✶ ▼Ư❈ ▲Ư❈ ▲í✐ ♥â✐ ✤➛✉ ✷ ✶ ✹ ✷ ỵ ổ t t t ❧✐➯♥ tö❝ ✶✳✶ ❈→❝ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ →♥❤ ①↕ ♠ð ❦❤ỉ♥❣ ❝➛♥ t➼♥❤ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝ ỵ →♥❤ ①↕ t→❝❤ ❝ì sð ✷✵ ✷✳✶ ⑩♥❤ ①↕ t→❝❤ ❝ì sð ✈➔ →♥❤ ①↕ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t ỡ s ỗ ♣❤æ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✾ ❑➳t ❧✉➟♥ ✸✹ ❚➔✐ ❧✐➺✉ t ỵ ởt tr ỳ ỵ ỡ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ♥â ❝â t❤➸ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ◆➳✉ ❳✱ ❨ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❢ ✿ ❳ →❨ ❧➔ t♦➔♥ →♥❤ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ t❤➻ ❢ ❧➔ →♥❤ ①↕ ♠ð✱ ♥❣❤➽❛ ❧➔ ❢✭●✮ ❧➔ ♠ð ♥➳✉ ● ♠ð tr♦♥❣ ❳✳ ởt q q trồ ỵ ▼ët s♦♥❣ →♥❤ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝ ❣✐ú❛ ❤❛✐ ❦❤ỉ♥❣ ởt ỗ ổ ứ õ ♥❛②✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ t➻♠ ❝→❝❤ t❤✐➳t ❧➟♣ ỵ tr ổ ợ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❦❤→❝ ♥❤❛✉✳ ❚r♦♥❣ ❝→❝ ♥➠♠ ✶✾✺✽✱ ✶✾✻✺✱ Ptr s tt ỵ →♥❤ ①↕ ♠ð ❝❤♦ t♦→♥ tû t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t❤ò♥❣✳ ◆➠♠ ✶✾✾✽ ✈➔ ✷✵✵✹✱ ◗✳ ❏✐♥❣❤✉✐✱ ❊✳ ❇❡❝❦❡♥st❡✐♥ r tt ỵ ♠ð ❝❤♦ t♦→♥ tû t✉②➳♥ t➼♥❤ ✤â♥❣ ✈➔ t♦→♥ tû t✉②➳♥ t➼♥❤ t→❝❤ ❝→❝ ✤✐➸♠✳ ●➛♥ ✤➙②✱ tr♦♥❣ ❤❛✐ ❜➔✐ ❜→♦ ❆♥ ♦♣❡♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠ ✇✐t❤♦✉t ❝♦♥t✐♥✉✐t② ❛♥❞ ❧✐♥❡❛r✐t② ❝õ❛ ▲✳ ❘♦♥❣❧✉✱ ❩✳ ❙❤✉❤✉✐✱ ❈✳ ❙✇❛rt③ ✈➔ ❆♥ ♦♣❡♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠ ❢♦r ❜❛s✐s s❡♣❛r❛t✐♥❣ ♠❛♣s ❝õ❛ ❊✳ ❇❡❝❦❡♥st❡✐♥✱ ▲✳ ◆❛r✐❝✐✱ ❝→❝ t→❝ ❣✐↔ ✤➣ t➻♠ ❝→❝❤ ♠ð rë♥❣ ❝❤♦ ❤å ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ ♥❤➡♠ t❤✐➳t ❧➟♣ ởt ỵ ợ r ỡ s tr ữợ sỹ ữợ ◆●×❚✳ P●❙✳ ❚❙✳ ❚r➛♥ ❱➠♥ ❹♥✱ ❝❤ó♥❣ tỉ✐ ✤➣ t✐➳♣ ữợ ự tỹ ợ t ởt số rở ỵ →♥❤ ①↕ ♠ð✑ ✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ■✳ ỵ ổ t t t ❧✐➯♥ ✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ♣❤➛♥ ✤➛✉ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❝→❝ ❤å t♦→♥ tû pl(X, Y )✱ P L(X, Y ) ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t õ ỗ tớ tử ữ r ❞ư ♠✐♥❤ ❤å❛✳ ❚✐➳♣ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ự ỵ ổ t t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝✳ ❚r♦♥❣ ♣❤➛♥ ❝✉è✐✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ →♥❤ ①↕ ♠ð ✈➔ ữỡ ỵ ♠ð ❝❤♦ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ✈➔ ❝❤♦ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ →♥❤ ①↕ t→❝❤ ❝ì sð✱ →♥❤ ①↕ ❣✐→✱ t➟♣ ❣✐→✱ t➟♣ tr✐➺t t✐➯✉✳ ❚✐➳♣ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ →♥❤ ①↕ t→❝❤ ❝ì sð ✈➔ →♥❤ ①↕ ❣✐→❀ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ →♥❤ ①↕ ❣✐→ ❧➔ ✤ì♥ →♥❤✳ ❈✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q tữỡ tỹ ữ q ỵ ①↕ ♠ð ✈➲ sü t÷ì♥❣ ✤÷ì♥❣ ❣✐ú❛ s♦♥❣ t→❝❤ ❝õ❛ s t t ợ ỗ ổ ữủ tỹ t rữớ ữợ sỹ ữợ t t t ì P ❚r➛♥ ❱➠♥ ❹♥✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ t❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝æ tr♦♥❣ ❚ê ●✐↔✐ t➼❝❤ ❦❤♦❛ ❚♦→♥ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣✱ ①✐♥ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍❈❙ ❈❛♦ ❳✉➙♥ ❍✉②✱ tê ❚♦→♥ tr÷í♥❣ ỗ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✶✼ ●✐↔✐ t➼❝❤ ✤➣ ❝ë♥❣ t→❝✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ú tổ rt ữủ ỳ ỵ õ ❣â♣ ❝õ❛ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❈❤ó♥❣ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ì ị é ❚➑◆❍ ❚❯❨➌◆ ❚➑◆❍✱ ▲■➊◆ ❚Ư❈ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❧➔♠ ❝ì sð ❝❤♦ ❧✉➟♥ ✈➠♥✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ❧➔ tr ởt rở ỵ ♠ð ❦❤ỉ♥❣ ❝➛♥ t➼♥❤ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝✳ ✶✳✶ ❈→❝ tự ỵ pa(R, R) t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝ tø R ✤➳♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❱ỵ✐ ♠å✐ x, u R, tỗ t , [0, 1] s❛♦ ❝❤♦ Ψ(x)+Ψ(u) = Ψ(αx+βu) ❉➵ t❤➜② r➡♥❣ ♠å✐ ❤➔♠ t✉②➳♥ t➼♥❤ T : R → R ✤➲✉ t❤✉ë❝ pa(R, R)✳ ❚✉② ♥❤✐➯♥✱ ♠ët sè ❤➔♠ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ ❝ơ♥❣ õ t t tr ỵ ✭❬✽❪✮✳ ●✐↔ sû Ψ : R → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â✱ Ψ ∈ pa(R, R) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Ψ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭✐✮ Ψ(0) = 0❀ ✭✐✐✮ Ψ(a).Ψ(b) ≤ ✈ỵ✐ ab < 0❀ ✭✐✐✐✮ min{Ψ(x) : a + b ≤ x ≤ 0} ≤ Ψ(a) + Ψ(b) ≤ max{Ψ(x) : a + b ≤ x ≤ 0} ✈ỵ✐ a ≤ b < ✈➔ min{Ψ(x) : ≤ x ≤ a + b} ≤ Ψ(a) + Ψ(b) ≤ max{Ψ(x) : ≤ x ≤ a + b} ✈ỵ✐ < a ≤ b ✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ●✐↔ sû Ψ ∈ pa(R, R)✳ õ ợ x, u R tỗ t α, β ∈ [0, 1] s❛♦ ❝❤♦ Ψ(x) + Ψ(u) = Ψ(αx + βu)✳ ❈❤ù♥❣ ♠✐♥❤✳ ✺ ✭✐✮ ▲➜② x = u = õ tỗ t , ∈ [0, 1] ✤➸ Ψ(0) + Ψ(0) = Ψ(0)✳ ❙✉② r❛ Ψ(0) = 0✳ ✭✐✐✮ ●✐↔ sû ✈ỵ✐ a < < b ♥❤÷♥❣ Ψ(a).Ψ(b) > 0✳ ❱➻ Ψ ∈ pa(R, R) tỗ t {cn} [a, b] s ❝❤♦ Ψ(cn) = kΨ(a) + (n + − k)Ψ(b) ✈ỵ✐ sè ♥➔♦ ✤â k ∈ {1, 2, , n}✳ ❉♦ ✤â✱ |Ψ(cn )| → ∞✳ ❱➻ Ψ ❧✐➯♥ tö❝ tr➯♥ [a, b] ♥➯♥ Ψ ❣✐ỵ✐ ♥ë✐ tr♦♥❣ [a, b]✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ |Ψ(cn)| → ∞✳ ❱➟② Ψ(a).Ψ(b) ≤ ✈ỵ✐ ab < 0✳ ✭✐✐✐✮ ●✐↔ sû a, b ∈ R s❛♦ ❝❤♦ a ≤ b < 0✳ õ tỗ t , [0, 1] s ❝❤♦ Ψ(a) + Ψ(b) = Ψ(αa + βb)✳ ❚❛ ❝â a + b ≤ αa + βb ≤ 0✳ ❙✉② r❛ min{Ψ(x) : a + b ≤ x ≤ 0} ≤ Ψ(αa + βb) = Ψ(a) + Ψ(b) ≤ max{Ψ(x) : a + b ≤ x ≤ 0}✳ ❚÷ì♥❣ tü✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ min{Ψ(x) : ≤ x ≤ a + b} ≤ Ψ(a) + Ψ(b) ≤ max{Ψ(x) : ≤ x ≤ a + b} ✈ỵ✐ < a ≤ b✳ ❱➟② Ψ t❤ä❛ ♠➣♥ ✭✐✐✐✮✳ ✣✐➲✉ ❦✐➺♥ ✤õ✳ ❱ỵ✐ x, u ❜➜t ❦ý t❤✉ë❝ R✱ ❦❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû x ≤ u✳ ❳➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ✲ ❚r÷í♥❣ ❤đ♣ ✶✳ x ≤ ≤ u✳ ❑❤✐ ✤â t❛ ❝â xu ≤ 0✳ ❙✉② r❛ Ψ(x).Ψ(u) ≤ 0✳ ●✐↔ sû Ψ(x) ≤ 0, Ψ(u) ≥ 0✳ ❑❤✐ ✤â✱ y∈[x,u] Ψ(y) ≤ Ψ(x) ≤ Ψ(x) + Ψ(u) ≤ Ψ(u) ≤ max Ψ(y)✳ ▼➦t ❦❤→❝✱ ❞♦ Ψ ❧✐➯♥ tö❝ ♥➯♥ tø ❜➜t ✤➥♥❣ t❤ù❝ tr s y[x,u] r tỗ t x0 [x, u] t❤ä❛ ♠➣♥ Ψ(x0) = Ψ(x) + Ψ(u)✳ ❍ì♥ ♥ú❛✱ x0 [x, u] tỗ t , [0, 1] ✤➸ x0 = αx + βu✳ ❙✉② r❛ tỗ t , [0, 1] s (x) + Ψ(u) = Ψ(αx + βu)✳ ✲ ❚r÷í♥❣ ❤đ♣ ✷✳ x ≤ u ≤ 0✳ ❑❤✐ ✤â t❛ ❝â min{Ψ(y) : x + u ≤ y ≤ 0} ≤ Ψ(x) + Ψ(u) ≤ max{Ψ(y) : x + u ≤ y 0} ỵ tữỡ tỹ tr s r tỗ t↕✐ α, β ∈ [0, 1] s❛♦ ❝❤♦ Ψ(x) + Ψ(u) = Ψ(αx + βu)✳ ✲ ❚r÷í♥❣ ❤đ♣ ✸✳ ≤ x ≤ u✱ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ ❱➟② Ψ pa(R, R) ủ pa(R, R) ỗ t t ụ ữ ỗ ổ ổ t✉②➳♥ t➼♥❤ tø R ❧➯♥ R✳ ex − 1, x ≥ 0, ❱➼ ❞ö✳ ❈→❝ ❤➔♠✿ Ψ(x) = x3 ❀ Ψ(x) = x, x < ✶✳✶✳✸ ◆❤➟♥ ①➨t ✻ ✭❬✽❪✮✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ K ✭t❤ü❝ ❤♦➦❝ ♣❤ù❝✮✳ ❍➔♠ : X → R ✤÷đ❝ ❣å✐ ❧➔ ♣❛r❛✲❝❤✉➞♥ ✭♣❛r❛♥♦r♠✮ ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭✶✮ ◆➳✉ x = 0✱ t❤➻ x = 0❀ ✭✷✮ − x = x ✈ỵ✐ ♠å✐ x ∈ X ❀ ✭✸✮ x + z ≤ x + z ✈ỵ✐ ♠å✐ x, z ∈ X ❀ ✭✹✮ ◆➳✉ {x, xn : n ≥ 1} ⊂ X s❛♦ ❝❤♦ xn − x → ✈➔ {t, tn : n ≥ 1} ⊂ K s❛♦ ❝❤♦ tn → t✱ t❤➻ tnxn − tx → 0✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X ❝ị♥❣ ✈ỵ✐ ♠ët ♣❛r❛✲❝❤✉➞♥ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❛r❛✲❝❤✉➞♥✳ ◆➳✉ ♣❛r❛✲❝❤✉➞♥ tr➯♥ X t❤♦↔ ♠➣♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥ ✭✺✮ ◆➳✉ x = 0✱ t❤➻ x = 0✱ t❤➻ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❛r❛✲❝❤✉➞♥ ❍❛✉s❞♦r❢❢✳ ◆➳✉ (X, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❛r❛✲❝❤✉➞♥ ✈➔ r > 0✱ t❛ ỵ Ur = {x X : x < r} ✭❬✽❪✮✳ ●✐↔ sû (X, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❛r❛✲❝❤✉➞♥ ❍❛✉s✲ ❞♦r❢❢ ✈➔ Y ❧➔ ❦❤æ♥❣ tỡ tổổ ỵ pl(X, Y ) = {f : X → Y : f (0) = ✈➔ f t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥(pl1 ), (pl2 ), (pl3 )} ợ (pl1) (pl2) (pl3) ữ s pl1 ợ x X, > tỗ t > s❛♦ ❝❤♦ f (x) + f (Uδ ) ⊂ f (x + Uε )❀ ✭pl2✮ ❱ỵ✐ ♠å✐ δ > 0, n N tỗ t m N s❛♦ ❝❤♦ f (nUδ ) ⊂ mf (Uδ )❀ ✭pl3✮ ợ > tỗ t () > s❛♦ ❝❤♦ −f (Uδ ) ⊂ f (Uη(δ)) ✈➔ lim η(δ) = 0✳ δ→0 ❚❛ ①➨t t❤➯♠ ✤✐➲✉ ❦✐➺♥ ✭pl4✮ ♥❤÷ s❛✉✿ ∞ ∞ ✭pl4✮ ●✐↔ sû xi < +∞ tr X f (xi) tử õ tỗ t↕✐ ✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛ i=1 {ui } ⊂ X ✈➔ c > s❛♦ ❝❤♦ i=1 ui ≤ c xi ✈ỵ✐ ♠å✐ i ✈➔ ∞ ∞ f (xi ) = f ( i=1 i=1 ui )✳ ✼ ❉➵ t❤➜② r➡♥❣ tr♦♥❣ ✭pl4✮ t❛ ❝â ∞ ui i=1 ❤ë✐ tö✳ ✭❬✽❪✮✳ ✭✐✮ pl(X, Y ) ❝❤ù❛ t➜t ❝↔ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ✭✐✐✮ ❈→❝ ✤✐➲✉ ❦✐➺♥ (pl1)✱ (pl2)✱ (pl3)✱ (pl4) ✤ë❝ ❧➟♣ ✈ỵ✐ t➼♥❤ t✉②➳♥ t➼♥❤ ❤♦➦❝ ❧✐➯♥ tư❝ ❝õ❛ →♥❤ ①↕✳ ❱➼ ❞ö✳❬❬✶❪❪ ●✐↔ sû f0 : R → R ❧➔ ❤➔♠ ❣✐→♥ ✤♦↕♥ ❤➛✉ ❦❤➢♣ ♥ì✐ ✈➔ f0 (G) = R ✈ỵ✐ ♠å✐ t➟♣ ♠ð ❦❤→❝ ré♥❣ G ⊆ R✱ ✤÷đ❝ ❝❤♦ ð ❱➼ ❞ư ✷✼ ❝❤÷ì♥❣ ✽ tr♦♥❣ ❬✶❪✳ ❑❤✐ ✤â✱ f0 t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (pl1)✱ (pl2)✱ (pl3)✱ (pl4)✳ ✶✳✶✳✻ ◆❤➟♥ ①➨t ✭❬✽❪✮✳ ●✐↔ sû (X, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❛r❛✲❝❤✉➞♥ ❍❛✉s✲ ❞♦r❢❢ ✈➔ Y ổ tỡ tổổ ỵ ✣à♥❤ ♥❣❤➽❛ P L(X, Y ) = {f ∈ pl(X, Y ) : f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (pl4)} ✭❬✽❪✮✳ ❛✮ ❉➵ t❤➜② r➡♥❣ ♥➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r➨❝❤❡t t❤➻ P L(X, Y ) ❝❤ù❛ t➜t ❝↔ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✳ ❜✮ ❚✉② ♥❤✐➯♥✱ ❝â ♥❤ú♥❣ ❦❤æ♥❣ ❣✐❛♥ X ♠➔ P L(X, Y ) ❝❤ù❛ →♥❤ ①↕ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤✳ ❱➼ ❞ư✳ ●✐↔ sû ej = (0, 0, , , 0, 0, )✱ ✈ỵ✐ j = 1, 2, ✈➔ ❣✐↔ sû ✶✳✶✳✽ ◆❤➟♥ ①➨t j C0 = {(tj )∞ ∈ R : tj → 0}✳ ❱ỵ✐ ∞ N j=1 tj ej ∈ C0 t❛ ✤➦t ∞ = supj |tj |✳ tj ej j=1 ∞ ❈è ✤à♥❤ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ ❦❤→❝ ❦❤æ♥❣ T : C0 → Y ✳ ●✐↔ sû pa(R, R) ởt ỗ ổ ổ t✉②➳♥ t➼♥❤ tø R ❧➯♥ R✳ ❚❛ ①→❝ ∞ ∞ ✤à♥❤ ❤➔♠ fT,ψ : C0 → Y ❝❤♦ ❜ð✐ fT,ψ ( tj ej ) = T ( ψ(tj )ej )✳ ❑❤✐ ✤â t❛ j=1 j=1 ❝â fT,ψ ∈ P L(C0, Y )✱ ♥❤÷♥❣ fT,ψ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤✳ ✶✳✶✳✾ ❇ê ✤➲ ✭❬✽❪✮✳ ◆➳✉ Y ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ✈➔ n ∈ N✱ t❤➻ ❤å t➜t ❝↔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ L(Rn , Y ) ⊂ P L(Rn , Y ) P L(Rn , Y ) ỗ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❝ơ♥❣ ♥❤÷ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤✳ ✽ ❉♦ Rn ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ♥➯♥ ♠å✐ t♦→♥ tû t✉②➳♥ t➼♥❤ T : Rn → Y ❧✐➯♥ tö❝✳ ❱➻ ✈➟②✱ L(Rn, Y ) ⊂ P L(Rn, Y )✳ ❳➨t T ∈ L(Rn, Y ) ✈➔ ψ ∈ pa(R, R) ởt ỗn ổ ổ t n t➼♥❤✳ ❚❛ ①→❝ ✤à♥❤ fT,ψ : Rn → Y ❜ð✐ fT,ψ ( tj ej ) = T ( ψ(tj )ej )✱ ✈ỵ✐ ♠å✐ ❈❤ù♥❣ ♠✐♥❤✳ j=1 j=1 tj ej ∈ Rn ✳ ❱➻ ❤❛✐ ❝❤✉➞♥ tj ej t2j = max |tj | ✈➔ tj ej = j=1 j=1 j=1 j=1 1≤j≤n ∞ t÷ì♥❣ ✤÷ì♥❣✱ ♥➯♥ ❝❤ó♥❣ t❛ ❝â t❤➸ ①❡♠ Rn ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ (C0, ∞)✳ n ✣➸ ✤ì♥ ❣✐↔♥✱ t❛ ✤➦t f = fT,ψ ✳ ❱ỵ✐ x = tj ej ∈ Rn ✈➔ ε > 0✳ ❚❛ ✤➦t n n n n j=1 n M= tj ej + j=1 Rn + 1✳ n ∞ t❛ ①→❝ ✤à♥❤ ψ(tj )ej j=1 ∞ n [ψ(tj ) + ψ(sj )]ej = j=1 j=1 n f( s❛♦ ❝❤♦ {αj } , {βj } ∈ [0, 1]N n [ψ(αj tj + βj sj )]ej ✳ n j=1 = f( j=1 [tj + (αj − 1)tj + βj sj ]ej ) (αj tj + βj sj )ej ) = f ( = f( j=1 n [(αj − 1)tj + βj sj ]ej ) tj e j + j=1 n j=1 n tj ej ∈ R j=1 ❝è ✤à♥❤✱ n [(αj − 1)tj + βj sj ]ej j=1 ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ t❛ s✉② r❛ n n ψ(tj )ej + j=1 [ψ(αj tj + βj sj )]ej ) n j=1 n sj ej ψ(sj )ej ) j=1 [ψ(tj ) + ψ(sj )]ej ) = T ( j=1 n j=1 ❑❤✐ ✤â✱ t❛ ❝â n = T( n j=1 j=1 ψ(tj )ej ) + T ( n ❱➻ x = ψ(sj )ej = ψ(tj )ej + n sj e j ) = T ( j=1 sj ej ∈ j=1 n n n tj e j ) + f ( j=1 ❱➻ ψ ∈ pa(R, R)✱ ✈ỵ✐ ♠é✐ n n ψ(sj )ej = j=1 n ψ(αj tj + βj sj )ej = j=1 ✈ỵ✐ ♥❤ú♥❣ ❞➣② ❦❤→❝ ♥❤❛✉ t❤✉ë❝ ❬✵✱✶❪✳ ψ(αj tj + β j sj )ej j=1 ✾ ❇➙② ❣✐í✱ t❛ ①→❝ ✤à♥❤ →♥❤ ①↕ ({αj } , {βj })✳ ✣➦t ξ( n n sj ej ) = j=1 j=1 n Φ( j=1 sj ej )✳ n → ([0, 1] ) ✈ỵ✐ n Φ( sj e j ) = j=1 [(αj −1)tj +βj sj ]ej ✱ tr♦♥❣ ✤â ({αj } , {βj }) = n tj ej ) + f ( j=1 ❱➻ Φ : R N ❑❤✐ ✤â✱ n f( n sj e j ) = f ( j=1 n tj ej + ξ( j=1 sj ej )) ✈ỵ✐ j=1 n sj ej Rn j=1 ỗ ổ tứ R R tỗ t (0, 1) s❛♦ ❝❤♦ ψ −1 (t) − ψ −1 (s) < 2ε ✈ỵ✐ t, s ∈ [−M, M ] , |t − s| < θ✳ ❈❤å♥ δ > s❛♦ n n ❝❤♦ |ψ(t)| < θ ♥➳✉ |t| < δ✳ ●✐↔ sû sj ej ∈ Uδ = rj ej : sup |rj | < δ ✳ j=1 j=1 ❱ỵ✐ ♠é✐ j ∈ N, |sj | < δ✱ t❤➻ |ψ(sj )| < θ < 1✳ ❑❤✐ ✤â✱ ψ |ψ(sj )| ≤ M − 1, |ψ(tj ) + ψ(sj )| < |ψ(tj )| + < M ✈➔ ε |(αj − 1)tj + βj sj | = |αj tj + βj sj − tj | = ψ −1 [ψ(tj ) + ψ(sj )] − ψ −1 [ψ(tj )] < ❉♦ ✤â✱ tä f ( n ξ( = sup |(αj − 1)tj + βj sj | ≤ sj ej ) j=1 n j ∞ n tj ej ) + f (Uδ ) ⊂ f ( j=1 j=1 tj ej + Uε )✳ ε < ε ✣✐➲✉ ♥➔② ❝❤ù♥❣ ❱➟② f t❤ä❛ ♠➣♥ (pl1)✳ ❈❤♦ δ > 0, n ∈ N ✈➔ tj ej ∈ Uδ ✳ ❱➻ ψ : R → R ❧✐➯♥ tư❝✱ s♦♥❣ →♥❤ ♥➯♥ ψ j=1 ✤ì♥ ✤✐➺✉ ♥❣➦t✳ ✣➦t M = max(|ψ(−nδ)| , |ψ(nδ)|), µ = min( ψ(− 2δ ) , ψ( 2δ ) ) ✈➔ số m > Mà ợ ộ j ∈ N✱ ✈➻ |tj | < δ✱ ♥➯♥ |ψ(ntm )| < Mm < µ✳ ❙✉② r❛ ψ(ntm ) ∈ ψ − 2δ , 2δ ✱ ψ(ntj ) = mψ(sj ) ✈ỵ✐ sj ∈ − 2δ , 2δ ✳ ❱➻ ψ−1 ❧✐➯♥ tö❝ ✈➔ tj → 0, sj = ψ−1 ψ(ntm ) → ❦❤✐ j → +∞✱ ♥➯♥ t❛ ❝â n j j j ✶✵ n sj e j ∈ C0 j=1 ✈ỵ✐ n j=1 n f n ≤ sj ej 0✱ t❛ ✤➦t η(δ) = ψ−1[−ψ(δ)] + ψ−1[−ψ(−δ)] ✳ ❱➻ lim ψ(t) = t→0 n s✉② r❛ lim η(δ) = 0✳ ●✐↔ sû tj ej ∈ Uδ ✳ ❱➻ ψ ❧➔ ✤ì♥ ✤✐➺✉ ♥❣➦t✱ ✈ỵ✐ δ→0 j=1 ♠é✐ j ∈ N ①↔② r❛ ♠ët tr♦♥❣ ❝→❝ ❦❤↔ ♥➠♥❣ s❛✉ ψ(−δ) < ψ(tj ) < ψ(δ) ❤♦➦❝ ψ(δ) < ψ(tj ) < ψ(−δ)✱ ❤♦➦❝ −ψ(δ) < −ψ(tj ) < −ψ(−δ)✱ ❤♦➦❝ −ψ(−δ) < −ψ(tj ) < −ψ(δ)✳ ❑❤✐ ✤â✱ ψ−1[−ψ(δ)] < ψ−1[−ψ(tj )] < ψ−1[−ψ(−δ)]✱ ❤♦➦❝ ψ−1[−ψ(−δ)] < ψ −1 [−ψ(tj )] < ψ −1 [−ψ(δ)] ♣❤↔✐ ✤ó♥❣ ✈ỵ✐ ♠é✐ j ∈ N✳ ❙✉② r❛✱ ψ −1 [−ψ(tj )] < max{ ψ −1 [−ψ(δ)] , ψ −1 [−ψ(−δ)] } < < ψ −1 [−ψ(δ)] + ψ −1 [−ψ(−δ)] = η(δ), ✈ỵ✐ ♠å✐ j ∈ N ▼➦t ❦❤→❝✱ tj → ♥➯♥ ψ [−ψ(tj )] → ✈➔ ψ−1[−ψ(tj )]ej ∈ Uη(δ)✳ ❍ì♥ j=1 ♥ú❛✱ n −1 n n −f tj ej = −T j=1 n ψ(tj )ej j=1 j=1 ψ[ψ −1 (−ψ(tj ))]ej [−ψ(tj )]ej = T =T n n j=1 ψ −1 (−ψ(tj )]ej ∈ f (Uη(δ) ) =f j=1 ❱➻ ✈➟②✱ f (Uδ ) ⊂ f (Uη(δ))✱ ❤❛② f t❤ä❛ ♠➣♥ (pl3)✳ ✷✶ x ∗ y = 0✱ ♥❣❤➽❛ ❧➔ x(n)y(n) = ✈ỵ✐ ♠å✐ n ∈ N✳ ✸✮ ❚❛ s➩ ỵ N = N {} õ ♠ët ✤✐➸♠ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ rí✐ r↕❝ N✳ ✹✮ ❱ỵ✐ k N t ỵ [k, ] [k, ) ❧➛♥ ❧÷đt ❧➔ ❝→❝ t➟♣ ❤đ♣ {n ∈ N∞ : n ≥ k}✱ {n ∈ N : n ≥ k}✳ ✺✮ [w1, , wn] ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ s✐♥❤ ❜ð✐ ❤➺ ✈❡❝tì {w1, , wn}✳ ✻✮ ●✐↔ sû A ❧➔ t➟♣ ❝♦♥ ❝õ❛ N∞✱ clA ❧➔ ❜❛♦ ✤â♥❣ tæ♣æ ❝õ❛ t➟♣ A tr♦♥❣ N∞ ✳ ❈❤✉é✐ an tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ n∈N ❤ë✐ tư ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥✱ ♥➳✉ ✈ỵ✐ ♠é✐ ❞➣② {εn } ❝→❝ sè ♥❤➟♥ ❣✐→ trà ❤♦➦❝ 1✱ t❛ ❝â ❝❤✉é✐ εnan ❤ë✐ tư✳ n∈N ❈ì sð {en} ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥✱ ♥➳✉ ♠å✐ ❝❤✉é✐ ❦❤❛✐ tr✐➸♥ t❤❡♦ ❝ì sð ♥➔② ❧➔ ❝❤✉é✐ ❤ë✐ tư ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥✳ ◆❤➟♥ ①➨t r➡♥❣ ❝→❝ ❝ì sð ✤è✐ ①ù♥❣ ❧➔ ❤å ❦❤æ♥❣ ✤✐➲✉ ❦✐➺♥✳ ✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✻❪✮✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝ì sð ❙❝❤❛✉❞❡r {xn }✱ x = x(n)xn ∈ X ✳ n∈N ❚➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ x ✭t❤❡ ③❡r♦ s❡t ♦❢ x✮✱ ❦➼ ❤✐➺✉ z(x)✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ z(x) = {n ∈ N : x(n) = 0}✳ ❚➟♣ ✤è✐ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ x ✭t❤❡ ❝♦③❡r♦ s❡t ♦❢ x✮✱ ❦➼ ❤✐➺✉ cozx✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ♣❤➛♥ ❜ị ❝õ❛ z(x)✳ ✷✳✶✳✺ ✣à♥❤ ♥❣❤➽❛ ✭❬✻❪✮✳ ●✐↔ sû X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ợ ỡ s r ữủt {xn} , {yn}✳ ▼ët →♥❤ ①↕ H : X → Y t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ H x(n)xn = H(x(n)yn ) ợ x = x(n)xn X ữủ nN nN n∈N ❣å✐ ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð ✭❜❛s✐s s❡♣❛r❛t✐♥❣✮ ✤è✐ ✈ỵ✐ ❝→❝ ❝ì sð {xn} , {yn}✳ ◆➳✉ H ✈➔ H −1 ❧➔ ❝→❝ →♥❤ ①↕ t→❝❤ ❝ì sð t❤➻ H ❣å✐ ❧➔ s♦♥❣ t→❝❤ ❝ì sð✳ ◆❤➟♥ ①➨t✳ ✲ ◆➳✉ H ❧➔ ♠ët →♥❤ ①↕ t→❝❤ ❝→❝ ❝ì sð ✈➔ ♥➳✉ x ∗ y = 0✱ t❤➻ Hx ∗ Hy = 0✱ ❤❛② ♥â✐ ❝→❝❤ ♥➳✉ cozx ∩ cozy = ∅ t❤➻ cozHx ∩ cozHy = ∅✳ ✷✳✶✳✹ ✣à♥❤ ♥❣❤➽❛ ✷✷ ✲ ❍✐➸♥ ♥❤✐➯♥✱ →♥❤ ①↕ t→❝❤ ❝ì sð H ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ t❤❡♦ tø♥❣ ✤✐➸♠✱ ♥❣❤➽❛ ❧➔ ợ n N ổ tỗ t x X s❛♦ ❝❤♦ Hx(n) = 0✳ ✷✳✶✳✻ ❱➼ ❞ö ✭❬✻❪✮✳ ●✐↔ sû π : N → N ❧➔ ♠ët →♥❤ ①↕ ❜➜t ❦➻✱ {xn } ❧➔ ♠ët ❝ì sð ❙❝❤❛✉❞❡r ❝õ❛ X ✳ ❑❤✐ ✤â ❝→❝ →♥❤ ①↕ W s❛✉ ✤➙② ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ✭✐✮ ●✐↔ sû W : X → X ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ s❛♦ ❝❤♦ W x(n)xn = n∈N w(n)x(π(n))xn ✱ tr♦♥❣ ✤â w(n) ❧➔ ♣❤➛♥ tû ❦❤→❝ ❦❤ỉ♥❣ ❝õ❛ ❤➻♥❤ ❝➛✉ ✤ì♥ n∈N ✈à tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ l∞✳ ✭✐✐✮ ❈❤å♥ w : N → R ✭❤♦➦❝ C✮ s❛♦ ❝❤♦ ❝❤✉é✐ w(n)x(π(n))yn ❤ë✐ tư n∈N ✈ỵ✐ ♠å✐ x ∈ X ✳ ❳→❝ ✤à♥❤ t♦→♥ tû W : X → X ❝❤♦ ❜ð✐ W x(n)xn = n∈N w(n)x(π(n))yn ✳ n∈N ◆❤ú♥❣ →♥❤ ①↕ W ð tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ❦❤❛✐ tr✐➸♥ trå♥❣ ❧÷đ♥❣ ✭✇❡✐❣❤t❡❞ ❝♦♠♣♦s✐t✐♦♥✮✳ ◆➳✉ π ❧➔ s♦♥❣ →♥❤ t❤➻ W ✤÷đ❝ ❣å✐ ❧➔ ❦❤❛✐ tr✐➸♥ ❣✐❛♦ ❤♦→♥ trå♥❣ ❧÷đ♥❣ ✭✇❡✐❣❤t❡❞ ♣❡r♠✉t❛t✐♦♥✮✳ ✭❬✻❪✮✳ ▼ët →♥❤ ①↕ ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð tị② t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ❝ì sð ❝õ❛ X, Y ✳ ❱➼ ❞ö✳ ❳➨t →♥❤ ①↕ I : L2 [−π, π] → L2 [−π, π]✳ ❳➨t ❝→❝ ❝ì sð {xn } = {1, cos t, sin t, } ✈➔ {yn } = 1, eit , e−it , ✳ ✲ ◆➳✉ ❝❤å♥ {xn} ✈➔ {xn} ❧➔ ❝→❝ ❝ì sð tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤ ✈➔ ♠✐➲♥ ❣✐→ trà✱ t❤➻ I ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ✲ ◆➳✉ ❝❤å♥ {xn} ✈➔ {yn} ❧➔ ❝→❝ ❝ì sð t÷ì♥❣ ù♥❣ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤ ✈➔ ♠✐➲♥ ❣✐→ trà✱ t❤➻ I ❦❤æ♥❣ ♣❤↔✐ ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ❚❤➟t ✈➟②✱ t❛ ❝â cos t ∗ sin t = (0, 1, 0, 0, ) ∗ (0, 0, 1, 0, ) = ♥❤÷♥❣ I(cos t) ∗ I(sin t) = 0, 21 , 12 , 0, 0, ∗ 0, 2i1 , − 2i1 , 0, = 0, 4i1 , − 4i1 , 0, = 0✳ ✷✳✶✳✼ ◆❤➟♥ ①➨t ✭❬✻❪✮✳ ●✐↔ sû H ❧➔ t ỡ s ợ số ữỡ m t ỵ m H ✤à♥❤ ❣✐→ ✭❤❛② ♣❤➨♣ ❝❤✐➳✉✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ [mˆ ◦ H] x = Hx(m) ✈ỵ✐ x ∈ X ✳ ✷✳✶✳✽ ✣à♥❤ ♥❣❤➽❛ ✷✸ ❚➟♣ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð U ⊂ N∞ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ tr✐➺t t✐➯✉ ✭t❤❡ ✈❛♥✐s❤✐♥❣ s❡t✮ ✤è✐ ✈ỵ✐ mˆ ◦ H ♥➳✉ tø cozx ⊂ U t❛ ❝â Hx(m) = 0✳ P❤➛♥ ❜ị ❝õ❛ ❤đ♣ t trt t ố ợ m H ữủ ❣å✐ ❧➔ t➟♣ ❣✐→ ✭t❤❡ s✉♣♣♦rt s❡t✮ ❝õ❛ mˆ ◦ H ✈➔ ❦➼ ❤✐➺✉ ❧➔ s✉♣♣mˆ ◦ H ✳ ❱➼ ❞ư✳ ◆➳✉ W ❧➔ ❦❤❛✐ tr✐➸♥ trå♥❣ ❧÷đ♥❣ ♥â✐ tr♦♥❣ ❱➼ ❞ư ✷✳✶✳✻✱ t❤➻ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ✈ỵ✐ mˆ ◦ W ✭m ∈ N✮ ❧➔ ♥❤ú♥❣ t➟♣ ✈ø❛ ✤â♥❣✱ ✈ø❛ ♠ð U ♠➔ π(m) ❦❤æ♥❣ t❤✉ë❝ U ✳ ✷✳✶✳✾ ❇ê ✤➲ ✭❬✻❪✮✳ ❍ñ♣ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ tr✐➺t t✐➯✉ ❧➔ tr✐➺t t✐➯✉✳ ●✐↔ sû H : X → Y ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✱ m ∈ N ✈➔ U1, , Uk ❧➔ ❝→❝ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ✈ỵ✐ mˆ ◦ W ✳ ❱➻ ♠ët t➟♣ ❝♦♥ ✈ø❛ ♠ð✱ ✈ø❛ ✤â♥❣ ❝õ❛ ♠ët t➟♣ tr✐➺t t✐➯✉ ❧➔ t➟♣ tr✐➺t t✐➯✉✱ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ U1, , Uk ✤ỉ✐ ♠ët rí✐ ♥❤❛✉✳ ●✐↔ sû x = x(n)xn ∈ X ✈➔ kU ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ K✲❣✐→ trà ❝õ❛ t➟♣ n∈N U ∩ N✳ ❍✐➸♥ ♥❤✐➯♥ r➡♥❣✱ xkU = x(n)xn ∈ X ✈ỵ✐ t➟♣ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð n∈U U ❜➜t ❦ý✳ k k ◆➳✉ cozx ⊂ Un✱ t❤➻ x = xkU ✳ ❱➻ t❤➳✱ ❈❤ù♥❣ ♠✐♥❤✳ n n=1 n=1 k Hx(m) = H k xkUn (m) = n=1 ❱➻ ✈➟② k Un n=1 H(xkUn )(m) = n=1 t trt t ỵ H : X → Y ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð t❤➻ ✈ỵ✐ ♠å✐ m ∈ N✱ t❤➻ t➟♣ s✉♣♣m H ỗ ởt sỷ m N rữợ t t ự sm H ❦❤→❝ ré♥❣✱ ♥❣❤➽❛ ❧➔ Us = N∞ ✈ỵ✐ Us, s ∈ S ❧➔ ❝→❝ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ✈ỵ✐ mˆ ◦ H ✳ s∈S ◆➳✉ ♥❣÷đ❝ ❧↕✐✱ Us = N∞✱ t tỗ t ỳ t trt t U1, , Uk ❈❤ù♥❣ ♠✐♥❤✳ s∈S ✷✹ ✤è✐ ✈ỵ✐ mˆ ◦ H s❛♦ ❝❤♦ Ui ♣❤õ N∞✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✾✱ Hx(m) = ✈ỵ✐ ♠å✐ i=1 x ∈ X ✳ t ợ t t ổ t tữớ t❤❡♦ tø♥❣ ✤✐➸♠ ❝õ❛ →♥❤ ①↕ H ✳ ❉♦ ✤â✱ s✉♣♣mˆ ◦ H ❦❤→❝ ré♥❣✳ ❇➙② ❣✐í✱ ❣✐↔ sû c, d ❧➔ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ s✉♣♣mˆ ◦ H ✈➔ U, V ❧➛♥ ❧÷đt ❧➔ ❝→❝ ❧➙♥ ❝➟♥ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð rí✐ ♥❤❛✉ ❝õ❛ c, d t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✱ U, V ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝→❝ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ✈ỵ✐ m ˆ ◦ H ✳ ❉♦ ✤â✱ tỗ t x, y X s cozx U, cozy ⊂ V ❦➨♦ t❤❡♦ Hx(m) = 0, Hy(m) = 0✳ ❙✉② r❛ cozx ∩ cozy = ∅ ♥❤÷♥❣ cozHx ∩ cozHy = {m} = ∅✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❝❤➜t t→❝❤ ❝ì sð ❝õ❛ →♥❤ ①↕ H t suppm H ỗ ởt ✤✐➸♠✳ k ✭❬✻❪✮✳ ❈❤♦ →♥❤ ①↕ t→❝❤ ❝ì sð H : X → Y ✳ ⑩♥❤ ①↕ h : N → N∞ ①→❝ ✤à♥❤ ❜ð✐ h(m) = s✉♣♣m ˆ ◦ H ợ m N ữủ ①↕ ❣✐→ ❝õ❛ H ✳ ◆❤➟♥ ①➨t✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ỵ t r ổ tỗ t ứ õ ứ ♠ð ♥➔♦ ❝õ❛ h(m) ❧➔ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ✈ỵ✐ mˆ ◦ H ✳ ❱➼ ❞ö✳ ❈❤♦ W ❧➔ ❦❤❛✐ tr✐➸♥ trå♥❣ ❧÷đ♥❣ tr♦♥❣ ❱➼ ❞ư ✷✳✶✳✻✳ ❚➟♣ tr✐➺t t✐➯✉ ❝õ❛ mˆ ◦ W ❧➔ t➟♣ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð U ♠➔ π(m) ∈/ U ✳ ❑❤✐ ✤â✱ →♥❤ ①↕ ❣✐→ h ❝õ❛ W ❧➔ π✳ ✷✳✶✳✶✶ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✷ ỵ sỷ H : X Y ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✱ m ∈ N ✈➔ x(n)xn ∈ X ✳ ❑❤✐ ✤â x= n∈N ✭❛✮ ◆➳✉ U ❧➔ t➟♣ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð ❝õ❛ N∞ ✈➔ U ∩ N ⊂ z(x)✱ t❤➻ h−1 (U ) ⊂ z(Hx)✳ ✭❜✮ ◆➳✉ h(m) < ∞ ✈➔ x(h(m)) = u(h(m)) ✈ỵ✐ u = t❤➻ Hx(m) = Hu(m)✳ u(n)xn ∈ X n∈N ✭❝✮ ◆➳✉ H t✉②➳♥ t➼♥❤ ✈➔ h(m) < ∞ t❤➻ Hxh(m) (m) = ✈➔ Hx(m) = x(h(m))Hxh(m) (m) ✈ỵ✐ ♠å✐ x ∈ X ✳ ✷✺ ✭❞✮ ◆➳✉ h(m) = n < ∞ t❤➻ Hxn (m) = ✈ỵ✐ m ❜➜t ❦ý ♠➔ h(m) = n✳ ❉♦ ✤â✱ Hxn = Hxn (m)ym ✳ h(m)=n ✭❡✮ ◆➳✉ h(m) = ∞ ✈➔ p ∈ N✱ t❤➻ Hup (m) = Hx(m) ✈ỵ✐ up = x(n)xn ✳ n≥p ✭❛✮ ●✐↔ sû U ❧➔ t➟♣ ❝♦♥ ✈ø❛ ✤â♥❣✱ ✈ø❛ ♠ð ❝õ❛ S∞ ✈ỵ✐ U ∩S ⊂ z(x)✳ ▲➜② ❜➜t ❦ý m ∈ h−1 (U )✱ ❦❤✐ ✤â t❛ ❝â h(m) ∈ U ✳ ◆❤í ◆❤➟♥ ①➨t tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✶✱ ✈➻ h(m) ∈ U ✱ ♥➯♥ U ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ tr✐➺t t✐➯✉ ✤è✐ ợ m H r tỗ t y ∈ X s❛♦ ❝❤♦ ✈ỵ✐ cozy ⊂ U ✱ ❦➨♦ t❤❡♦ Hy(m) = 0✳ ▼➦t ❦❤→❝✱ U ∩ N ⊂ z(x) ♥➯♥ cozy ∩ cozx = ∅✳ ❑➳t ❤ñ♣ ✈ỵ✐ t➼♥❤ t→❝❤ ❝ì sð ❝õ❛ →♥❤ ①↕ H s✉② r❛ Hx(m) = 0✳ ❉♦ ✤â✱ m ∈ z(Hx)✳ ❱➟② h−1(U ) ⊂ z(Hx)✳ ✭❜✮ ❚ø x(h(m)) = u(h(m)) t❛ ❝â h(m) ∈ z(x − u)✳ ✣➦t U = {h(m)}✳ ⑩♣ ❞ö♥❣ ✭❛✮ t❛ ❝â h−1(h(m)) ⊂ z(H(x − u))✳ ❙✉② r❛ m ∈ z(H(x − u))✳ ❚ø ✤â t❛ ❝â H(x − u)(m) = 0✳ ❱➻ H ❝ë♥❣ t➼♥❤ ♥➯♥ t❛ s✉② r❛ Hx(m) = Hu(m)✳ ✭❝✮ ❱ỵ✐ ❜➜t ❦ý x = x(n)xn ∈ X ✳ ❚ø t➼♥❤ ❝❤➜t ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ t❤❡♦ n∈N tø♥❣ ✤✐➸♠ ❝õ❛ H ✱ s r tỗ t z = z(n)xn X s ❝❤♦ Hz(m) = 0✳ n∈N ✣➦t u = z(h(m))xh(m)✳ ❱➻ u(h(m)) = z(h(m))✱ ♥➯♥ →♣ ❞ö♥❣ ✭❜✮ t❛ s✉② r❛ Hz(m) = Hu(m) = z(h(m))Hxh(m) (m)✳ ❱➻ ✈➟②✱ Hxh(m) (m) = t v = x(h(m))xh(m) ỵ tữỡ tỹ t❛ ❝â Hx(m) = Hv(m) = x(h(m))Hxh(m) (m)✳ ✭❞✮ ❱ỵ✐ U = z(xn) = C{n}✱ t❤❡♦ ❦❤➥♥❣ ✤à♥❤ ✭❛✮ t❛ ❝â h−1(U ) ⊂ z(Hxn)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♥❤ú♥❣ m ♠➔ h(m) = n t❤➻ h(m) ∈ z(xn) ❤❛② m ∈ h−1(U )✳ ❙✉② r❛ m ∈ z(Hxn)✳ ❉♦ ✤â✱ Hxn(m) = 0✳ ❚ø ✤â t❛ ❝â Hxn = Hxn (m)ym ✳ ❈❤ù♥❣ ♠✐♥❤✳ h(m)=n ✭❡✮ ❚❛ ❝â [p, ∞] ∩ N ⊂ z(x − up)✳ ▲↕✐ ✈➻ h(m) = ∞ ∈ [p, ∞]✳ ⑩♣ ❞ö♥❣ ✭❛✮ ✈➔ t➼♥❤ ❝❤➜t ❝ë♥❣ t➼♥❤ ❝õ❛ H s✉② r❛ Hup(m) = Hx(m)✳ ✷✳✶✳✶✸ ỵ sỷ H : X Y ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✱ t→❝❤ ❝ì ✷✻ sð✳ ❑❤✐ ✤â✱ ♥➳✉ {yn } ❧➔ ❝ì sð ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ t❤➻ ✈ỵ✐ ♠å✐ x ∈ X t❛ ❝â h(m) 1✳ ❚❛ ①➙② ❞ü♥❣ ♠ët ♣❤➨♣ ❤♦→♥ ✈à π ❝õ❛ N s❛♦ ❝❤♦ m = π(nm) ✈ỵ✐ ♠å✐ m ∈ N✳ ❱➻ ❝ì sð {xn } ❧➔ ✤è✐ ①ù♥❣✱ ♥➯♥ t❛ ❝â x(π(n))xn = z ∈ X ✳ ❍ì♥ ♥ú❛ t❛ ❝â n∈N x(h(π(nm )))Hxh(n ) (nm ) = x(h(m))Hxh(n ) (nm ) > ✈ỵ✐ ♠å✐ m ∈ N✳ ❈❤ù♥❣ ♠✐♥❤✳ m m m ✸✶ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❤ë✐ tư ❝õ❛ x(h((n))Hxh(n)(n)yn = Hz nN tỗ t K > ✤➸ Hxh(n)(n) < K ✈ỵ✐ ♠å✐ n ∈ N✳ ❇➙② ❣✐í✱ ❣✐↔ sû inf Hxh(n)(n) : n ∈ N = õ tỗ t (nm )mN ỗ ❝→❝ sè ♥❣✉②➯♥ ♣❤➙♥ ❜✐➺t nm s❛♦ ❝❤♦ Hxh(n ) (m) < ∞✳ m∈N ❉♦ ✤â✱ Hxh(n )(nm)yn ∈ Y t H t tỗ t x = m∈N x(n)xn ∈ X s❛♦ ❝❤♦ Hx = x(h(n))Hxh(n) (n)yn = Hxh(n ) (nm )yn ✳ n∈N n∈N n∈N ❙✉② r❛ xh(n )Hxh(n )(nm) = Hxh(n )(nm) ✈ỵ✐ ♠å✐ m ∈ N✱ ❤❛② xh(n ) = ✈ỵ✐ ♠å✐ m ∈ N✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ limnx(n) = 0✳ ❱➻ ✈➟② t❛ ❝â inf Hxh(n) (n) : n N > õ tỗ t D > ✤➸ Hxh(n) (n) > D✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝❤➾ r❛ r➡♥❣ {xn} ✈➔ {yn} ❧➔ ❤❛✐ ❝ì s tữỡ ữỡ ợ sỷ x(n)xn tử ❱➻ |Hx (n)| < D1 ♥➯♥ t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❝ì n∈N sð ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ t❛ ❝â Hx (n) x(h(n))Hxh(n)(n)yn = x(h(n))yn n∈N n∈N ❤ë✐ tö✳ ❉♦ h(N) = N ỵ t s r x(n)yn ❤ë✐ tö✳ ❙û n∈N ❞ö♥❣ H −1 ❧➔ s♦♥❣ →♥❤✱ t✉②➳♥ t➼♥❤✱ t→❝❤ ❝ì sð✱ ❜➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â y(n)yn ❤ë✐ tư ❦➨♦ t❤❡♦ y(n)xn ❤ë✐ tư✳ ❱➟② ❤❛✐ ❝ì sð {xn} n∈N n∈N ✈➔ {yn} tữỡ ữỡ ợ ố ũ t ự H ỗ ổ {xn} {yn} ❤❛✐ ❝ì sð ❙❝❤❛✉❞❡r ✤è✐ ①ù♥❣ t÷ì♥❣ ✤÷ì♥❣ ♥➯♥ ♠ët ❤♦→♥ ✈à ❜➜t ❦ý ❝õ❛ {xn } s➩ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ♠ët ❤♦→♥ ✈à ❜➜t ❦ý ❝õ❛ {yn }✳ ❱➻ t❤➳✱ ✈ỵ✐ →♥❤ ①↕ A : X → Y ❝❤♦ ❜ð✐ A x(n)xn = x(n)yn s➩ ❧✐➯♥ tö❝ ❤❛✐ ❝❤✐➲✉✳ nN nN {xn} tữỡ ữỡ ợ {xh(n)} →♥❤ ①↕ B : X → X ①→❝ ✤à♥❤ ❜ð✐ B x(n)xn = B x(h(n))xh(n) = x(h(n))xn s➩ ❧✐➯♥ tö❝ ❤❛✐ n∈N n∈N n∈N ❝❤✐➲✉✳ ❱➻ ✈➟②✱ →♥❤ ①↕ A ◦ B : X → Y, x(n)xn → x(h(n))yn ❧✐➯♥ tö❝ m m m m m m m m m h(n) h(n) nN õ tỗ t L > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x = x(n)xn ∈ X ✳ n∈N n∈N x(h(n))yn ≤ L n∈N x(n)xn n∈N ✸✷ ▲ó❝ ✤â✱ t❛ ❝â x(h(n))yn ❤ë✐ tư✳ ❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❝ì sð ❙❝❤❛✉❞❡r n∈N ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ✈ỵ✐ M ❧➔ ❤➡♥❣ sè ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝ì sð {yn} t❛ ❝â x(h(n))Hxh(n) (n)yn ≤ 2M sup Hxh(n) (n) Hx = n∈N x(h(n))yn n∈N ✈ỵ✐ ♠å✐ x ∈ X ✳ ❚ø ✤â t❛ ❝â Hx ≤ 2M KL x(n)xn ✈ỵ✐ ♠å✐ x ∈ X ✳ n∈N ❙✉② r❛ H ❧✐➯♥ tư❝✳ ❑➳t ❤đ♣ ✈ỵ✐ H ❧➔ s♦♥❣ →♥❤✱ t t t q ỵ t s r H ỗ ổ ▼ët s♦♥❣ →♥❤✱ t✉②➳♥ t➼♥❤✱ ❣✐→♥ ✤♦↕♥ ❝â t❤➸ ❦❤æ♥❣ ♣❤↔✐ ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ❱➼ ❞ư✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ ❝ì sð trü❝ ❝❤✉➞♥ {xn }✳ ❚❛ ①→❝ ✤à♥❤ →♥❤ ①↕ H t✉②➳♥ t➼♥❤ tr➯♥ s♣❛♥[xn] s❛♦ ❝❤♦ Hx = x✳ ●✐↔ sû a x= an xn ∈ X tr♦♥❣ ✤â an = ✈ỵ✐ ✈ỉ ❤↕♥ n✳ ❳➨t ♣❤➛♥ tû z = n xn ∈ n∈N n∈N X ✳ ✣➦t Hx = z, Hz = x ✈➔ ♠ð rë♥❣ H t✉②➳♥ t➼♥❤ tr➯♥ [{xn } ∪ {x, z}] = s♣❛♥({xn} ∪ {x, z})✳ ●✐↔ sû W ⊂ X ❧➔ ♣❤➛♥ ❜ò ✤↕✐ sè ❝õ❛ [{xn} ∪ {x, z}]✳ ✣➦t Hw = w ✈ỵ✐ ♠å✐ w ∈ W ✈➔ ♠ð rë♥❣ t✉②➳♥ t➼♥❤ H tr➯♥ X ✳ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② H ❧➔ s♦♥❣ →♥❤✱ t✉②➳♥ t➼♥❤✳ ❍ì♥ ♥ú❛✱ H ❣✐→♥ ✤♦↕♥ ✈➻ m m a lim H an xn = lim an xn = an xn ✳ ❱➟② H ❧➔ n xn = H m n=1 m n=1 n∈N n∈N s♦♥❣ →♥❤✱ t✉②➳♥ t➼♥❤✱ ❣✐→♥ ✤♦↕♥ ✈➔ ❦❤ỉ♥❣ t→❝❤ ❝ì sð✳ ✷✳✷✳✺ t tr ỵ ✷✳✷✳✸ ❝❤ó♥❣ t❛ ❝â ♠ët sè ♥❤➟♥ ①➨t s❛✉✳ ✭✐✮ ▼ët →♥❤ ①↕ t♦➔♥ →♥❤✱ ❣✐→♥ ✤♦↕♥✱ t→❝❤ ❝ì sð H : X → X t❤➻ ❝â t❤➸ ❦❤æ♥❣ ❧➔ ✤ì♥ →♥❤✳ ❱➼ ❞ư✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ ❝ì sð trü❝ ❝❤✉➞♥ {xn }✳ ●✐↔ sû U = {2n − 1; n ∈ N} ✈➔ Γ ❧➔ s✐➯✉ ❧å❝ ❝❤ù❛ ❧å❝ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ ∞ tr♦♥❣ N∞✳ ❑❤✐ ✤â✱ Γ ∩ U = {B ∩ U : B ∈ Γ}✳ ●✐↔ sû XU , XE ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ X s✐♥❤ ❜ð✐ {x2n−1 : n ∈ N} ✈➔ {x2n : n ∈ N}✱ t÷ì♥❣ ù♥❣✳ ●✐↔ sû f ❧➔ →♥❤ ①↕ t♦➔♥ →♥❤✱ t✉②➳♥ t➼♥❤✱ ❣✐→♥ ✤♦↕♥ tr➯♥ XU ✤÷đ❝ ❝❤♦ ð ✈➼ ✷✳✷✳✹ ◆❤➟♥ ①➨t n n ✸✸ ❞ö ✸✳✻ tr♦♥❣ ❬✺❪✳ ◆➳✉ x = x(n)xn t❤➻ x = xU +xE = x(2n − 1)x2n−1+ n∈N n∈N x(2n)x2n ✈ỵ✐ xU ∈ XU ✈➔ xE ∈ XE ✳ ❳→❝ ✤à♥❤ H : X → X ✈ỵ✐ Hx = n∈N f (xU )x1 + x(2n − 2)xn ✳ ❑❤✐ ✤â✱ t❛ t❤➜② r➡♥❣ H ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✱ n≥2 t→❝❤ ❝ì sð ✈➔ ❣✐→♥ ✤♦↕♥✳ ▲↕✐ ✈➻ f ❧➔ t♦➔♥ →♥❤ ợ ổ ữợ b t ý tỗ t {x(2n−1)} s❛♦ ❝❤♦ f x(2n − 1)x2n−1 = b✳ ❉♦ ✤â✱ ✈ỵ✐ ❜➜t ❦ý z = z(n)xn ∈ X t❛ n∈N n∈N ❝â t❤➸ ❝❤å♥ xU ∈ XU ✤➸ f (xU ) = z(1) ✈➔ xE ∈ XE ✤➸ H(xE ) = z(n)xn✳ n≥2 ❑❤✐ ✤â✱ Hx = z(1)x1 + z(n)xn = z(n)xn = z✳ ❱➻ ✈➟②✱ H ❧➔ t♦➔♥ n≥2 n∈N →♥❤✳ ◆❤÷♥❣ H ❦❤ỉ♥❣ ✤ì♥ →♥❤ ✈➻ ❣✐↔ sỷ H ỡ t t ỵ H ♣❤↔✐ ❧✐➯♥ tư❝✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❧➟♣ ❧✉➟♥ tr➯♥✳ ✭✐✐✮ ▼ët ♣❤➨♣ ✤➥♥❣ ❝ü t❤➻ ❦❤ỉ♥❣ t→❝❤ ❝ì sð✳ ❍ì♥ ♥ú❛ ♠ët ♣❤➨♣ ✤➥♥❣ ❝ü✱ t♦➔♥ →♥❤ ❝ơ♥❣ ❦❤ỉ♥❣ t→❝❤ ❝ì sð✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ❤♦➦❝ ♣❤ù❝ ✈ỵ✐ ❝ì sð trü❝ ❣✐❛♦ {xn}✳ ❱ỵ✐ x ∈ X t❛ ❜✐➸✉ ❞✐➵♥ x = x(2n − 1)x2n−1 + x(2n)x2n ✈ỵ✐ n∈N n∈N x(2n − 1), x(2n) ❧➔ ❝→❝ ❤➺ sè t❤ü❝✳ ❚❛ ①→❝ ✤à♥❤ →♥❤ ①↕ V : X → X ❝❤♦ ❜ð✐ Vx = (x(2n) + ix(2n − 1))x2n ✳ ❱➻ |x(2n) + ix(2n − 1)|2 = |x(2n)|2 + n∈N |x(2n − 1)|2 s✉② r❛ V x = x ✳ ❱➻ ✈➟② V ❧➔ ♣❤➨♣ ✤➥♥❣ ❝ü✳ ◆❤÷♥❣ V ❦❤ỉ♥❣ ❧➔ →♥❤ ①↕ t→❝❤ ❝ì sð✱ ✈➻ ♥➳✉ t❛ ❧➜② z, w ∈ X s❛♦ ❝❤♦ z(2n − 1) = ✈➔ w(2n) = 0✱ t❤➻ t❛ ❝â (z(n)) (w(n)) = ữ (V (z(n))) (V (w(n)) = ợ n ∈ N✳ ❱➻ V (X) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ ❝ì sð trü❝ ❝❤✉➞♥ {x2n}✱ ♥➯♥ sû ❞ư♥❣ ♣❤➨♣ t÷ì♥❣ ù♥❣ ✶✲✶ ❣✐ú❛ ❝→❝ ❝ì sð ❝õ❛ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ X ✈➔ V (X) t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ♣❤➨♣ ✤➥♥❣ ❝ü✱ t✉②➳♥ t➼♥❤✱ t♦➔♥ →♥❤ ✈➔ s♦♥❣ t→❝❤ W : V (X) → X s❛♦ ❝❤♦ ♥➳✉ x = x(2n)x2n t❤➻ W x = x(2n)xn✳ ❑❤✐ ✤â →♥❤ ①↕ ❤ñ♣ n∈N n∈N t❤➔♥❤ W◦V ❧➔ ♠ët ♣❤➨♣ ✤➥♥❣ ❝ü✱ t♦➔♥ →♥❤ ♥❤÷♥❣ ❦❤ỉ♥❣ t→❝❤ ❝ì sð✳ ✸✹ ❑➌❚ ▲❯❾◆ ❙❛✉ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤❛♠ ❦❤↔♦ ♥❤✐➲✉ t➔✐ ữợ sỹ ữợ t t ◆●×❚✳ P●❙✳ ❚❙✳ ❚r➛♥ ❱➠♥ ❹♥✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤✉ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉ ✶✳ ❍➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥✱ ♠è✐ q✉❛♥ ❤➺ ❝õ❛ ❤å pa(R, R)✱pl(X, Y )✱P L(X, Y )✳ ❑❤↔♦ s→t ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ →♥❤ ①↕ ♠ð✱ →♥❤ ①↕ ❤➛✉ ♠ð✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët ♠ð rë♥❣ ❝õ❛ ✤à♥❤ ỵ ổ t t ❧✐➯♥ tö❝✳ ❚r➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ →♥❤ ①↕ t→❝❤ ❝ì sð✱ →♥❤ ①↕ ❣✐→ ✈➔ ỗ ổ r ự ỵ →♥❤ ①↕ ♠ð ❝❤♦ →♥❤ ①↕ t→❝❤ ❝ì sð✳ ✸✳ ự tt ởt số ỵ ❤➺ q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♠➔ ❝→❝ t➔✐ ❧✐➺✉ ✤÷❛ r❛ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ❤♦➦❝ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t✳ ỵ ỵ ỵ ỵ ỵ ỵ ❑❍❷❖ ❬✶❪ ❇✳ ●❡❧❜❛✉♠✱ ❏✳ ❖❧♠st❡❞ ✭✶✾✽✷✮✱ ❈→❝ ♣❤↔♥ ✈➼ ❞ö tr♦♥❣ ❣✐↔✐ t➼❝❤✱ ◆❤➔ ①✉➜t ❜↔♥ ✣❍✫❚❍❈◆✳ ❬✷❪ ❚r➛♥ ❱➠♥ ❹♥✱ ❇➔✐ ❣✐↔♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✸❪ ❚r➛♥ ❱➠♥ ❹♥✱ ❚↕ ❑❤➢❝ ❈÷ ✭✷✵✵✺✮✱ ❑❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ t✉②➳♥ t➼♥❤✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✹❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯✱ ▲➯ ▼➟✉ ỡ s ỵ tt t ❤➔♠ ❚➟♣ ■■✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✳ ❬✺❪ ❊✳ ❇❡❝❦❡♥st❡✐♥✱ ▲✳ ◆❛r✐❝✐✱ ❆✳❚♦❞❞ ✭✶✾✽✽✮✱ ❆✉t♦♠❛t✐❝ ❝♦♥t✐♥✉✐t② ♦❢ ❧✐♥✲ ❡❛r ♠❛♣s ♦♥ s♣❛❝❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳✱ ✻✷✱ ✷✺✼✲✷✼✺✳ ❬✻❪ ❊✳ ❇❡❝❦❡♥st❡✐♥✱ ▲✳ ◆❛r✐❝✐ ✭✷✵✵✹✮✱ ❆♥ ♦♣❡♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠ ❢♦r ❜❛s✐s s❡♣❛r❛t✐♥❣ ♠❛♣s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳✱ ✶✸✼✱ ✸✾✲✺✵✳ ❬✼❪ ❲✳ ❘✉❞✐♥ ✭✶✾✾✶✮✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ▼❝ ●r❛✇✲❍✐❧❧✳ ❬✽❪ ▲✳ ❘♦♥❣❧✉✱ ❩✳ ❙❤✉❤✉✐✱ ❈✳ ❙✇❛rt③ ✭✷✵✶✵✮✱ ❆♥ ♦♣❡♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠ ✇✐t❤♦✉t ❝♦♥t✐♥✉✐t② ❛♥❞ ❧✐♥❡❛r✐t②✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳✱ ✶✺✼✱ ✷✵✽✻✲✷✵✾✸✳ ... ❚r➯♥ ❝ì sð ❤❛✐ ❜➔✐ tr ữợ sỹ ữợ ì P r ú tổ t ữợ ❝ù✉ ♥➔② ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ✈ỵ✐ ✤➲ t➔✐ ởt số rở ỵ ợ ự ữủ tr tr ữỡ ữỡ ỵ ①↕ ♠ð ❦❤æ♥❣ ❝➛♥ t➼♥❤ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ ✳ ❚r♦♥❣ ❝❤÷ì♥❣... f (Uδ ) ∈ N (Y ) ✈ỵ✐ ♠å✐ δ > 0✳ ❈❤♦ δ > ✈➔ y ∈ f U ✳ ❱➻ f (Ur ) ∈ N (Y ) ✈ỵ✐ ♠å✐ r > tỗ t ởt số n1 < n2 < s❛♦ ❝❤♦ V ⊆ f U ✈ỵ✐ ♠å✐ k ∈ N✳ ❉♦ X ❦❤↔ ❧② ♥➯♥ t❛ ❝❤å♥ x1 ∈ U s❛♦ ❝❤♦ y − f (x1 )... δ}✳●✐↔ sû c > ❧➔ ❤➡♥❣ sè ♥â✐ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ (pl4)✳ ●✐↔ sû y ∈ f U ✳ ❱➻ f ❧➔ ❤➛✉ ♠ð t↕✐ ∈ X tỗ t số n1 < n2 < s❛♦ ❝❤♦ V = y ∈ Y : y ≤ n1 ⊂ f U ✈ỵ✐ ♠å✐ k ∈ N✳ ❉♦ X ❦❤↔ ❧② ♥➯♥ t❛ ❝❤å♥ x1 ∈ U s❛♦