗ t ởt t t ψ : X → R s❛♦ ❝❤♦✿ ✭✐✮ ψ|Y = φ❀ ✭✐✐✮ ψ(x) ≤ q(x) x ∈ X ❈❤ù♥❣ ♠✐♥❤✳ rữợ t t ự ỵ trữớ ủ ✤➦❝ ❜✐➺t ❞✐♠X/Y = 1✳ ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔✱ tỗ t x0 X s X = {y + sx0 | y ∈ Y, s ∈ R} ❈❤ó♥❣ t❛ ❝➛♥ ①→❝ ✤à♥❤ ❣✐→ trà ψ(x0 )✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ ❝❤ó♥❣ t❛ ❝➛♥ t➻♠ α∈R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤➸ ❤➔♠ sè ✈ỵ✐ ♠å✐ ψ:X→R ψ(y + sx0 ) = φ(y) + sα✱ y ∈ Y, s ∈ R t❤ä❛ ♠➣♥ ✭✐✐✮✳ ❱ỵ✐ s > ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ✤÷đ❝ ❤✐➸✉ ❧➔✿ φ(y) + sα ≤ q(y + sx0 ) ∀y ∈ Y, s > 0, ❝❤✐❛ ❤❛✐ ✈➳ ❝❤♦ s ✭✤➦t z = s−1 y ✮✱ ✤✐➲✉ tr tữỡ ữỡ ợ q(z + x0 ) − φ(z) ∀z ∈ Y ❱ỵ✐ s < 0✱ ✭✶✳✶✮ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ❧➔✿ φ(y) − tα ≤ q(y − tx0 ) ∀y ∈ Y, t > 0, ❝❤✐❛ ❤❛✐ ✈➳ ❝❤♦ t ✭✤➦t w = t1 y tr tữỡ ữỡ ợ ≥ φ(w) − q(w − x0 ) − ∀w ∈ Y ✹ ✭✶✳✷✮ ❳➨t ❝→❝ t➟♣ s❛✉ Z = {q(z + x0 ) − φ(z) | z ∈ Y } ⊂ R W = {φ(w) − q(w − x0 ) | w ∈ Y } ⊂ R ❈→❝ ✤✐➲✉ ❦✐➺♥ tữỡ ữỡ ợ t tự sup W ≤ α ≤ inf Z ✣✐➲✉ ✤â ❝â t t số tữỡ ữỡ ợ ♣❤↔✐ ❝❤➾ r❛ α ✭✶✳✸✮ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t tr➯♥✳ ✣✐➲✉ ♥➔② sup W ≤ inf Z ✱ ❤❛② ❧➔ φ(w) − q(w − x0 ) ≤ q(z + x0 ) − φ(z) ∀z, w ∈ Y ✭✶✳✹✮ ◆❤÷♥❣ ✤✐➲✉ tữỡ ữỡ ợ (z + w) q(z + x0 ) + q(w − x0 ), ✤✐➲✉ ♥➔② ❤✐➸♥ ♥❤✐➯♥ t❤ä❛ ♠➣♥ ✈➻ φ(z + w) ≤ q(z + w) = q((z + x0 ) + (w − x0 )) ≤ q(z + x0 ) + q(w − x0 ) ự ữủ ỵ tr tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ♥➔②✳ ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ỵ tr trữớ ủ tờ qt tt Z (Z, ) ợ ởt ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ •ν:Z→R ✭✐✮ ✭✐✐✮ X s❛♦ ❝❤♦ Z ⊃Y✳ ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ s❛♦ ❝❤♦ ν |Y = φ❀ ν(z) ≤ q(z) ∀z ∈ Z ✳ ✺ Ξ ❧➔ t➟♣ ❜➔② s❛✉ ✤➙② ✤÷đ❝ ❣å✐ ỵ õ ữủ s r trỹ t tứ ỵ ợ A := h, g(a) = a f := h |A ỵ ✷✳✸✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r t❤ü❝ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ✈➔ ❧➔ ♠ët tü❛✲♥û❛ ❝❤✉➞♥✱ h ∈ P C(X) ✈➔ −h ≤ S tr➯♥ X ✳ ❑❤✐ ✤â✱ tỗ t ởt t t L tr X s❛♦ ❝❤♦ −h ≤ L ≤ S tr➯♥ X ✳ S:XR ỵ s r ỵ q tở ỵ ỵ ỵ rr ỵ q X ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✱ A ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✱ S : X → R ❧➔ tü❛✲♥û❛ ❝❤✉➞♥✱ M : A → R ❧➔ t✉②➳♥ t➼♥❤ ✈➔ M ≤ S tr➯♥ A✳ ❑❤✐ õ tỗ t ởt t t L tr X s❛♦ ❝❤♦ L ≤ S tr➯♥ X ✈➔ L\A = M ✳ ❍➺ q✉↔ ✷✳✷✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✱ S : X → R tỹỷ A ởt t ỗ rộ X õ tỗ t ởt ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ L tr➯♥ X s❛♦ ❝❤♦ L ≤ S tr➯♥ X ✈➔ inf A L = inf A S ỵ õ t sỷ ✤➸ ✤÷❛ r❛ ♠ët ❝❤ù♥❣ ♠✐♥❤ r➜t ✤ì♥ ❣✐↔♥ ❝õ❛ ỵ ữủ t ữợ ỵ ởt t ỗ rộ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r ✈➔ f1, , fm ❧➔ ❝→❝ tỹ ỗ tr A õ tỗ t , , λm ≥ s❛♦ ❝❤♦ λ1 + + λm = ✈➔ A inf [f1 ∨ ∨ fm ] = inf [λ1 f1 + + λm fm ] A A ✷✹ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t α := inf A[f1 ∨ ∨fm] ◆➳✉ α = −∞✱ ❦➳t q✉↔ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø ❜➜t ❦ý λ1 , , λm t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❦❤→❝✱ ♥➯♥ t❛ ❣✐↔ sû α ∈ R✳ ❑➳t q✉↔ ✤÷đ❝ s✉② r tứ ỵ ợ X := Rm , S(µ1 , , µm ) := µ1 ∨ ∨ µm , g(a) := (f1 (a), , fm (a)) ✈➔ f (a) := ỵ õ t r ởt ự ỵ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❑✳ ❋❛♥ tr♦♥❣ ❬✷❪ ✷✳✷✳✷✳ tr t ỗ X ởt ổ tr tổổ tỹ ỗ ữỡ s r ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ ❧✐➯♥ tư❝ tr➯♥ X✳ ◆➳✉ X ∗✳ ❑➼ ❤✐➺✉ f ∈ P C(X) t❤➻ S(X) ❧➔ ❤å t➜t ❝↔ ❝→❝ ♥û❛ ❧✐➯♥ ❤ñ♣ ❋❡♥❝❤❡❧ f ∗ ❝õ❛ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ f ∗ (x∗ ) := sup(x∗ − f ) ∀x∗ ∈ X ∗ X ❚ø ✤à♥❤ ♥❣❤➽❛ ❞➵ ❞➔♥❣ s✉② r❛✱ ✈ỵ✐ ♠å✐ y∈X f (y) ≥ sup(y − f ∗ ) ✭✷✳✸✮ X∗ ▼ët tr♦♥❣ ♥❤ú♥❣ ❦➳t q✉↔ ❝ì t ỗ r f P C(X) ❝æ♥❣ t❤ù❝ ❋❡♥❝❤❡❧✲ ❧➔ ♠ët ❤➔♠ ♥û❛ ❧✐➯♥ tử ữợ t t ổ õ tự tr ✭✷✳✸✮✳ ❇➙② ❣✐í✱ ❣✐↔ sû ❧➔ ♠ët f ❦❤ỉ♥❣ ♥❤➜t tt ỷ tử ữợ õ yX r ❝õ❛ f ♥➳✉ ①↔② r❛ ❞➜✉ ✤➥♥❣ t❤ù❝ tr♦♥❣ ✭✷✳✸✮✳ ỵ ữ r ởt trữ ỡ ❝❤♦ ❝→❝ ✤✐➸♠ ❋❡♥❝❤❡❧✲▼♦r❡❛✉ ❝õ❛ f ✷✺ ◆❤➟♥ ①➨t ✷✳✶✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ✣➦t ✈➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ L tr➯♥ X✳ x∗ ∈ X \{0} ✣à♥❤ ♥❣❤➽❛   ∞, x, x∗ < 1; f (x) := L(x), x, x∗ ≥ ❘ã r➔♥❣✱ f ∈ P C(X) ✈➔ ♠ët ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ t↕✐ u, v ∈ X f ởt ỷ tử ữợ t ✵✳ ▲➜② X ∗✳ ❱➻ x∗ ✈➔ y∗ − L R t t tỗ s❛♦ ❝❤♦ u, x∗ = 1, v, x∗ = 0, (y ∗ − L)(u) = 0, ▲➜② y∗ ✈➔ ✤➦t f (x) = Lf (x)✳ x := u + λv ✳ ❑❤✐ ✤â ✈➔ (y ∗ − L)(v) = x, x∗ = u, x∗ = ✈➔ ♥❤÷ ✈➟② ❉♦ ✤â f ∗ (y ∗ ) ≥ x, y ∗ − f (x) = (y ∗ − L)(x) = λ(y ∗ − L)(v) = λ ❱➻ t❛ ❝è ✤à♥❤ ✈ỵ✐ ♠å✐ λ ∈ R, f ∗ (y ∗ ) = ∞✳ ❉♦ ✤â t❛ ❝â f (0) = ∞ > −∞ = sup(0 − f ∗ ), X∗ ✈➔ ♥❤÷ ✈➟② ✵ ❧➔ ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ ❋❡♥❝❤❡❧✲▼♦r❡❛✉ ❝õ❛ f ✭❱➼ ❞ư ♥➔② ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ sỷ ỵ rr tr ủ tờ ỗ ú t ữ r ♠ët ✤➦❝ tr÷♥❣ ①→❝ ✤à♥❤ ♠ët ✤✐➸♠ ❋❡♥❝❤❡❧✲ ▼♦r❡❛✉✳ ✣à♥❤ ỵ f P C(X) y X ✳ ❑❤✐ ✤â✱ f (y) ≤ supX (y − f ∗) ∗ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ λ < f (y) tỗ t S S(X) s x ∈ ❞♦♠ f ⇒ f (x) + S(x − y) ≥ λ ✷✻ ✭✷✳✹✮ ❈❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ ✣à♥❤ ỵ ợ A := f L tr LS X tr➯♥ g(x) := x−y ✱ ✭✷✳✹✮ λ < f (y)✱ tỗ t S S(X) ởt tữỡ ữỡ ợ ợ t t s X ✈➔ x ∈ ❞♦♠f ⇒ f (x) + L(x − y) tữỡ ữỡ ợ < f (y) tỗ t x X s❛♦ ❝❤♦ x ∈ ❞♦♠ ⇒ f (x) + (y − x, x∗ ) ≥ λ, ❤❛② ❧➔✿ ✈ỵ✐ ♠å✐ < f (y) tỗ t x X ∗ ) − f ∗ (x∗ ) s❛♦ ❝❤♦(y, x ≥ λ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✶✳ ◆➳✉ f ∈ P C(X) ❧➔ ♥û❛ ❧✐➯♥ tö❝ ữợ ợ y f r f t❤➻ y ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✹✮ t ủ ợ ỵ t ữủ t q✉↔ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ▲➜② λ < f (y)✳ ❈❤å♥ S ∈ S(X) s❛♦ ❝❤♦ S(u − y) ≤ f (y) − λ ⇒ f (u) > λ ▲➜② x ❧➔ ♠ët ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ ❞ư♥❣ ✭✷✳✺✮ ✈ỵ✐ u ✤÷đ❝ t❤❛② ❜ð✐ X✳ ◆➳✉ S(x − y) ≤ f (y) − λ ✭✷✳✺✮ t❤➻ ✭sû x✮ f (x) + S(x − y) ≥ f (x) ≥ λ ▼➦t ❦❤→❝✱ ♥➳✉ ✈➔ S(x − y) > f (y) − λ✱ ✤➦t γ := (f (y) − λ)/S(x − y) ∈ (0, 1) u := γx + (1 − γ)y ✳ ❑❤✐ ✤â S(u − y) = γS(x − y) = f (y) − λ ✭✷✳✺✮ λ < f (u) = f (γx + (1 − γ)y ≤ γf (x) + (1 − γ)f (y), ✷✼ ✈➔ tø s✉② r❛ γf (x) + f (y) − λ > γf (y) > γλ ❈❤✐❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ γ t❛ ✤÷đ❝ ❦➳t q✉↔ f (x) + S(x − y) > λ ❉♦ ✤â ✭✷✳✹✮ t❤ä❛ ♠➣♥ ✈➔ ❇ê ✤➲ ✤÷đ❝ ự ỵ f P C(X) ỷ tử ữợ t z f ✈➔ t↕✐ y ∈ X t❤➻ y ❧➔ ♠ët ✤✐➸♠ ❋❡♥❝❤❡❧✲▼♦r❡❛✉ ❝õ❛ f ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ t ủ ợ ỵ t t ✤÷đ❝ ❦➳t q✉↔ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ▲➜② t↕✐ T ∈ S(X) λ < f (y) ✳ ❚ø ❇ê ✤➲ ✷✳✶✱ s r tỗ s x X f (x) + T (x − z) ≥ f (z) − 1, ✈➔ ❜➡♥❣ ❝→❝❤ ❧➔♠ T ❧ỵ♥ ❤ì♥ ♥➳✉ ❝➛♥✱ t❛ ❝â t❤➸ ❣✐↔ sû f (x) ≤ λ ⇒ T (x − y) ≥ ❈❤å♥ M s❛♦ ❝❤♦ ✭✷✳✻✮ M ≥1 ✈➔ ✭✷✳✼✮ M ≥ λ − f (z) + T (y − z) + ✈➔ ✤à♥❤ ♥❣❤➽❛ S := M T ∈ S(X)✳ ▲➜② x ❧➔ ởt tỷ tũ ỵ X f (x) ≥ λ t❤➻ ❦❤✐ ✤â ❤✐➸♥ ♥❤✐➯♥ f (x) + S(x − y) ≥ λ ▼➦t ❦❤→❝✱ ♥➳✉ f (x) < λ✱ ✭✷✳✼✮ s✉② r❛ ✷✽ T (x − y) t ủ ợ t ữủ f (x) + S(x − y) = f (x) + M T (x − y) = f (x) + T (x − y) + (M − 1)T (x − y) ≥ f (x) + T (x − z) − T (y − z) + (M − 1)T (x − y) ≥ f (z) − − T (y − z) + (M − 1) ≥ λ ❉♦ ✤â ✭✷✳✹✮ ✤÷đ❝ ú t t ự ỵ r r t tr ởt ỵ tt t ố tỷ r ỵ tt ✤â♥❣ ♠ët ✈❛✐ trá r➜t q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ỗ õ t sỷ ỵ ữ r ởt ự ỵ s ỵ X ởt ổ ✈❡❝t♦r t❤ü❝✱ U ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r tæ♣æ t❤ü❝✱ ỗ ữỡ ợ ổ ố U ✈➔ F : X × U → [−∞, ∞] ❧➔ ởt ỗ (P ) t ố ❜➔✐ t♦→♥ t➻♠ ❣✐→ trà β := x∈X inf F (x, 0) ✈➔ (D) ❧➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✱ ❤❛② ❜➔✐ t♦→♥ t➻♠ sup h(u∗)✱ ✈ỵ✐ h : U ∗ → [−∞, ∞] ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ u∗ ∈U ∗ h(u∗ ) := [F (x, u) + u, u∗ ] inf (x,u)∈X×U ❈✉è✐ ❝ò♥❣✱ ❝❤♦ α ∈ [−∞, ∞]✳ ❈❤ó♥❣ t t ố s tỗ t u ∈ U ∗ s❛♦ ❝❤♦ inf [F (x, u) + u, u ] (x,u)XìU tỗ t S S(U ) s (x,u)XìU tỗ t S ∈ S(U ) s❛♦ ❝❤♦ (x,u)∈X×U (iv) inf(P ) = sup(D) inf [F (x, u) + S(u)] ≥ α inf [F (x, u) + S(u)] tỗ t u∗ ∈ U ∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ (D) ✷✾ ❑❤✐ ✤â✱ t❛ ❝â✿ ✭❛✮ (i) ⇔ (ii) ✭❜✮ (iii) ⇔ (iv) ✭❝✮ ●✐↔ sû r➡♥❣ ❤➔♠ ❣✐→ trà tè✐ ÷✉ u → x∈X inf F (x, u) ❧➔ ❜à ❝❤➦♥ tr➯♥ tr➯♥ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ 0✳ ❑❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ (iv) ✤÷đ❝ ♥❣❤✐➺♠ ✤ó♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭❛✮ ✭⇒✮ ✣✐➲✉ ♥➔② ❝â ✤÷đ❝ ✈➻ ♠ët ♣❤➛♥ tû ❜➜t ❦➻ ❝õ❛ U ∗ ✤÷đ❝ ❧➔♠ trë✐ ❜ð✐ ♠ët ♣❤➛♥ tû ❝õ❛ ✭⇐✮ ❉♦ ✭✐✮ ❧➔ ❤✐➸♥ ♥❤✐➯♥ ♥➳✉ S(U )✳ α = −∞✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ α > −∞✳ ❚ø ✭✐✐✮ s✉② r❛ F : X ×U (, +] F ỗ t + t ✭✐✮ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❇ð✐ ✈➟②✱ t❛ ❝â t❤➸ ❣✐↔ sỷ F P C(X ì U ) ỵ ✷✳✶ ✈ỵ✐ t➼♥❤ tr➯♥ U ◆➳✉ A := domF ✈➔ g(x, u) := u✱ tø ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ ✤÷đ❝ ❧➔♠ trë✐ ❜ð✐ ♠ët ♣❤➛♥ tû ❝õ❛ u∗ ∈ U ∗ t❤➻ S(U ) ❧➔ ❧✐➯♥ tö❝✳ h(u∗ ) ≤ inf x∈X [F (x, 0) + 0, u∗ ] = β ✳ ❇ð✐ ✈➟② ✭❜✮ s✉② r❛ trü❝ t✐➳♣ tø tt tr õ tỗ t T ∈ S(U ) M ∈R ✈➔ s❛♦ ❝❤♦ T (u) < tỗ t zX ú t s r r tỗ t s S S(U ) F (z, u) ≤ M, t❤ä❛ ♠➣♥ ✭✐✐✐✮✳ ❑❤➥♥❣ ✤à♥❤ ♥➔② ✤÷đ❝ s✉② r❛ tø ✭❜✮✳ ❘ã r➔♥❣✱ ❜➡♥❣ ❝→❝❤ t❤❛② ❝â β ≤ M✳ ◆➳✉ t❤➸ ❣✐↔ sû r➡♥❣ β = −∞ β ∈ R✳ t❤➻ ✭✐✐✐✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ❈❤♦ S u=0 ✈➔ λ > T (u)✳ ❧➔ ♥û❛ ❝❤✉➞♥ ❧✐➯♥ tö❝ ❚ø T (−u/λ) < 1✱ ✸✵ tr♦♥❣ ✭✷✳✽✮ t❛ S := 0✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✭✐✐✐✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ❣✐→ trà ♥➔② ❝õ❛ (x, u) ∈ X × U ✭✷✳✽✮ ✈➻ ✈➟② t❛ ❝â (M − β)T ✿ S✳ ❝❤ó♥❣ ❇➡♥❣ ❝→❝❤ ❧➜② ✭✷✳✽✮ s✉② r❛ z∈X s❛♦ ❝❤♦ F (z, −u/λ) ≤ M ✳ ◆❤÷♥❣ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ x + λz ,0 1+ t ỗ F u F (x, u) + F z, − 1+λ 1+λ λ λ F (x, u) + M ≤ 1+λ 1+λ β≤F ❚ø ✤â s✉② r❛ ≤ F (x, u) + (M − β)λ ≥ β ✳ ❇➙② ❣✐í✱ ♥➳✉ ❝❤♦λ → T (u)+ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔②✱ t❛ ♥❤➟♥ ✤÷đ❝ ❝â ♥❣❤➽❛ ❧➔ F (x, u) + (M − β)T (u) ≥ β ✳ ✣✐➲✉ ♥➔② F (x, u)+S(u) ≥ β ✳ ❚ø ✤✐➲✉ ♥➔② ✤ó♥❣ ✈ỵ✐ ♠å✐ (x, u) X ìU t ữủ ✈➔ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✭❝✮✳ ✷✳✷✳✸✳ ▼ët →♣ t tỷ tr ỡ ỵ ✷✳✽✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❦❤æ♥❣ t tữớ ợ ổ ố X S : X → 2X ❧➔ ♠ët t♦→♥ tû ✤❛ tr ỡ ỹ ợ ỗ t G sỷ r tỗ t M s ợ (x, x∗ ) ∈ G, x∗ ≤ M ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ X t❛ ❝â S(x) = ∅✳ ự ứ t ỡ S tỗ t ởt t ỗ A ởt ổ ✈❡❝tì A→R V ✈➔ →♥❤ ①↕ δ : G → A, p : A → X, q : A → X ∗ ✈➔ r: s❛♦ ❝❤♦ ✭✐✮ ❆ ❧➔ ❜❛♦ ỗ p, q (x, x ) G ⇒ p◦δ(x, x∗ ) = x, q◦δ(x, x∗ ) = x∗ ✈➔ r δ(G) ❧➔ ❛❢❢✐♥❡✱ ✈➔ r◦δ(x, x∗ ) = (x, x∗ ) ✭✐✈✮ a ∈ A ⇒ r(a) ≥ (p(a), q(a))✳ ❚❛ ❝â t❤➸ ❧➜② V ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ✸✶ X × X∗ ❝õ❛ R✱ tø (x, x∗ ) ∈ G, δ(x, x∗ ) ∈ V ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐  1, ∗ ∗ δ(x, x )(s, s ) := 0, ✈➔ p, q ✈➔ r ữủ (s,s )XìX à(s, s )s∗ ▼ö❝ ✾✱ ♣✳ ✸✷✲✸✸❪✳ ▲➜② x ✭✐✮✱ ✭✐✐✮ ✈➔ ✭✐✐✐✮✱ ✈ỵ✐ ♠å✐ ✈➔ (x, x∗ ) = (s, s∗ ) (x, x∗ ) = (s, s∗ ) p(µ) := r(à) := (s,s )XìX (s,s )XìX à(s, s )(s, s∗ ) ①❡♠ ❬✶✸✱ ❧➔ ♠ët ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ a ∈ A, q(a) ≤ M µ(s, s∗ )s, q(µ) := X✳ ❑❤✐ ✤â✱ sû ❞ư♥❣ ✈➔ sû ❞ö♥❣ ✭✐✈✮✱ r(a) − (x, q(a)) + M x − p(a) ≥ r(a) − (x, q(x)) + (x − p(a), q(a)) r(a) − (p(a), q(a)) ≥ = ❚ø ✣à♥❤ ỵ ợ tỗ t x X S := M · , f (a) := r(a)−(x, q(a)) s❛♦ ❝❤♦ ( x∗ ≤ M ) ✈➔g(a) := x−p(a)✱ ✈➔ a ∈ A ⇒ r(a) − (x, q(a)) + (x − p(a), x∗ ) ≥ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ✤➦t a = δ(s, s∗ )✱ tø ✭✐✐✐✮ ❝â (s, s∗ ) ∈ G ⇒ (s, s∗ ) − (x, s∗ ) + (x − s, x∗ ) ≥ ⇔ (x − s, x∗ − s∗ ) ≥ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ S s✉② r❛ (x, x∗ ) ∈ G✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷✳✹✳ ỵ tỗ t ổ õ ữ t ỵ A ởt t ỗ ❦❤→❝ ré♥❣ ❝õ❛ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤ü❝✱ g : A → X ❧➔ →♥❤ ①↕ ❛❢❢✐♥❡ ✈➔ f : A → R ỗ õ a ∈ A ⇒ f (a) + g(a) ✭✐✮ ≥0 ỗ t x X s a A f (a) 2(g(a), x) ú ỵ r➡♥❣ ✭✐✮ ❝â t❤➸ ✈✐➳t ❧↕✐ ❧➔ ψ:R→R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✈➔ tr♦♥❣ ✭✐✮ ❝â t❤➸ ✈✐➳t ❧➔ ♥❣❤➽❛ ❜ð✐ ψ(λ) := λ✳ inf A [f + ψ ◦ S ◦ g] ≥ 0✱ ψ(λ) := λ2 , S := · inf A [f + ψ ◦ S ◦ g]✱ ✈ỵ✐ ✈➔ x∗ ✳ tr♦♥❣ ✤â inf A [f + S ◦ g]✱ ψ:R→R ✤÷đ❝ ✤à♥❤ ❉♦ ✤â✱ ♠ët ❝➙✉ ❤ä✐ tü ♥❤✐➯♥ ❧➔ ❧✐➺✉ ❝â ♠ët ❦➳t q ỗ tớ qt ỵ ỵ t tỗ t ởt t q✉↔ ♥❤÷ ✈➟② t❤➻ ❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ✣à♥❤ ỵ ố ợ ỵ ữủ tỹ tr ỹ ỵ rữợ t ú t t số ỵ ✷✳✷✳ ❈❤♦ S : X → R ✈➔ g : A → X✳ S, g ✲t÷ì♥❣ t❤➼❝❤ ♥➳✉ ψ ∈ P C(R) ✈➔ ❞♦♠ψ ∩ ψ:R→R a∈A (S ◦ g(a), ∞) t❤➼❝❤ ❜➜t ❦ý ❣✐→ trà ❝õ❛ S ✈➔ ψ = ỗ ữ trữớ ủ ✈ỵ✐ ❤❛✐ ✈➼ ❞ư ♥➯✉ tr➯♥✮ t❤➻ S ✱ g ✲t÷ì♥❣ r➔♥❣ ❧➔ ❚❛ ♥â✐ r➡♥❣ ψ rã g ❇ê ✤➲ ✷✳✷✳ ❈❤♦ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ✈➔ S : ❧➔ tü❛✲♥û❛ ❝❤✉➞♥✳ ❈❤♦ A ởt t ỗ rộ ởt ổ tỡ g : A X S ỗ f : A R ỗ sû✱ ❤ì♥ ♥ú❛ ψ ❧➔ S, g✲t÷ì♥❣ t❤➼❝❤ ✈➔ X→R ✭✐✮ a∈A ✈➔ S ◦ g(a) ≤ α ⇒ f (a) + ψ(α) ≥ 0✳ ✣➦t M := inf f (b) + ψ(β) : b ∈ A, β ∈ ❞♦♠ψ, β > S ◦ g(b) β − S ◦ g(b) õ tỗ t [0, M ] ✈➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ L tr➯♥ X s❛♦ ❝❤♦✿ L ≤ γS ✭✐✐✮ tr➯♥ X ✈➔ f + L g () tr A ự rữợ t✐➯♥ t❛ ①➨t ✭✐✮ ✈➔ S, g✲t÷ì♥❣ t❤➼❝❤ ❝õ❛ ψ [0, ∞]✳ ◆➳✉ ✤â M ∈ ❚❛ ❝❤➾ r❛ r➡♥❣✿ ✭✐✐✐✮ a∈A ▲➜② a ✈➔ α∈ ❞♦♠ψ ⇒ f (a) + ψ(α) + M (S ◦ g(a) − α)+ ≥ 0✳ ❧➔ ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ S ◦ g(a) ≤ α A✱ ✈➔ α ❧➔ ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ ❞♦♠ψ ✳ t❤➻ tø ✭✐✮ s✉② r❛ f (a) + ψ(α) + M (S ◦ g(a) − α)+ = f (a) + ψ(α) ≥ ●✐↔ sû✱ ♠➦t ❦❤→❝ S g(a) > tỷ tũ ỵ ợ := tỹỷ S b tỷ tũ ỵ > S ◦ g(b)✳ ✤÷đ❝ t❤❛② ❜ð✐ 0≤f β ❧➔ S ◦ g(a) − α > β − S ◦ g(b) ✈➔ t➼♥❤ ❉♦ ✤â✱ sû ❞ö♥❣ ✭✐✮ ❧➛♥ t❤ù ❜❛ ợ t S ỗ S g(a) + µS ◦ g(b) α + µβ = 1+µ 1+µ µ ≥S g(a) + g(b) 1+µ 1+µ ✈➔ A✱ a g ❝❤➾ r❛ r➡♥❣ ≥S◦g ✤÷đ❝ t❤❛② t❤➳ ❜ð✐ a + µb 1+µ (a + µb)/(1 + µ) (α + µb)/(1 + µ)✮✱ α + µβ a + µb +ψ g(b) 1+µ 1+µ ≤ f (a) + ψ(α) + µf (b) + µψ(β) 1+µ ❉♦ ✤â✿ ≤ f (a) + ψ(α) + µ(f (b)) + ψ(β) = f (a) + ψ(α) + f (b) + ψ(β) (S ◦ g(a) − α) β − S ◦ g(b) ữợ ú tr b s r❛ ≤ f (a) + ψ(α) + M (S ◦ g(a) − α), ✈➔ ✭✐✐✐✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➙② ❣✐í✱ ✤➦t ˜ := X × R, S˜ : X ˜ → R X ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ˜ λ) := M (S(x) − λ)+ , A˜ := A × ❞♦♠ψ, g˜ : A˜ → X ˜ S(x, ❜ð✐ g˜(a, α) := (g(a), α) f (a) + ψ(α)✳ f˜ : A˜ → R ✈➔ ❚❛ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ tr ữủ r g Sỗ ữủ ✤÷đ❝ ①→❝ ❞à♥❤ ❜ð✐ S f˜(a, α) := ❧➔ tü❛✲♥û❛ ❝❤✉➞♥ tr➯♥ ˜ X ✈➔ ✭✐✐✐✮ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ♥❤÷ s❛✉ f˜ + S˜ ◦ g˜ ≥ tr➯♥ A, õ tứ ỵ r r tỗ t ởt tử L s❛♦ ❝❤♦✿ ˜ ≤ S˜ L ✭✐✈✮ ˜ ◦ g˜ f˜ + L tr➯♥ ▲➜② L ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr➯♥ X tr➯♥ ˜ X ✤➸ ✈ỵ✐ ♠å✐ tr➯♥ ˜ X ✈➔ ˜ A ˜ L(x, ˜ λ) = L(x) − γλ✳ (x, λ) ∈ X, ˜ 1) = ✈➔ γ = L(0, ˜ −1) ≤ S(0, ˜ −1) = M S(0, ❧➔ ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ X t❤➻ a∈A ˜ g˜ L, ✈➔ ✈➔ γ ∈ R ✤÷đ❝ ❧ü❛ ❝❤å♥ ❇➙② ❣✐í✱ ✈➔ ❞♦ ✤â ˜ 1) ≤ −γ = L(0, γ ∈ [0, M ]✳ ◆➳✉ x ˜ S(x)) ≤ S(x, ˜ S(x)) = 0✳ L(x)−γS(x) = L(x, ❑❤✐ ✤â t❛ ♣❤↔✐ ❝❤ù♥❣ r tự f rỗ L γS tr➯♥ X✳ ❈✉è✐ ❝ò♥❣✱ ❝❤✐❛ ❜➜t t❤❛② ✈➔♦ ✭✐✈✮ t❛ ✤÷đ❝ α ∈ ❞♦♠ψ ⇒ f (a) + ψ(α) + L ◦ g(a) − γα ≥ 0, ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ✭✐✐✮ ✈➔ ❞♦ ✤â ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤ ỵ X ởt ổ ❣✐❛♥ ✈❡❝tì ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ✈➔ S:X →R ❧➔ tü❛✲♥û❛ A ởt t ỗ rộ ❝õ❛ ✸✺ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ g : A → X S ỗ f : A R ỗ ỳ ỵ ✷✳✷✮❍ì♥ ♥ú❛✱ ❣✐↔ sû ψ ❧➔ S, g✲t÷ì♥❣ t❤➼❝❤ ✈➔ ✭✐✮ ψ ❧➔ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ a∈A [S ◦ g(a), ∞]✳ ❑❤✐ ✤â✱ ✭✐✐✮ ⇔ ✭✐✐✐✮✳ ✭✐✐✮ f +ψ◦S◦g ≥0 tr A ỗ t ởt ❤➔♠ t✉②➳♥ t➼♥❤ L tr➯♥ X s❛♦ ❝❤♦✿ L ≤ γS tr➯♥ X ✈➔ f + L ◦ g ≥ ψ∗(γ) tr➯♥ A✳ ✭✐✐✐✮ ✸✻ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✤➣ tr ởt số ỵ ởt ợ ỵ ổ ❦❤ê ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ♠ët sè ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ởt số ỵ ổ ữủ tr ✣ë❝ ❣✐↔ ♠✉è♥ ①❡♠ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝õ❛ ỵ õ t tr t ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤÷đ❝ ❧✐➺t ❦➯ ð ❝✉è✐ ❦❤â❛ ❧✉➟♥✳ ◆❣♦➔✐ r❛✱ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❝á♥ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ qt ự ỵ tr ♠ët sè ❧➽♥❤ ✈ü❝ ❝õ❛ ❣✐↔✐ t➼❝❤ t♦→♥ ❤å❝✳ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❍✳ ❇r❡③✐s✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ❙♦❜♦❧❡✈ ❙♣❛❝❡s ❛♥❞ P❛rt✐❛❧ ❉✐❢❢❡r✲ ❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✷✵✶✶✳ ❬✷❪ ❑✳ ❋❛♥✱ ■✳ ●❧✐❝❦s❜❡r❣✱ ❛♥❞ ❆✳ ❏✳ ❍♦❢❢♠❛♥✱ ❙②st❡♠s ♦❢ ■♥❡q✉❛❧✐t✐❡s ■♥✈♦❧✈✐♥❣ ❈♦♥✈❡① ❋✉♥❝t✐♦♥s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ❱♦❧✳ ✽✱ ♣♣✳ ✻✶✼✕✻✷✷✱ ✶✾✺✼✳ ❬✸❪ ❙✳ ❙✐♠♦♥s✱ ❆ ♥❡✇ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛❤♥✕❇❛♥❛❝❤ t❤❡♦r❡♠✱ ❆r❝❤✳ ▼❛t❤✳ ✽✵ ✭✷✵✵✸✮✱ ♣♣✳ ✻✸✵✕✻✹✻✳ ❬✹❪ ❘✳ ❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥❥✉❣❛t❡ ❉✉❛❧✐t② ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ❙■❆▼ P✉❜✲ ❧✐❝❛t✐♦♥s✱ P❤✐❧❛❞❡❧♣❤✐❛✱ P❡♥♥s②❧✈❛♥✐❛✱ ✶✾✼✹✳ ❬✺❪ ❲✳ ❘✉❞✐♥✱ ❋✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✸✳ ✸✽