▼Ư❈ ▲Ư❈ ▼Ư❈ ▲Ư❈✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶ ▲❮■ ▼Ð ✣❺❯✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷ ❈❤÷ì♥❣ ■✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✺ ❈❤÷ì♥❣ ■■✳ ỵ ỵ ổPổ t ỗ ❝♦♠♣❛❝t ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✺ ❑➌❚ ▲❯❾◆ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ỵ t ổ õ ỵ t tró❝ ❝õ❛ ❣✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤ ❤➔♠ r➜t ❦❤→❝ s♦ ợ ữ t t ỵ ởt tr ỵ q trồ ỡ ❜↔♥ ♥❤➜t ❝õ❛ ●✐↔✐ t➼❝❤ ❤➔♠✱ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ỵ ỡ s õ ỵ tt r ỵ sỹ tỗ t ♠➔ ❞↕♥❣ ❝õ❛ ♥â ✤➦❝ ❜✐➺t t❤➼❝❤ ❤ñ♣ ♥❤ú♥❣ ✈➜♥ t t ợ ởt ữủ ợ ỳ ự tỹ t q trồ ỵ ởt ỵ rt ữủ t ữ ▼ö❝ ✤➼❝❤ ❝õ❛ ❦❤♦→ ❧✉➟♥ ❦❤➥♥❣ ✤à♥❤ ❝❤➢❝ ❝❤➢♥ sü tỗ t ởt t t ũ ợ ♥❤ú♥❣ ✤➦❝ t➼♥❤ ♥➔♦ ✤â✳ ◆❤➟♥ t❤➜② t➛♠ q✉❛♥ trå♥❣ ỵ ữủ sỹ ữợ t❤➛② ❣✐→♦ ❚❤✳❙ ▲÷ì♥❣ ◗✉è❝ ❚✉②➸♥✱ t→❝ ❣✐↔ ✤➣ q✉②➳t ự t ỵ ố t s ỵ ợ ự ữ tr t ữủ ữỡ ợ ữ s❛✉✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥✳ ❚r➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ❝õ❛ tỉ♣ỉ ✤↕✐ ❝÷ì♥❣ ✤➸ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ự ỵ q ữỡ s ữỡ ỵ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣✱ ❝❤✉➞♥✱ ♥û❛ ❝❤✉➞♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✱ ♥❣♦➔✐ r❛ ✤÷❛ r❛ ♠ë sè ✤à♥❤ ỵ ỡ ự ởt số t t õ q ỵ ✤à♥❤ ♥❣❤➽❛ ❚æ♣æ ②➳✉✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â✳ ❈✉è✐ ❝ị♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✷✱ ♥➯✉ ❧↕✐ ❝→❝ t➼♥❤ ❝❤➜t ỗ ự ữủ ởt số ỵ q ố tữủ ự ỵ ❍❛❤♥✲❇❛♥❛❝❤✳ ✹✳ P❤↕♠ ✈✐ ✤➲ t➔✐✳ ●✐↔✐ t➼❝❤ ❤➔♠✳ ✺✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐✳ ❑❤♦→ ❧✉➟♥ s➩ ❧➔ ♠ët t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤➸ s ỡ ỵ ự õ ự ỵ ❦❤→❝✳ ❉♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ♥➯♥ ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❦❤♦→ ❧✉➟♥ ♥➔② ❝❤➾ tr➼❝❤ ð ❞↕♥❣ ❜ê ✤➲✱ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷♥❣ ❝â ❝❤ó t❤➼❝❤ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❙❛✉ ✤➙② ❧➔ ♠ët số ỵ ữủ q ữợ tr t ✈✐➳t ❚r♦♥❣ t♦➔♥ ❜ë ❜➔✐ ✈✐➳t ❦❤✐ ❝❤♦ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ X t❤➻ t❛ ❤✐➸✉ ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ t q ữợ tt ổ ❧➔ τ1 ✈➔ ❝❤➼♥❤ q✉② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t❤✉➟t ♥❣ú ❦❤→❝ ♥➳✉ ❦❤ỉ♥❣ ♥â✐ ❣➻ t❤➯♠ t❤➻ ✤÷đ❝ ❤✐➸✉ tổ tữớ sỷ E t ỗ ổ ỗ ữỡ X õ õ E ữủ ỵ E õ ỵ E sỷ X ổ tổổ õ ổ ỗ ữỡ ỵ Xw ố Xw ữủ ỵ X ∗ ✳ ✇ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ♥➔② t→❝ ❣✐↔ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï r➜t ♥❤✐➲✉ ❝õ❛ ❣✐❛ ✤➻♥❤✱ t❤➛② ❝æ ✈➔ ❜↕♥ ❜➧✳ ✣➛✉ t✐➯♥ ❝❤♦ ♣❤➨♣ t→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝→♠ ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ❚❤✳❙ ▲÷ì♥❣ ◗✉è❝ ❚✉②➸♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ð tr♦♥❣ s✉èt q✉→ tr➻♥❤ ✤➸ ❤♦➔♥ t❤➔♥❤ ❦❤♦→ ❧✉➟♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕②✳ ❈✉è✐ ❝ò♥❣ t→❝ ❣✐↔ ①✐♥ ❝→♠ ì♥ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ✈➔ t➜t ❝↔ ✸ ❜↕♥ ❜➧✱ ✤➦❝ ❜✐➺t ❧➔ ❇❛ ♠→ ✈➔ ❝→❝ ❜↕♥ ❧ỵ♣ ✵✽❈❚❚✷ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ✈➻ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤♦→ ❧✉➟♥ ✈➝♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât ✈➲ ♠➦t ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝✳ ❱➻ ✈➟②✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ✤÷đ❝ ữủ sỹ õ õ qỵ t ổ ỳ õ ỵ ✣➔ ◆➤♥❣✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✷✳ ❚→❝ ❣✐↔✳ ✹ ❈❤÷ì♥❣ ■✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❇❷◆ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ♠ët t➟♣ ❤ñ♣✱ K ❧➔ ♠ët tr÷í♥❣✳ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥ tr÷í♥❣ K ✱ ♥➳✉ ✶✳ ✣÷❛ ✤÷đ❝ ♣❤➨♣ ❝ë♥❣ ❝→❝ ♣❤➛♥ tû ✈➔♦ X ✱ tù❝ ❧➔ ù♥❣ ♠é✐ x, y) X ì X ợ ởt tỷ X ỵ x + y s❛♦ ❝❤♦ a✮ x + y = y + x, ∀ x, y ∈ X b✮ ✭x + y) + z = x + (y + z), ∀ x, y, z X c ỗ t tỷ ổ X ỵ s x + = x, ∀x ∈ X ✳ d✮ ❱ỵ✐ ♠å✐ x X õ tỗ t tỷ ố x ỵ x s x + (x) = 0, ∀x ∈ X ✷✳ ✣÷❛ ✤÷đ❝ ♣❤➨♣ ♥❤➙♥ ❝→❝ ♣❤➛♥ tû ❝õ❛ K ✈ỵ✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✱ tù❝ ❧➔ ù♥❣ ✈ỵ✐ ♠é✐ ❝➦♣ ✭α, x K ì X ợ ởt tỷ X ỵ x t ✈ỵ✐ x✱ s❛♦ ❝❤♦ a✮ ✶✳x = x, ∀x ∈ X ✱ ✶ ❧➔ ✤ì♥ ✈à ❝õ❛ tr÷í♥❣ K ✳ b✮ α(βx) = (αβ)x, ∀α, β ∈ K, ∀x ∈ X ✳ c✮ (α + β)x = αx + βx, ∀α, β ∈ K, ∀x, y ∈ X ✳ d✮ α(x + y) = αx + αy, ∀α ∈ K, x, y X ú ỵ t ❝❤➾ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ ❧➔ tr÷í♥❣ R ❝→❝ sè t❤ü❝ ✈➔ ❦❤✐ ✤â X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝✳ a K ❧➔ tr÷í♥❣ C ❝→❝ sè ♣❤ù❝ ✈➔ ❦❤✐ ✤â X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝✳ ❱➻ t❤➳ t❛ s➩ ❣å✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ K ❧➔ ❝→❝ sè ✈➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝tì ❤♦➦❝ ✤✐➸♠✳ b K ✺ ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❝♦♥ M ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✱ ♥➳✉ ✈ỵ✐ ♠å✐ x, sè α, β t❛ ✤➲✉ ❝â αx + βy ∈ M ✳ y ∈ M✱ ✈➔ ♠å✐ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ Φ ✭Φ ❧➔ t❤ü❝ ❤♦➦❝ ♣❤ù❝✮✳ ❚❛ ♥â✐ r➡♥❣ tæ♣æ τ tr➯♥ X tữỡ t ợ trú số tr X ❝→❝ ♣❤➨♣ t♦→♥ ✤↕✐ sè tr➯♥ X ✭❝ë♥❣ ✈➔ ♥❤➙♥ ổ ữợ tử t tổổ õ a b ❱ỵ✐ ∀x, y ∈ X ✱ ✈➔ ♠å✐ W x + y tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x ✈➔ V ❝õ❛ y s❛♦ ❝❤♦ U + V ⊂ W ✳ ❱ỵ✐ ∀x ∈ X, ∀α ∈ Φ✱ ♠å✐ ❧➙♥ ❝➟♥ W ❝õ❛ x tỗ t ởt U x ✈➔ sè r > s❛♦ ❝❤♦ βU ⊂ W, ∀β ∈ Φ ♠➔ |β − α| < r✳ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X tr➯♥ tr÷í♥❣ Φ ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✭❤❛② ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ t✉②➳♥ t➼♥❤ ✮ ♥➳✉ tr➯♥ ✤â ✤➣ ❝❤♦ ♠ët tæ♣æ tữỡ t ợ trú số tr X s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ ❝õ❛ X ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣✳ ✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛ ●✐↔ sû X, Y ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ tr➯♥ ❝ị♥❣ ♠ët tr÷í♥❣ Φ✱ →♥❤ ①↕ Λ : X → Y ✤÷đ❝ ❣å✐ ❧➔ t✉②➳♥ t➼♥❤ ♥➳✉ Λ(αx + βy) = αΛ(x) + βΛ(y), ∀x, y ∈ X, ∀α, β ∈ Φ✳ ✯ ▼ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ Λ : X → Φ ✤÷đ❝ ❣å✐ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✳ ✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✳ a ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❤ót✱ ♥➳✉ ∀x ∈ X, ∃Λ > s❛♦ ❝❤♦ x ∈ αA, ∀α ∈ Φ b ♠➔ |α| ≥ Λ ❚➟♣ A ữủ ỗ x, y A, ✻ ∀Λ ∈ [0, 1] t❛ ❝â Λx + (1 − Λ)y ∈ A✳ c ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❝➙♥ ♥➳✉ ∀x ∈ A✱ t❛ ❝â αx ∈ A, ∀α ∈ Φ ♠➔ |α| ≤ d e ❚➟♣ A ữủ tt ố ỗ A ứ ỗ ứ A ữủ ♥➳✉ ✈ỵ✐ ♠å✐ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ ∈ X tỗ t số > s A ⊂ tV, ∀t > s✳ ✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ♣❤õ tr X tỗ t ỳ ✭♥❣❤➽❛ ❧➔✿ ◆➳✉ u = {uα , α ∈ λ} ởt ỗ t X t tỗ t n , , , n s❛♦ ❝❤♦ X ⊂ uαi ✮✳ i=1 ✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ K ⊂ X ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➳✉ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ K ❝ị♥❣ ✈ỵ✐ tỉ♣ỉ ❝↔♠ s✐♥❤ tr➯♥ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t✱ ♥❣❤➽❛ ❧➔ ♠å✐ ♣❤õ ❝õ❛ K ỗ t tr ổ K tỗ t ỳ sỷ A t ỗ út ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✱ ❤➔♠ t❤ü❝ ❦❤ỉ♥❣ ➙♠ tt ố ỗ àA : X R ❝ỉ♥❣ t❤ù❝ µA (x) = inf{t > : t−1 x ∈ A}, ∀x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❝ï ❤♦➦❝ ♣❤✐➳♠ ❤➔♠ ▼✐♥❦♦✇s❦✐ ❝õ❛ A✳ ✶✳✶✳✶✵ ◆❤➟♥ ①➨t✳ a ❚➟♣ A ⇔ ∀x1, x2, , xn ∈ X, n N tỗ t số > s❛♦ ❝❤♦ {x1 , x2 , , xn } ⊂ αA, ∀α ∈ Φ ✼ ♠➔ |α| ≥ Λ b ❚➟♣ A ❝➙♥ ⇔ αA ⊂ A, ∀α ∈ Φ, |α| ≤ ✶✳✶✳✶✶ ▼➺♥❤ ✤➲✳ ●✐↔ sû Λ : X → Y ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â a Λ(0) = 0✳ b c d ◆➳✉ A ❧➔ ởt ổ ỗ X t❤➻ Λ(A) ❝ơ♥❣ ✈➟②✳ ◆➳✉ B ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ỗ Y t (B) ❝ô♥❣ ✈➟②✳ ✣➦❝ ❜✐➺t✱ Λ−1(0) = {x ∈ X : ❣✐❛♥ ❝♦♥ ❝õ❛ X ✳ Λ(x) = 0} = N (Λ) ❧➔ ♠ët ❦❤æ♥❣ ✶✳✶✳✶✷ ✣à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët ❝➦♣ ✭X, ρ✮✱ tr♦♥❣ ✤â X ❧➔ ♠ët t➟♣ ❤đ♣✱ ρ : X × X → R ❧➔ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ X × X t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✶✳ ρ(x, y) ≥ 0✱ ✈ỵ✐ ♠å✐ x, y ∈ X ✱ ρ(x, y) = ⇔ x = y ❀ ✷✳ ρ(x, y) = ρ(y, x)✱ ✈ỵ✐ ♠å✐ x, y, z ∈ X ❀ ✸✳ ρ(x, z) ∈ ρ(x, y) + ρ(y, z)✱ ✈ỵ✐ ♠å✐ x, y, z ∈ X ✶✳✶✳✶✸ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ❧➔ ♠ët ❝➦♣ ✭X, τ ✮✱ tr♦♥❣ ✤â X ❧➔ ♠ët t➟♣ ❤ñ♣✱ τ ❧➔ ♠ët ❤å ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X t❤♦↔ ♠➣♥ ✶✳ ∅ ∈ τ, X ∈ τ ✳ ✷✳ U1, U2 ∈ τ ⇒ U1 ∩ U2 ∈ τ ✳ ✸✳ ◆➳✉ {Uα}α∈I t❤➻ Uα ∈ τ α∈I ❚➟♣ ❤ñ♣ X ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥✱ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ❣å✐ ❧➔ ❝→❝ ✤✐➸♠ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✈➔ ♠é✐ ♣❤➛♥ tû ❝õ❛ τ ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ❤đ♣ ♠ð ❝õ❛ X ✳ ◆❤÷ ✈➟②✱ ✽ ❚➟♣ ❤đ♣ ré♥❣ ✈➔ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ ❧➔ ♥❤ú♥❣ t➟♣ ❤đ♣ ♠ð✳ ii ●✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ❤ñ♣ ♠ð ❧➔ ♠ët t➟♣ ủ iii ủ ởt tý ỵ ỳ t➟♣ ❤ñ♣ ♠ð ❧➔ ♠ët t➟♣ ❤ñ♣ ♠ð✳ ✶✳✶✳✶✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✭X, τ ✮ ❣å✐ ❧➔ ♠ët T 1−❦❤ỉ♥❣ ❣✐❛♥ ♥➳✉ ✈ỵ✐ ❤❛✐ ♣❤➛♥ tû ❦❤→❝ ♥❤❛✉ t ý x1 x2 X tỗ t ♠ët t➟♣ ❤đ♣ ♠ð U ❝❤ù❛ x1 ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤ù❛ x2 i ✶✳✶✳✶✺ ✣à♥❤ ♥❣❤➽❛✳ X ✤÷đ❝ ❣å✐ ❧➔ T 2−❦❤æ♥❣ ❣✐❛♥ ✭❤❛② ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦s❢❢ ✮ ♥➳✉ ∀x, y ∈ X, ❝õ❛ y s❛♦ ❝❤♦ U ∩ V = x = y tỗ t U x V ú ỵ ộ T 2ổ ❣✐❛♥ ✤➲✉ ❧➔ ♠ët T 1−❦❤æ♥❣ ❣✐❛♥✳ ✶✳✶✳✶✻ ✣à♥❤ ♥❣❤➽❛ sỷ A t õ tũ ỵ ổ tỡ tổổ X a ỗ A ỵ convA tờ ủ tt tê ❤ñ♣ t✉②➳♥ k t➼♥❤ ❤ú✉ ❤↕♥ Λixi✱ tr♦♥❣ ✤â i=1 ∗ k Λi ≥ 0, xi ∈ A, i = 1, 2, , k, k ∈ N , Λi = i=1 b tt ố ỗ A ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ k ❤ú✉ ❤↕♥ Λixi✱ tr♦♥❣ ✤â i=1 ∗ k Λi ∈ Φ, xi ∈ A, i = 1, 2, , k, k ∈ N , i=1 Λ i ≤ 1✳ t ỗ A tt t ủ ỗ tr X ự A õ t ỗ ọ t ự A ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ Φ ✭Φ t❤ü❝ ❤❛② ♣❤ù❝✮✳ ❍➔♠ p : t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✳ X→R ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ X ♥➳✉ ✾ a p(x + y) ≤ p(x) + p(y), ∀x, y ∈ X b p(λx) = |λ| p(x), ∀λ ∈ Φ ▼ët ♥û❛ ❝❤✉➞♥ p t❤♦↔ ♠➣♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥ c p(x) = ⇒ x = t❤➻ p ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ ①→❝ ✤à♥❤ tr➯♥ X õ t ỵ t p(x) ◆❤÷ ✈➟② t❤♦↔ ♠➣♥ ✸ ✤✐➲✉ ❦✐➺♥ ✶✳ x ≥ 0, ∀x ∈ X ✷✳ λx = |λ| x , ∀x ∈ X, ∀λ ∈ Φ✳ ✸✳ x + y ≤ x + y , ∀x, y ∈ X x x = ⇔ x = 0✳ ✶✳✶✳✶✽ ✣à♥❤ ♥❣❤➽❛✳ ❍å P ❝→❝ ♥ú❛ ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X ✤÷đ❝ ❣å✐ ❧➔ t→❝❤ ❝→❝ ✤✐➸♠ tr➯♥ X ♥➳✉ ợ ộ x X x = tỗ t↕✐ ♥û❛ ❝❤✉➞♥ p ∈ P s❛♦ ❝❤♦ p(x) = ✶✳✶✳✶✾ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ ✭❤❛② ự ũ ợ ởt tr X ữủ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤ü❝ ✭❤❛② ♣❤ù❝✮✳ ú ỵ r ổ ổ ỗ ữỡ sỷ A ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X tr➯♥ tr÷í♥❣ Φ✳ ữủ ổ ỗ ữỡ õ ❝â ♠ët ❝ì sð ✤à❛ ♣❤÷ì♥❣ B s❛♦ ❝❤♦ ♠å✐ tỷ B t ủ ỗ a X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❜à ❝❤➦♥ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ ♥â ❝â ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ ✵ ❧➔ t➟♣ ❜à ❝❤➦♥✳ b X ✶✵ V = {x ∈ X : |Λi x| < ri , ∀i = 1, n} (1✮✳ ❑❤✐ ✤â✱ ❤å t➜t ❝↔ ❝→❝ t➟♣ ❝â ❞↕♥❣ ✭✶✮ ❧➟♣ t❤➔♥❤ ❝ì sð ✤à❛ ♣❤÷ì♥❣ ❝õ❛ (X, ) t õ V ỗ t ❞♦ ❝→❝ ♣❤➛♥ tû ❝õ❛ X t✉②➳♥ t➼♥❤ ♥➯♥ V ỗ V + V = ( + )V ; α, β ∈ ΦR+ +✮ V ❈➙♥✳ ▲➜② x ∈ V, α ∈ Φ, |α| ≤ 1✳ ❑❤✐ ✤â |Λi (αx)| = |α| |Λi x| < ri , ∀i = 1, n + V ỗ t [0, 1]; x, y ∈ V |Λi (tx + (1 − t)y)| = |tΛi x + (1 − t)Λi y| ≤ t |Λi x| + (1 − t) |Λi y| ≤ tri + (1 − t)ri = ri , ∀i = 1, n +✮ αV + βV = (α + β)V ✳ ❍✐➸♥ ♥❤✐➯♥ (α + β)V ⊂ αV + βV αV + βV ⊂ (α + β)V ▲➜② f = αx + βy✳ |Λi (αx + βy)| = |Λi (αx)| + |Λi (βy)| = |α| |Λi x| + |β| |Λi y| ≤ (|α| + |β|)ri = (α + β)ri , ∀α, β ∈ R+ +✮ P❤➨♣ ❝ë♥❣ ❧✐➯♥ tö❝ ✈➻ V = 12 V + 21 V +✮ P❤➨♣ ♥❤➙♥ ổ ữợ tử ợ x t ý tở X, α ❜➜t ❦ý t❤✉ë❝ Φ ✈➔ αx + V ❧➔ x õ tỗ t s > s❛♦ ❝❤♦ x ∈ sV ✳ ▲➜② r > s❛♦ ❝❤♦ r(s + r) + |α| r < ✭❉ò♥❣ t❛♠ t❤ù❝ r2 + (|α| + s)r − < 0✱ ❞♦ a.c < 0✮✱ ❦❤✐ ✤â ∀β ∈ Φ t❤ä❛ ♠➣♥ |β − α| < r ✈➔ ∀y ∈ x + rV ✱ t❛ ❝â ✷✾ βy − αx = (β − α)y + α(y − x)✳ ▼➔ t❛ ❧↕✐ ❝â i (β − α)y ∈ (β − α)(x + rV ) ⊂ (β − α)(sV + rV ) = (β − α)(s + r)V ii α(y − x) ∈ αrV ✳ ❚ø (i) ✈➔ (ii) s✉② r❛ (β − α)y + α(y − x) ∈ (β − α)(r + s)V + αrV ⊂ (|β − α| (r + s) + |α| r)V ⊂ V ❱➟② βy − αx ∈ V ✱ s✉② r❛ βy ∈ x + V õ tỗ t w = x + rV ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ y ✈➔ ∀β ∈ B(α, r)✱ t❛ ❝â βw ⊂ αx + V r ổ ữợ tử X ∗ =④f ✿ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ X ⑥✳ ❈❤ù♥❣ ♠✐♥❤ X∗ = X ✳ ❘ã r➔♥❣ X ⊂ X ∗✳ ✭iii✮ ❇➙② ❣✐í ❧➜② Λ ❜➜t ý tở X õ tỗ t V õ ❞↕♥❣ ✭✶✮ s❛♦ ❝❤♦ |Λx| < 1, ∀x ∈ V tỗ t 1, , n X ✈➔ ❝→❝ sè ❞÷ì♥❣ r1, , rn s❛♦ ❝❤♦ |Λx| < 1✱ ✈ỵ✐ ♠å✐ x t❤ä❛ ♠➣♥ |Λi x| < ri , ∀i = 1, n✳ ❙✉② r❛ |Λi x| < γ = max{ri : i = 1, n} s✉② r❛ ✸✵ γ |Λi x| < 1, ∀i = 1, n s✉② r❛ γ max{|Λi x| : i = 1, n} < s✉② r❛ |Λx | < γ max |Λi x| , ∀x ∈ X ✳ i=1,n ❙✉② r❛ Λ t❤ä❛ ♠➣♥ ❇ê ✤➲ ✷✳✷✳✹ ✭b✮ ♥➯♥ t❤❡♦ a s r tỗ t , , αn ∈ Φ✱ s❛♦ ❝❤♦ Λ = α1 Λ1 + α2 Λ2 + + αn Λn ❱➻ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì Λi ∈ X s✉② r❛ Λ ∈ X ✳ ❉♦ ✤â✱ X ∗ ⊂ X ✳ ✭iv✮ ❚ø ✭iii✮ ✈➔ ✭iv ✮ s✉② r❛ X ∗ = X ✳ ✷✳✷✳✻ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈ỵ✐ tỉ♣ỉ τ s❛♦ ❝❤♦ ✤è✐ ♥❣➝✉ X ∗ ❝õ❛ ♥â t→❝❤ ✤÷đ❝ ❝→❝ ✤✐➸♠ tr➯♥ X ✳ ❚æ♣æ ✤➛✉ tr➯♥ X s✐♥❤ ❜ð✐ ❤å X ữủ tổổ X ỵ ❤✐➺✉ ❧➔ τw ✳ ✯ ❑❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ tỉ♣ỉ ②➳✉ τw ✤÷đ❝ ❦➼ ❤✐➺✉ Xw ✳ ✷✳✷✳✼ ◆❤➟♥ ①➨t✳ ✶✳ ứ ỵ t s r r Xw ởt ổ ỗ ữỡ ố ♥â ❝❤➼♥❤ ❧➔ X ∗✳ ✷✳ ❱➻ τw ❧➔ tæ♣æ ②➳✉ ♥❤➜t tr➯♥ X s❛♦ ❝❤♦ t➜t ❝↔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ Λ ∈ X ∗ ❧✐➯♥ tö❝ ✈➔ ✈➻ ∀Λ ∈ X ∗ ❧✐➯♥ tư❝ t❤❡♦ tỉ♣ỉ ①✉➜t t tổổ trữợ w τ ❣å✐ ❧➔ tæ♣æ ①✉➜t ♣❤→t✳ ✸✳ ❚❛ ♥â✐ r➡♥❣ {xn} X tử ỵ ❤✐➺✉ ❧➔ xn →ω ♥➳✉ ✈ỵ✐ ♠å✐ ❧➙♥ ❝➟♥ ❝õ❛ ✵ t❤❡♦ tỉ♣ỉ ②➳✉ ✤➲✉ ❝❤ù❛ ❝→❝ xn✱ ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❚❛ ❝â xn → ⇔ Λxn → 0, ∀Λ ∈ X ∗ ✳ ω n→∞ ✸✶ ✹✳ ❚➟♣ ❝♦♥ E ❝õ❛ X ❧➔ ❜à ❝❤➦♥ ②➳✉ ⇔ ∀Λ ∈ X ∗✱ ✤➲✉ ∃ γ(Λ) < ∞✳ ❙❛♦ ❝❤♦ |Λx| < γ(Λ), ∀x ∈ E ✳ ✺✳ ◆➳✉ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈ỉ ❤↕♥ ❝❤✐➲✉ t❤➻ ♠å✐ ❧➙♥ ❝➟♥ ②➳✉ ❝õ❛ ✵ ✤➲✉ ❝❤ù❛ ♠ët ổ ổ ứ ỵ s✉② r❛ r➡♥❣ (Xw )w = Xw ✳ ✷✳✷✳✽ ✣à♥❤ ỵ E ởt t ỗ ổ ỗ ữỡ X õ õ ②➳✉ Ew ❝õ❛ E trị♥❣ ✈ỵ✐ ❜❛♦ ✤â♥❣ E ❝õ❛ ♥â t❤❡♦ tæ♣æ ①✉➜t ♣❤→t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â Ew ❧➔ ♠ët t➟♣ ✤â♥❣ t❤❡♦ tæ♣æ ②➳✉ ♥➯♥ X\E ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ τw ✱ s✉② r❛ X\Ew ∈ τ s✉② r❛ E ✤â♥❣ t❤❡♦ tæ♣æ ①✉➜t ♣❤→t✳ ❉♦ ✤â✱ E ⊂ Ew ✭✶✮✳ ▲➜② x0 ∈ X s❛♦ ❝❤♦ x0 ∈/ E ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x0 ∈/ Ew ✳ ❚❤➟t ✈➟②✱ {x0} ✲❝♦♠♣❛❝t ✳ E ❧➔ ♠ët t➟♣ ✤â♥❣✱ {x0} ∩ E = ∅✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ỗ ữỡ X ỵ b ❙✉② r❛ ∃Λ ∈ X ∗ , γ ∈ R : ReΛx0 < γ < ReΛx, ∀x ∈ E ✳ ❉♦ ✤â ✇ U = {x ∈ X : ReΛx < γ} ❧➔ ♠ët ❧➙♥ ❝➟♥ ♠ð ❝õ❛ x0 t❤❡♦ tæ♣æ ②➳✉✱ U ∩ E = ∅✳ ❙✉② r❛ x0 ∈/ Ew ✳ ❙✉② r❛ Ew ⊂ E ✭✷✮ ❚ø ✭1✮ ✈➔ ✭2✮ s✉② r❛ Ew = E ✳ ✷✳✷✳✾ ỵ sỷ X ổ ỗ ♣❤÷ì♥❣ ❦❤↔ ♠❡tr✐❝✳ ◆➳✉ ❞➣② {xn} tr♦♥❣ X ❤ë✐ tư ②➳✉ tỵ✐ ♣❤➛♥ tû x ♥➔♦ ✤â t❤✉ë❝ X ✱ t❤➻ tr♦♥❣ X ❝â ❞➣② {yi} s❛♦ ❝❤♦ ✸✷ a ộ yn tờ ủ ỗ ỳ xn✳ i→∞ b yi → x t❤❡♦ tæ♣æ ①✉➜t ♣❤→t✳ ự a H ỗ ❝õ❛ {xn : n = 1, } ∞ ∞ H= α x , α ≥ 0, α = 1✱ ❝❤➾ ❝â ❤ú✉ ❤↕♥ αin = ✐♥ ♥ ✐♥ n=1 ✐♥ n=1 ●å✐ K ❧➔ ❜❛♦ ✤â♥❣ ②➳✉ ❝õ❛ H ✳ ❑❤✐ ✤â✱ ✈➻ {xn} ⊂ H ✈➔ xn →ω x ♥➯♥ x ∈ K ✳ ❱➻ H ỗ t ỵ t õ H = K ❉♦ ✤â x ∈ H ✯ ✭b✮✳ ❱➻ X ợ tổổ t t tr tỗ t ❞➣② {yi} ⊂ H s❛♦ ❝❤♦ yi i→∞ → x ✭t❤❡♦ tæ♣æ ①✉➜t ♣❤→t✮✳ ✷✳✷✳✶✵ ▼➺♥❤ ✤➲✳ ●✐↔ ①û X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✱ X ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ ♥â✳ ❱ỵ✐ ♠é✐ x ∈ X ✱ t❛ ①→❝ ✤à♥❤ →♥❤ ①↕ fx : X ∗ → Φ ❝❤♦ ❜ð✐ fx(Λ) = Λx, ∀Λ ∈ X ∗✳ ❑❤✐ ✤â✱ {fx : x ∈ X} ❧➔ ❤å ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t→❝❤ ✤÷đ❝ ❝→❝ ✤✐➸♠ ❝õ❛ X ∗✳ ❈❤ù♥❣ ♠✐♥❤✳ fx t✉②➳♥ t➼♥❤✱ ∀x ∈ X ✳ ❱ỵ✐ Λ, Λ ❜➜t ❦ý t❤✉ë❝ X ∗; ∀α, β ∈ Φ t❛ ❝â fx (αΛ + βΛ ) = (αΛ + βΛ )(x) = αΛx + βΛ x = αfx (Λ) + βfx (Λ ) ▲➜② Λ, Λ ❜➜t ❦ý t❤✉ë❝ X ∗ ✈➔ Λ = Λ ✱ ❦❤✐ ✤â✱ ∃x ∈ X : Λx = Λ x s✉② r❛ fx(Λ) = Λx = Λ x = fx(Λ ✮✳ ❱➟② ❤å {fx}x∈X t→❝❤ ✤÷đ❝ ❝→❝ ✤✐➸♠ ❝õ❛ X ∗✳ ✷✳✷✳✶✶ ✣à♥❤ ♥❣❤➽❛✳ ❚æ♣æ ✤➛✉ tr➯♥ X ∗ s✐♥❤ ❜ð✐ ❤å {fx : x ∈ X ⑥ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ ②➳✉∗ tr➯♥ X ∗✳ ✷✳✷✳✶✷ ▼ët sè ♥❤➟♥ ①➨t✳ ✶✳ ỵ t tổổ ởt tổổ ỗ ữỡ tr ổ X ▼é✐ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ②➳✉∗ tr➯♥ X ∗ ❝â ❞↕♥❣ Λ → Λx ✈ỵ✐ ♠é✐ x ♥➔♦ õ tở X ự ỵ t❛ s✉② r❛ ✤è✐ ♥❣➝✉ ❝õ❛ X ∗ t❤❡♦ tæ♣æ ②➳✉ ❧➔ X ✱ ♥❣❤➽❛ ❧➔ ♠é✐ ♣❤✐➳♥ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ②➳✉∗ ✤➲✉ ❝â ❞↕♥❣ fx ✳ ✸✹ t ỗ t ỵ ổrố ❚r❛♥❣ ✾✵✮ a b ◆➳✉ t➼❝❤ ✣➲❝→❝ Dx ❝õ❛ ❤å ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ❦❤→❝ ré♥❣ Dx ❧➔ x∈X ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t t❤➻ Dx ❧➔ ❝♦♠♣❛❝t ✈ỵ✐ ♠å✐ x ∈ X ✳ ❚➼❝❤ ❝õ❛ ♠ët ❤å ❦❤æ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t ❧➔ ởt ổ t ỵ ◆➳✉ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✵ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✱ t❤➻ t➟♣ ❤đ♣ K = {Λ ∈ X ∗ : |Λx| ≤ 1, ∀x ∈ V } ❧➔ ❝♦♠♣❛❝t ②➳✉∗✳ ✭K ✤÷đ❝ ❣å✐ ❧➔ ♣ỉ❧❛ ❝õ❛ V ỵ V ự sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✱ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ ✵ ∈ X ✳ ❇ð✐ ✈➻ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ ✵ ❧➔ t➟♣ ❤ót✱ ♥➯♥ ♠é✐ x X tỗ t (x) tở x γ(x) < ∞ s❛♦ ❝❤♦ x ∈ γ(x)V ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ X, Λ ∈ K ✳ ❚❛ ❝â |Λx| ≤ γ(x), (Λx ∈ γ(x).Λ(V ), |Λ(y)| ≤ 1, ∀y ∈ V ) ✣➦t Dx = {α : |α| ≤ γ(x)}, x ∈ X ✳ Dx = {f : X → P = x∈X Dx , f (x) ∈ Dx } x∈X = {f : X → Dx ; |f(x)| ≤ γ(x)} x∈X ❱➻ Dx ❝♦♠♣❛❝t tr♦♥❣ , x X t ỵ ổrố t ❝â P ❝♦♠♣❛❝t ✈ỵ✐ tỉ♣ỉ t➼❝❤ τ ✱ ✈➔ t❛ ❝â K ⊂ X∗ ∩ P ✳ ✸✺ ❙✉② r❛ tr➯♥ K ❝â ❤❛✐ tỉ♣ỉ ✤÷đ❝ ❝↔♠ s✐♥❤ tø X ∗ ✭tæ♣æ ②➳✉∗✮ ✈➔ tø P ✭tæ ♣æ τ ✮✳ ❉♦ ✈➟②✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ K ✲ ❝♦♠♣❛❝t ②➳✉∗ t❛ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ✤✐➲✉ s❛✉ a ❚r➯♥ K ✱ tæ♣æ ②➳✉∗ ✈➔ tỉ♣ỉ t➼❝❤ trị♥❣ ♥❤❛✉✳ ✤â♥❣ tr♦♥❣ ✭P, τ ✮✳ ✭●✐↔✐ t❤➼❝❤✳ ❱➻ K ✤â♥❣ tr♦♥❣ ✭P, τ ✮ ✲ ❝♦♠♣❛❝t ♥➯♥ K ❧➔ t➟♣ ❝♦♠♣❛❝t t❤❡♦ τ ✳ ❚r➯♥ K, τ trị♥❣ ✈ỵ✐ tỉ♣ỉ ②➳✉∗✳ ❉♦ ✤â✱ K t➟♣ ❝♦♠♣❛❝t ②➳✉∗✮✳ ∗ ❈❤ù♥❣ ♠✐♥❤ ✭a✮✳ ●✐↔ sû Λ0 ❜➜t ❦ý tr♦♥❣ K ✳ ❱ỵ✐ x1 , x2 , , xn ∈ X ✈➔ δ > 0✳ ❚❛ ✤➦t W1 = {Λ ∈ X ∗ : |Λxi − Λ0 xi | < δ, i = 1, n} (1✮✳ b K W2 = {f ∈ P : |f (xi ) − Λ0 xi | < δ, i = 1, n} (2✮✳ ❉♦ ✤è✐ ♥❣➝✉ ❝õ❛ X ∗ ❧➔ X ((X ) = X) t õ t ỗ t fx ✈ỵ✐ x, ∀x ∈ X ✱ tù❝ ❧➔ fx = x✳ ❑❤✐ ✤â✱ t❛ ❝â • n {Λ ∈ X ∗ : |Λxi − Λ0 xi | < δ} W1 = i=1 n = {Λ ∈ X ∗ : |xi (Λ) − xi (Λ0 )| < δ} i=1 ✭❉♦ fx(Λ) = Λx ⇔ x(Λ) = Λ(x)✮✳ ❙✉② r❛ t❤❡♦ ✣à♥❤ ♥❣❤➽❛ tæ♣æ ✤➛✉ t❤➻ W1 ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ Λ0 ✈➔ ❤å ❝→❝ t➟♣ ❝â ❞↕♥❣ ✭1✮ t↕♦ t❤➔♥❤ ♠ët ❝ì sð ❧➙♥ ❝➟♥ ②➳✉∗ t↕✐ Λ0 tr♦♥❣ ❳∗✳ • ❚❛ ❧↕✐ ❝â n W2 = i=1 p−1 x0 B(Λ0 xi, δ) ♥➯♥ ❤å ❝→❝ t➟♣ ❝â ❞↕♥❣ ✭✷✮ t↕♦ t❤➔♥❤ ♠ët ❝ì sð ❧➙♥ ❝➟♥ ❝õ❛ τ t↕✐ Λ0 tr♦♥❣ P ✳ • ❇ð✐ ✈➻ W1 ∩ K = W2 ∩ K ✳ ✣✐➲✉ ✤â ❝❤ù♥❣ tä ✭a✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ∗ ❈❤ù♥❣ ♠✐♥❤ ✭b✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ K ✤â♥❣ tr♦♥❣ ✭P, τ ✮ t❛ ❝❤ù♥❣ ♠✐♥❤ τ τ K = K ✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f0 ∈ K ✳ ❑❤✐ ✤â✱ ∀ε > 0; x, y ∈ X; α, β ∈ Φ✱ t❛ ❝â t➟♣ ❤ñ♣ ✸✻ U = {f ∈ P : |f(zi ) − f0 (zi )| < ε, i = 1, 2, 3} tr♦♥❣ ✤â z1 = x, z2 = y, z3 = αx + βy ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ f0 tr♦♥❣ ✭P, τ ✮✳ ❉♦ f0 ∈ K τ ♥➯♥ U ∩ K = ∅✳ ❙✉② r❛ f˜ ∈ U ∩ K ❇ð✐ ✈➻ f ❧✐➯♥ tö❝ ♥➯♥ t❛ ❝â f0 (αx + βy) − αf0 (x) − βf0 (y) = (f0 − f˜)(αx + βy) + α(f˜ − f0 )(x) + β(f˜ − f0 )(y) s✉② r❛ |f0 (αx + βy) − αf0 (x) − βf0 (y)| < (1 + || + ||) > tũ ỵ s✉② r❛ f0 (αx + βy) = αf0 (x) + βf0 (y)✳ ❉♦ ✤â f0 t✉②➳♥ t➼♥❤✳ ❚❛ ❧↕✐ ❝â f˜(x) − f (x0 ) < ε s✉② r❛ |f0 (x)| < ε + f˜(x) ≤ ε + 1, ∀x ∈ V s✉② r❛ |f0 (x)| ≤ 1, ∀x ∈ V ❱➟② f0 t✉②➳♥ t➼♥❤ ✈➔ |f0(x)| ≤ 1, ❙✉② r❛ ✭b✮ t❤ä❛ ♠➣♥✳ ✭❞♦ ε > ❜➨ tò② þ✮✳ ∀x ∈ V s✉② r❛ f0 ∈ K ⇒ K = K ỵ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ❦❤↔ ❧②✱ ♥➳✉ K ⊂ X ∗ ✈➔ K ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉∗ t❤➻ K ❦❤↔ ♠❡tr✐❝ t❤❡♦ tæ♣æ ②➳✉∗✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ X ❦❤↔ tỗ t t ữủ trũ t {xn : n ∈ N ∗ }✳ ✣➦t ✸✼ fn ✿ X ∗ → Φ Λ → fn (Λ) = Λxn ❚ø ✤à♥❤ ♥❣❤➽❛ tæ♣æ ②➳✉∗ t❛ ❝â fn ❧✐➯♥ tö❝✱ ∀n = 1, 2, ▼➦t ❦❤→❝ {fn} t→❝❤ ❝→❝ ✤✐➸♠ ❝õ❛ X ∗✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ Λ, Λ ∈ X ∗ s❛♦ ❝❤♦ Λ = Λ ✳ ●✐↔ t❤✐➳t ♣❤↔♥ ❝❤ù♥❣ fn(Λ) = fn(Λ ), ∀n = 1, 2, ✱ tù❝ ❧➔ Λ(xn ) = Λ (xn ), ✈ỵ✐ ♠å✐ n = 1, 2, ❇ð✐ = tỗ t x X s (x) = (xn) r tỗ t U, V ❧➛♥ ❧÷đt ❧➔ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ Λx, Λ x✱ s❛♦ ❝❤♦ U ∩ V = ∅✳ ❑❤✐ ✤â✱ x ∈ Λ−1 (U ) ∩ Λ −1 (V ) = W ✳ ❙✉② r❛ W ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x✱ ✈➻ {xn, n ∈ N ∗} = X ♥➯♥ W ❝❤ù❛ ♠ët xn ♥➔♦ ✤â✳ ❙✉② r❛ Λxn ∈ U, Λ xn ∈ V, U ∩ V = ∅✳ ✣✐➲✉ ✤â ❞➝♥ ✤➳♥ ♠➙✉ t❤✉➝♥ ♥➯♥ Λxn = Λ xn✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✷✳✸ ✭K = X ✮ s✉② r❛ K ❦❤↔ ♠❡tr✐❝ t❤❡♦ tæ♣æ ②➳✉∗✳ ✷✳✸✳✹ ❍➺ q✉↔✳ ◆➳✉ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✵ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ❦❤↔ ❧② X ✈➔ ♥➳✉ {Λn} ❧➔ ❞➣② tr♦♥❣ X ∗ s❛♦ ❝❤♦ |Λnx| ≤ 1, x V t tỗ t {n } ⊂ {Λn} ✈➔ Λ ∈ X ∗ s❛♦ ❝❤♦ i Λx = lim Λni x, ∀x ∈ X ✳ ni →∞ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ K = {Λ ∈ X∗ : |x| 1, x V} ỵ t❛ ❝â K ❧➔ t➟♣ ❝♦♠♣❛❝t t❤❡♦ ②➳✉∗✳ ❚❤❡♦ ✣à♥❤ ỵ t K tr ứ tt t s✉② r❛ {Λn} ⊂ K ❧➔ t➟♣ ♠❡tr✐❝ ❝♦♠♣❛❝t ♥➯♥ tỗ t {n } {n} s i ni → Λ ∈ K, (K = Λ−1 [B(0, 1)] ❚ø ✣à♥❤ ♥❣❤➽❛ tæ♣æ ②➳✉ t❛ ❝â Λn i →Λ s✉② r❛ K ✤â♥❣✮✳ t❤❡♦ tæ♣æ ②➳✉∗ ⇔ Λni x → Λx, ∀x ∈ X ✳ ✸✽ ❱➟② Λx = lim ni x, x X i ỵ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ s❛♦ ❝❤♦ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤ñ♣ X ∗ ❧➔ t→❝❤ ❝→❝ ✤✐➸♠ ❝õ❛ X sỷ A, B t ỗ ❝♦♠♣❛❝t ❦❤→❝ ré♥❣ rí✐ ♥❤❛✉ ❝õ❛ X ✳ ❑❤✐ ✤â✱ tỗ t ởt X s ❝❤♦ sup Re Λx < inf Re Λy ✭✶✮ y∈B x∈A ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû Xw ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈ỵ✐ tỉ♣ỉ ②➳✉✳ ❑❤✐ ✤â✱ A, B ❧➔ ❝→❝ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Xw (τw ⊂ τ ; A, B ❝♦♠♣❛❝t t❤❡♦ τ ✮✳ ❱➻ Xw ❧➔ T −❦❤æ♥❣ ❣✐❛♥ ✭tæ♣æ ②➳✉ s✐♥❤ ❜ð✐ ❤å t→❝❤ ❝→❝ ✤✐➸♠ t❤➻ ♥â ❧➔ T −❦❤æ♥❣ ❣✐❛♥ ✮ ♥➯♥ A, B ❝ô♥❣ ❧➔ ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ tr♦♥❣ Xw ✳ ❱➻ Xw ổ ỗ ữỡ t ữỡ t ỵ b ữỡ t X Xw t s r tỗ t ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ Λ ∈ X ∗ t❤ä❛ ♠➣♥ ✭✶✮✳ ❇ð✐ ✈➻ (X )∗ = X ∗ ✭◆❤➟♥ ①➨t ✷✳✷✳✼ ✭✶✮ ❈❤÷ì♥❣ ■■✮✳ ❚❛ s✉② r❛ Λ ∈ X ∗ s❛♦ ❝❤♦ ✇ ✇ sup Re Λx < inf Re λy y∈B x∈A ✷✳✸✳✻ ❇ê ✤➲✳ ●✐↔ sû W ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ ✵ tr♦♥❣ X ✳ ❑❤✐ õ tỗ t ởt ố ự U ✵ s❛♦ ❝❤♦ U + U ⊂ W ❈❤ù♥❣ ♠✐♥❤✳ ◆❤í t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ ♣❤➨♣ ❝ë♥❣ t↕✐ ✤✐➸♠ ✭✵✱ ✵✮ ♥➯♥ ✈ỵ✐ ❧➙♥ ❝➟♥ W ❝õ❛ ✤✐➸♠ ✵ tr♦♥❣ X tỗ t V1, V2 ✵ ð tr♦♥❣ X s❛♦ ❝❤♦ V1 + V2 ⊂ W ✳ ✣➦t U = V1 ∩ V2 ∩ (−V1 ) ∩ (−V2 ) ❑❤✐ ✤â✱ U ❧➔ ❧➙♥ ❝➟♥ t ỵ X ổ ỗ ữỡ H ỗ t ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ E ⊂ X ✱ t❤➻ H ❧➔ t➟♣ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥✳ ✸✾ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ∈ X ✳ õ tỗ t ởt ỗ ∈ X s❛♦ ❝❤♦ V + V ⊂ U ✳ X ỗ ữỡ E t tỗ t t ỳ ❤↕♥ F ⊂ X s❛♦ ❝❤♦ E ⊂F +V✳ ●✐↔ sû F = {x1, , xn}✳ ●å✐ S ❧➔ ✤ì♥ ❤➻♥❤ ❝❤ù❛ Rm m S = (t1 , , tm ), ti ≥ 0, ∀i = 1, m, ti = i=1 m ❦❤✐ ✤â✱ →♥❤ ①↕✿ (t1, , tm) → tixi ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ tø t➟♣ ❝♦♠♣❛❝t i=1 S ❧➯♥ ❝♦♥vF ❙✉② r❛ ❝♦♥vF ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ H = convE ⊂ convFn + V ✳ n ❱ỵ✐ ♠é✐ x ∈ H ⇒ ∃x1 xn ∈ E : x = αixi, αi ≥ 0, αi = i=1 i=1 ❱➻ xi ∈ E ♥➯♥ xi ∈ F + V s✉② r❛ ∃yi ∈ F : xi ∈ yi + V ⇒ xi − yi V V ỗ s r n i (xi − yi ) ∈ convV = V i=1 ▼➦t ❦❤→❝ n n αi (xi − yi ) + x= i=1 αi yi ∈ convF + V i=1 ❱➻ convF ❧➔ t➟♣ ❝♦♠♣❛❝t✱ s✉② r❛ convF ❧➔ t➟♣ ❤♦➔♥ t♦➔♥ s r tỗ t F ỳ convF ⊂ F + V s✉② r❛ H ⊂ F + V + V ⊂ F + U ❙✉② r❛ ✤✐➲✉ ự ỵ sỷ H ❝♦♥v(K), K ❝♦♠♣❛❝t ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✳ ❑❤✐ ✤â✱ ✹✵ a ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r➨❝❤❡t✱ t❤➻ H ❧➔ t➟♣ ❝♦♠♣❛❝t✳ b ◆➳✉ X = Rn t❤➻ H ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❈❤ù♥❣ ♠✐♥❤✳ a) + ❇❛♦ ✤â♥❣ ❝õ❛ ♠ët t➟♣ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ ❧➔ t➟♣ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝✳ + tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t❤➻ t➟♣ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ ⇔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✳ ❱➟② t❛ ❝â✱ K ❝♦♠♣❛❝t tr♦♥❣ X− ❋r➨❝❤❡t s r K t X ỗ ữỡ t ỵ v(K) t ❜à ❝❤➦♥ s✉② r❛ H = conv(K) ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✳ ❙✉② r❛ H ❝♦♠♣❛❝t✳ n+1 ∗b✮✳ ●✐↔ sû S = {(t1 , , ts+1 ) : t1 ≥ 0, ti = 1} ⊂ Rn+1 , K ❧➔ t➟♣ i=1 n ❝♦♠♣❛❝t ❝❤ù❛ R ❙✉② r❛ n+1 ti xi , t ∈ S, xi ∈ K, i ∈ 1, n + x ∈ conv(K) ⇔ x = i=1 ❳➨t →♥❤ ①↕✿ f : S × K n+1 → conv(K) (t; x1 , x2 , , xn+1 ) → t1 x1 + t2 x2 + + tn+1 xn+1 ❙✉② r❛ ❝♦♥v(K) ❝♦♠♣❛❝t ❤❛② H ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ✹✶ ❑➌❚ ▲❯❾◆ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ♥➔②✱ t→❝ ❣✐↔ ✤➣ t❤✉ ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ ♥❤÷ s❛✉✿ ✭✶✮ ◆❤➢❝ ❧↕✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ●✐↔✐ t➼❝❤ ❤➔♠✱ ✤÷❛ r❛ ♠ët tr♦♥❣ ♥❤ú♥❣ ỵ q trồ t ỵ ỵ t ự tt ởt số ỵ ỡ ổổ ỗ t ❤✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✸❪ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ❧➔♠ ❝ì sð ❝❤♦ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉✳ ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶ ❪✳ ❲✳ ❘✉❞✐♥✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧✐s✐s✱ ✶✾✾✶✳ ❬✷ ❪✳ ❆✳P✳ ❘♦❜❡rts♦♥✱ ❲✳❏✳ ❘♦❜❡rts♦♥✱ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ✈➔ ❚r✉♥❣ ❝➜♣ ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ✶✾✼✼✳ ❬✸ ❪✳ ❍✳❍✳ ❙❝❤❛❝❢❢❡r✱❚♦♣♦❣✐❝❛❧ ✈❡❝tì s♣❛❝❡s✱ ✶✾✻✻✳ ❬✹ ❪✳ ◆❣✉②➵♥ ❳✉➙♥ ▲✐➯♠✱ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣ ✲ ✣ë ✤♦ ✈➔ t➼❝❤ ♣❤➙♥✱ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝✱✶✾✾✹✳ ❬✺ ❪✳ P●❙✳P❚❙✳ ✣é ❱➠♥ ▲÷✉✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❤➔ ①✉➜t ❜↔♥ ❑❤♦❛ ❤å❝ ✈➔ ❑ÿ t❤✉➟t ❍➔ ◆ë✐✱ ✶✾✾✾✳ ✹✸