▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶ ▲í✐ ♥â✐ ✤➛✉ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷ ❈❤÷ì♥❣ ■✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ổổ ữỡ Đ ỵ ✲ ❇❛♥❛❝❤ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✺ § ✷✳ ⑩♥❤ ①↕ ❧✐➯♥ tư❝ Đ ỵ ữỡ số Đ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✸ § ✷✳ P❤ê ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✣↕✐ sè ❇❛♥❛❝❤ ✳✳✳✳✳✳✳✳✳ ✷✽ ❈❤÷ì♥❣ ■■■✿ ✣↕✐ sè ❇❛♥❛❝❤ ●✐❛♦ ❤♦→♥✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✸ § ✶✳ P❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✸ § ✷✳ P❤➨♣ ✤è✐ ❤đ♣ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✼ § ✸✳ ❇✐➯♥ ❙❤✐❧♦✈ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✵ ❑➳t ❧✉➟♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✻ ✶ ▲❮■ ◆➶■ ✣❺❯ ❚r♦♥❣ ❧✉➟♥ →♥ ✤÷đ❝ ✈✐➳t ✈➔♦ ♥➠♠ ✶✾✷✵ ❝õ❛ ❙t❡❢❛♥ ❇❛♥❛❝❤✱ æ♥❣ ✤➣ ❤➻♥❤ t❤ù❝ ❤â❛ ❦❤→✐ ♥✐➺♠ ❜➙② ❣✐í ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤÷ ❧➔ ổ ự ỵ ỡ sð ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ❚r♦♥❣ ✤â✱ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◗✉❛ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ t→❝ ❣✐↔ ✤➣ ✤÷đ❝ ❤å❝ q✉❛ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣✱ t→❝ ❣✐↔ ❜✐➳t r➡♥❣ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ❧➔ ♠ët ✈➜♥ ✤➲ ✤❛♥❣ ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❚æ♣æ ✣↕✐ sè q✉❛♥ t➙♠✳ ◆❤➟♥ t❤➜② t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✈➔ sỹ ữợ t ữỡ ố t→❝ ❣✐↔ q✉②➳t ✤à♥❤ ❝❤å♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✏✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✑ ✳ ✷✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐✿ ❚➻♠ ❤✐➸✉ ❦❤❛✐ ✈➲ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✳ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ tr♦♥❣ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✳ ✹✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐✿ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ●✐↔✐ t➼❝❤ ❤➔♠✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ tr➯♥✱ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ t➟♣ ❤ñ♣ ♠ð✱ t➟♣ ❤ñ♣ ✤â♥❣✱ →♥❤ ①↕ tử ự ởt số ỵ q ữ ỵ ỵ ữỡ số ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ♥❤÷✿ ✣↕✐ sè ♣❤ù❝✱ ✣↕✐ số số ỗ ự õ ❧➔ ♥➯✉ ♠ët sè ❦❤→✐ ♥✐➺♠ ♥❤÷✿ P❤ê✱ ❣✐↔✐✱ ✳✳✳❈✉è✐ ❝ị♥❣ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ỵ q ữỡ số r ữỡ trữợ t ú tổ ữ r ❦❤→✐ ♥✐➺♠ ✈➲ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞✱ ♣❤➨♣ ✤è✐ ❤đ♣✱ ❜✐➯♥ ❙❤✐❧♦✈✳ ❈✉è✐ ❝ị♥❣ ❝❤ó♥❣ tổ tr ự ởt số ỵ ❧✐➯♥ q✉❛♥✳ ❉♦ ❦❤✉æ♥ ❦❤ê ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ ❜➔✐ ❜→♦ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ ❦❤â❛ ú tổ ữủ tr ữợ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè ❦➼ ❤✐➺✉ ✤÷đ❝ ✈✐➳t tr♦♥❣ ❦❤â❛ ❧✉➟♥ ●✐↔ sû A, B, E ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ✳ ❑❤✐ ✤â A ❧➔ E ❜❛♦ ✤â♥❣ ❝õ❛ A tr♦♥❣ X ✱ A ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ A tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ E ✱ ✐♥tA ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ A tr♦♥❣ X ✱ A\B ❧➔ ❤✐➺✉ ❝õ❛ A ✈➔ B ✳ ●✐↔ sû V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ x ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ V ✱ x ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ x✳ ◆❤➙♥ ❞à♣ ♥➔②✱ ❝❤♦ ♣❤➨♣ t→❝ ❣✐↔ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ t❤➛② ❣✐→♦ ❚❤✳❙ ▲÷ì♥❣ ◗✉è❝ ❚✉②➸♥✳ ❚❤➛② ✤➣ t➟♥ t➻♥❤ ữợ t tr sốt q tr ự ỗ tớ t t ỡ ❝❤õ ◆❤✐➺♠ ❑❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕②✱ tr✉②➲♥ t❤ö ❦✐➳♥ t❤ù❝ tr♦♥❣ s✉èt ✹ ♥➠♠ ❤å❝ q✉❛✳ ✸ ❈✉è✐ ❝ò♥❣ t→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ t➜t ❝↔ ❝→❝ ❜↕♥ ❜➧ tr♦♥❣ ❧ỵ♣ ✵✽❈❚❚✷ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ✈➻ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ❝↔ ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝✳ ❱➻ ✈➟② t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ♥❤ú♥❣ ❧í✐ ❝❤➾ qỵ t ổ ỳ õ ỵ t ✷✵✶✷ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ■ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ P ì Đ ị ❍❆❍◆ ✲ ❇❆◆❆❈❍ ✶✳✶✳✶ ❙ì ❝❤✉➞♥✱ ♥û❛ ❝❤✉➞♥✳●✐↔ sû p : E → R ❧➔ →♥❤ ①↕✱ ✈ỵ✐ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❛✳ p ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ ♥➳✉ p (αx) = αp (x) , ∀α ≥ 0, ∀x ∈ E p (x + y) ≤ p (x) + p (y) ❜✳ p ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝❤✉➞♥ ♥➳✉ p (x) ≥ 0, ∀x ∈ E p (αx) = | α | p (x) , ∀α ≥ 0, ∀x ∈ E p (x + y) ≤ p (x) + p (y) ỵ ổ t t➼♥❤ t❤ü❝ ✮✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝✱ p ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ X ✈➔ M ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ X ✳ ◆➳✉ f : M → R ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ tr➯♥ M ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x) ≤ p (x) , x M t tỗ t ❤➔♠ t✉②➳♥ t➼♥❤ λ : X → R ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X s❛♦ ❝❤♦ i✳ λ (x) = f (x) , ∀x ∈ M ii✳ λ (x) ≤ p (x) , ∀x ∈ M ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❣å✐ ♠ët s✉② rë♥❣ ❝õ❛ f ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ g ①→❝ ✤à♥❤ tr➯♥ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ Dg ⊃ M t❤ä❛ ♠➣♥ i✳ g (x) = f (x) , ∀x ∈ M ✺ ii✳ g (x) ≤ p (x) , ∀x ∈ M ●å✐ F ❧➔ t➟♣ t➜t ❝↔ ❝→❝ s✉② rë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ F = ∅ ❜ð✐ ✈➻ f ∈ F ✳ ❚❛ ①→❝ ✤à♥❤ ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr♦♥❣ F ♥❤÷ s❛✉✱ ♥➳✉ g1 , g2 ∈ F t❤➻ g1 ≤ g2 ⇔ Dg1 ⊂ Dg2 ✈➔ g1 (x) = g2 (x) , ∀x ∈ Dg1 ●✐↔ sû D ❧➔ ♠ët t➟♣ ❝♦♥ s➢♣ t❤➥♥❣ ❝õ❛ F ✳ ❑➼ ❤✐➺✉ D∗ = Dg ✳ g∈D ❑❤✐ ✤â✱ D∗ ⊃ M ✈➔ D∗ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ ❳✳ ❱ỵ✐ ♠å✐ x ∈ D∗ tỗ t g D s x Dg ✳ ❚❛ ①→❝ ✤à♥❤ ♣❤✐➳♠ ❤➔♠ g ∗ tr➯♥ D∗ ❜➡♥❣ ❝→❝❤ ✤➦t g ∗ (x) = g (x)✳ ❚❛ ✤÷đ❝ g ∗ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ tr➯♥ D∗ ✈➔ g ∗ ≥ g, ∀g ∈ D ◆❤÷ ✈➟②✱ ♠å✐ t➟♣ ❝♦♥ s➢♣ t❤➥♥❣ ❤➔♥❣ D ❝õ❛ F ✤➲✉ ❝â ♠ët ❧➙♥ ❝➟♥ tr➯♥ tr♦♥❣ F r tỗ t ởt tỷ ❝ü❝ ✤↕✐ λ ❝õ❛ F ✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ✤â ❧➔ ♣❤✐➳♠ ❤➔♠ ❝➛♥ t➻♠✳ ▼✉è♥ t❤➳✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♠✐➲♥ ①→❝ ✤à♥❤ D ❝õ❛ λ ❧➔ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X ✳ P❤↔♥ ❝❤ù♥❣✳ ●✐↔ sỷ D = X õ tỗ t x0 ∈ X s❛♦ ❝❤♦ x0 ∈/ D s✉② r❛ x0 = 0✳ ❑➼ ❤✐➺✉ [ x0 ] ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ♠ët ❝❤✐➲✉ ❝õ❛ X ❣➙② ♥➯♥ ❜ð✐ x0 ✳ ❚❛ ❝â D ∩ [x0 ] = {0} ❑➼ ❤✐➺✉ Z = D + [x0 ]✳ ▼é✐ ✈❡❝tì z ∈ Z ✤➲✉ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ❞✉② ♥❤➜t ữợ z = x + x0 , x D ▲➜② z, z ∈ D✱ t❛ ❝â λ(x) − λ(x ) = λ(x − x ) ≤ p(x − x ) = p(x + x0 − x − x0 ) ≤ p(x + x0 + p(−x − x0 )) ❙✉② r❛ ✻ sup [−p (−x − x0 ) − λ (x)] ≤ inf [p (x + x0 ) − (x)] xD xD õ tỗ t số c ∈ p(x + x0 ) − λ(x), ∀x ∈ D s❛♦ ❝❤♦ −p (−x − x0 ) − λ (x) ≤ c, ∀x ∈ D ❱ỵ✐ z ∈ Z ✈➔ z = x + αx0 , x ∈ D t❛ ✤➦t G(z) = λ(x) + αc✳ ❑❤✐ ✤â G ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr➯♥ Z ✈➔ ♥➳✉ x ∈ D t❤➻ G(x) = λ(x) ≤ p(x) ◆➳✉ z ∈ Z \ D✱ t❤➻ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ z = x + αx0 ✱ t❛ ❝â α = 0✳ ✯ ❳➨t tr÷í♥❣ ❤đ♣ α > 0✳ ❱➻ c ∈ p(x + x0 ) − λ(x) ✈ỵ✐ ♠å✐ x ∈ D✱ ♥➯♥ c≤p x x + x0 − λ α α ❙✉② r❛ αc ≤ αp x x + x0 − αλ = p (x + αx0 ) − λ (x) α α ❉♦ ✤â λ (x) + αc ≤ p (x + αx0 ) ❙✉② r❛ G(z) ≤ p(z)✳ ✯ ❳➨t tr÷í♥❣ ❤đ♣ α < 0✳ ❱➻ −p (−x − x0 ) − λ (x) ≤ c, ∀x ∈ D ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ D ♥➯♥ −p − x x − x0 − λ ≤c α α ❙✉② r❛ − (−α) p − x x − x0 + αλ ≤ −αc α α ❉♦ ✤â −p (x + αx0 ) + λ (x) ≤ −αc ✼ ❙✉② r❛ G(z) ≤ p(z)✳ ❇ð✐ ✈➟② G ∈ F ✈➔ G ≥ λ, G = λ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ λ ❧➔ ♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ F ✳ ❈❤ù♥❣ tọ D = X ỵ ❇❛♥❛❝❤ ✭❝❤♦ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝ ✮✳ ●✐↔ ①û X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ ♣❤ù❝✱ p ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ tr➯♥ X ✈➔ M ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ X ✳ ◆➳✉ f : X → C ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ tr➯♥ M ✈➔ t❤♦➣ ♠➣♥ ✤✐➲✉ ❦✐➺♥ |f (x)| ≤ p(x), ∀x ∈ M t tỗ t t t : X → C ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X s❛♦ ❝❤♦ i λx = f (x), ∀x ∈ M ii |λx| ≤ p(x), ∀x ∈ X ❈❤ù♥❣ ♠✐♥❤✳ f (x) ữợ f (x) = u1 (x) + iu2 (x) ✈➔ ♥❤➟♥ ✤÷đ❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ f1 (x) ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t t tỹ M ỗ tớ f1 (x) |f1 (x)| ≤ |f (x)| ≤ p(x), ∀x ∈ M ỵ tỗ t t t t❤ü❝ λ1 (x) ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ X s❛♦ ❝❤♦ ❛ λ1 (x) = f1 (x), ∀x ∈ M ❜ |λ1 (x)| ≤ p(x), ∀x ∈ X ❉♦ p ❧➔ ♠ët ♥ú❛ ❝❤✉➞♥✱ tø |λ1 (x)| ≤ p(x), ∀x ∈ X t❛ s✉② r❛ −λ1 (x) = λ1 (−x) ≤ p (−x) = p (x) ❙✉② r❛ ✽ |λ1 (x)| ≤ p (x) , ∀x ∈ X ✣➦t λ (x) = λ1 (x) − iλ2 (ix) , ∀x ∈ X ❑❤✐ ✤â✱ λ ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ♣❤ù❝ ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝ X ✳ ▼➔ t❛ ❝â f (x) = u1 (x) − iu1 (ix) ♥➯♥ s✉② r❛ λ (x) = f (x) , ∀x ∈ M ▼➦t ❦❤→❝✱ ♥➳✉ λ(x) = 0✱ t❤➻ λ(x) = λ(x)eiθ ✳ ❱➻ ✈➟② |λ (x)| = e−iθ λ (x) = λ xe−iθ ❱➻ λ xe−iθ ❧➔ sè t❤ü❝ ✈➔ λ (x) = λ1 (x) − iλ2 (ix) , ∀x ∈ X ♥➯♥ t❛ ❝â λ xe−iθ = λ1 xe−iθ ❇ð✐ ✈➟② |λ (x)| = λ1 xe−iθ ≤ p xe−iθ = p (x)✳ ✶✳✶✳✹ ❈→❝ ❤➺ q✉↔✳ ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ F ⊂ E ✳ ❑❤✐ ✤â✱ ♥➳✉ f : F → K ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tử t tỗ t t t tử f : E → K t❤ä❛ ❛✳ ❍➺ q✉↔ ✶✳ f |F = f f = f ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✱F ⊂ E ✈➔ v ∈ E\F s❛♦ ❝❤♦ d (v, F ) = > õ tỗ t ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ f : E → K t❤ä❛ ❜✳ ❍➺ q✉↔ ✷✳ ✾ f |F = f (v) = δ ❈❤ù♥❣ ♠✐♥❤✳ ❛✳ ✣➦t p : E → R x → p (x) = f x ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ p ❧➔ ♥û❛ ❝❤✉➞♥✳ ❍ì♥ ♥ú❛ ✈➻ f ❧➔ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝ tr➯♥ F | f (x) | ≤ f x = p (x) , x F ỵ tỗ t t t tử f : E → K t❤ä❛ ♠➣♥ f |F = f f (x) ≤ p (x) , ∀x ∈ E ❚❛ ❝â f (x) ≤ f x = p (x) , ∀x ∈ E ✳ ❉♦ ✤â f t✉②➳♥ t➼♥❤ ✈➔ ❜à ❝❤➦♥✳❙✉② r❛ f ❧✐➯♥ tö❝ ✈➔ f ≤ f ✳ ❍ì♥ ♥ú❛✱ t❛ ❝â f = sup f (x) ≥ x∈E, x =1 ❉♦ ✈➟② f sup f (x) = x∈F, x =1 sup | f (x) | = f x∈E, x =1 = f ✳ ❜✳ ✣➦t G = v, F = {λv + z : z ∈ F, λ ∈ K}✳ ❑❤✐ ✤â G ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ E ✳ ✣➦t g : G → K ①→❝ ✤à♥❤ ❜ð✐ x = λv + z → g (x) = λ.δ ❙✉② r❛ g ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ✯ ❈❤ù♥❣ ♠✐♥❤ g ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ◆➳✉ λ = 0✱ t❤➻ ✶✵ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x ∈ A✱ x ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A✳ ❑❤✐ ✤â x ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ B ✳ ❉♦ ✈➟②✱ G(A) ⊂ G(B)✳ ❚ø ✤â s✉② r❛ G(A) ✈➔ A ∩ G(B) ❧➔ ❝→❝ t➟♣ ❝♦♥ ♠ð ð tr♦♥❣ A ♠➔ t❤♦↔ ♠➣♥ G(A) ⊂ A ∩ G(B)✳ ●✐↔ sû y ❧➔ ✤✐➸♠ ❜✐➯♥ ❝õ❛ G(A)✳ ❑❤✐ õ tỗ t {xn } G(A) s xn → y ❦❤✐ n → ∞✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✻ t❛ ❝â x−1 → ∞ ❦❤✐ n → ∞✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐✱ y ∈ G(B)✳ ❑❤✐ ✤â✱ ✈➻ x x1 n ỗ ổ tø G(B) ❧➯♥ G(B) ♥➯♥ s✉② r❛ xn −1 → y −1 ∈ G(B)✳ ❙✉② r❛ ❞➣② x−1 xn −1 ❧➔ ❞➣② ❜à ❝❤➦♥✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ → ∞✳ ❇ð✐ ✈➟②✱ y ∈ / G(B)✳ ❉♦ ✤â t❛ ❝â G(A)✱ A ∩ G(B) ❧➔ ❝→❝ t➟♣ ❝♦♥ ♠ð ❝õ❛ A✱ G(A) ⊂ A ∩G(B) ✈➔ A ∩ G(B) ❦❤ỉ♥❣ ❝❤ù❛ ✤✐➸♠ ❜✐➯♥ ❝õ❛ G(A)✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✷✳✺ t❛ s✉② r❛ G(A) ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ♥➔♦ ✤â ❝õ❛ A ∩ G(B)✳ ✸✷ ì Đ PP ❇■➌◆ ✣✃■ ●❊▲❋❆◆❉ ✸✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✶✳ ✣↕✐ sè ❇❛♥❛❝❤ A ✤÷đ❝ ❣å✐ ❧➔ ❣✐❛♦ ❤♦→♥ ♥➳✉ f g = gf, ∀f, g ∈ A✳ ✷✳ ✣↕✐ sè ❇❛♥❛❝❤ A ữủ õ ỡ tỗ t tû ∈ A s❛♦ ❝❤♦ ef = f e, ∀f ∈ A ✭ð ✤➙② t❛ ❤✐➸✉ e ∈ A s❛♦ ❝❤♦ e = e✮ ✸✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû ủ tt ỗ ự ✤↕✐ sè ❣✐❛♦ ❤♦→♥ A✳ ❱ỵ✐ ♠é✐ x ∈ A t❛ ①→❝ ✤à♥❤ ❤➔♠ x : ∆ → C✱ ❝❤♦ ❜ð✐ x (h) = h (x) , ∀h ∈ ∆✳ ✲ x : ∆ → C ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ ❝õ❛ ♣❤➛♥ tû x ∈ A✳ ❑➼ ❤✐➺✉ A = {x : x ∈ A}✳ ✲ ❚æ♣æ ②➳✉ ♥❤➜t tr➯♥ ∆ ❧➔♠ ❝❤♦ t➜t ❝↔ ❝→❝ x ∈ A ✲ ❧✐➯♥ tư❝ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ ●❡❧❢❛♥❞✳ ủ ợ tổổ ữủ ❝→❝ ✤↕✐ ❝õ❛ ✤↕✐ sè A✳ ❦❤æ♥❣ ❣✐❛♥ ■✤➯❛♥ ❝ü❝ ✲ ⑩♥❤ ①↕ tø A ✈➔♦ A ❝❤♦ ❜ð✐ x → x, x ∈ A ✤÷đ❝ ❣å✐ ❧➔ ✤ê✐ ●❡❧❢❛♥❞ ❝õ❛ ✤↕✐ sè A✳ ♣❤➨♣ ❜✐➳♥ ✲ ●✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ■✤➯❛♥ ❝ü❝ ✤↕✐ ❝õ❛ ✤↕✐ sè A ✤÷đ❝ ❣å✐ ❧➔ ✭❤❛② ❝➠♥✮ ❝õ❛ ✤↕✐ sè A ✈➔ ❦➼ ❤✐➺✉ ❧➔ radA✳ ◆➳✉ rad A = {0}✱ t❤➻ A ữủ ỷ ỡ ữ ỵ ợ tæ♣æ ●❡❧❢❛♥❞ tr➯♥ ∆ t❛ ❝â A ⊂ C (∆) ✈ỵ✐ C (∆) = {f ❧✐➯♥ tư❝ tø ∆ ✈➔♦ C } ỵ sỷ ổ ❣✐❛♥ ❝→❝ ■✤➯❛♥ ❝ü❝ ✤↕✐ ❝õ❛ ✤↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ A✳ ❑❤✐ ✤â✱ ❛✳ ❑❤æ♥❣ ❣✐❛♥ ∆ ❧➔ ❍❛✉s❞♦r❢❢ t P ỗ ✤↕✐ sè A ❧➯♥ ✤↕✐ sè A ❝õ❛ C (∆)✳ ỡ ỳ t ỗ trũ ợ radA✳ ❉♦ ✤â✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ ❧➔ ✤➥♥❣ ❝➜✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤↕✐ sè A ❧➔ ♥û❛ ✤ì♥✳ ❝✳ ❱ỵ✐ ♠é✐ ♣❤➛♥ tû x ∈ A✱ t❛ ❝â x (A) = σ (x)✳ ❱➻ t❤➳✱ x tr♦♥❣ ✤â x ∞ ∞ = sup |x (h)| h∈∆ = f (x) ≤ x ✳ ✈➔ x ∈ rad A ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❛✳ ❑➼ ❤✐➺✉ A∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ A ✈➔ tr➯♥ A∗ t❛ ❧➜② ❝❤✉➞♥ ❝❤♦ ❜ð✐ x = sup |x (u)|✳ u ≤1 ❑➼ ❤✐➺✉✱ K = {x ∈ A∗ : x ≤ 1} ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ tr♦♥❣ A∗ ✳ ❚❤❡♦ ✣à♥❤ ỵ t K t t ỵ t õ K ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ∆ ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ②➳✉✯ tr♦♥❣ A∗ ✈➔ tr➯♥ ∆ ❚ỉ♣ỉ ●❡❧❢❛♥❞ trị♥❣ ✈ỵ✐ ❚æ♣æ ②➳✉✯✳ ✯ ●✐↔ sû Λ0 ❧➔ ✤✐➸♠ t❤✉ë❝ ❜❛♦ ✤â♥❣ ②➳✉✯ ❝õ❛ ∆✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ Λ0 ∈ ∆ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ Λ0 (xy) = Λ0 x Λ0 y, ∀x, y ∈ A Λ0 (e) = ✸✹ ❚❤➟t ✈➟②✱ ❣✐↔ sû ∀x, y ∈ A, > tý ỵ õ t t W = {Λ ∈ A∗ : |Λzi − Λ0 zi | < ε, i = 1, 2, 3, 4} ❚r♦♥❣ ✤â z1 = e, z2 = x, z3 = y, z4 = xy ✳ ❑❤✐ ✤â W ❧➔ ♠ët ❧➙♥ ❝➟♥ ②➳✉✯ ❝õ❛ Λ0 ✳ ❱➻ Λ0 t❤✉ë❝ ❜❛♦ ✤â♥❣ ②➳✉ ✯ ❝õ❛ ∆ ♥➯♥ W ∩ A = ∅✱ tỗ t h s h ∈ W✳ ▲ó❝ ✤â t❛ ❝â |h (e) − Λ0 (e)| < ε ⇔ |1 − Λ0 (e)| < ε✳ < tũ ỵ (e) = 1✳ ▲↕✐ ✈➻ h ∈ ∆ ♥➯♥ t❛ ❝â h(xy) = h(x).h(y)✳ ❉♦ ✤â✱ t❛ ❝â Λ0 (xy) − Λ0 (x) Λ0 (y) = [Λ0 (xy) − h (xy)] + [h (x) h (y) − Λ0 (x) Λ0 (y)] = [Λ0 (xy) − h (xy)] + [h (y) − Λ0 (y)] h (x) + [h (x) − Λ0 (x)] Λ0 (y) ❉♦ h ≤ ♥➯♥ s✉② r❛ |λ0 (xy) − λ0 (x) λ0 (y)| ≤ ε (1 + x + y ) ✳ ❇ð✐ ✈➻ ε > ❜➨ tý ỵ t s r (xy) = (x) λ0 (y) ⇒ λ0 ∈ ∆✳ ❉♦ ✈➟② ∆ ⊂ K ✳ ❱➻ K ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉ ✯ ♥➯♥ tø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ s✉② r❛ ∆ ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉✯ tr♦♥❣ A∗ ✳ ▼➦t ❦❤→❝✱ ✈➻ A ⊂ K ✱ K ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉ ✯ ♥➯♥ tø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ s✉② r❛ ∆ ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉✯✳ ✯ ❱➻ tæ♣æ ●❡❧❢❛♥❞ ❧➔ tæ♣æ ❍❛✉s❞♦r❢❢ ✈➔ ②➳✉ ❤ì♥ tỉ♣ỉ ②➳✉ ✯ ♥➯♥ s✉② r❛ tỉ♣ỉ ●❡❧❢❛♥❞ trị♥❣ ✈ỵ✐ tỉ♣ỉ ②➳✉✯ tr➯♥ ∆ ❱➟② ∆ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ❝♦♠♣❛❝t✳ ❜✳ ●✐↔ sû x, y ∈ A, α ∈ C, h ∈ ∆✳ ❑❤✐ ✤â✱ t❛ ❝â αx (h) = h (αx) = α.h (x) = α.x (h) = (α.x) (h)✳ ❙✉② r❛ αx = α.x ✯ x + y (h) = h (x + y) = h (x) + h (y) = x (h) + y (h) = (x + y) (h) ✸✺ ✯ x.y (h) = h (x.y) = h (x) h (y) = x (h) y (h) = (x.y) (h) ❉♦ ✤â✱ t❛ ❝â αx = αx x+y =x+y x.y = x.y ❙✉② r❛ A ❧➔ ✤↕✐ sè ❝♦♥ ❝õ❛ C (∆) ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ x x A ởt ỗ ❤✐➺✉ J ❧➔ ❤↕t ♥❤➙♥ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞✳ ❑❤✐ ✤â✱ t❛ ❝â x ∈ J ⇔ x (h) = 0, ∀h ∈ ∆ ⇔ h (x) = 0, ∀h ∈ ∆ ⇔ x ∈ rad A✳ ❙✉② r❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤↕✐ sè A ❧➔ ♥û❛ ✤ì♥✳ ❝✳ ❚❛ ❝â λ ∈ x (∆) ⇔ λ = x (h) ✈ỵ✐ h ♥➔♦ ✤â t❤✉ë❝ ∆ ⇔ λ = h (x) ✈ỵ✐ h ♥➔♦ ✤â t❤✉ë❝ ∆ ⇔ λ ∈ σ (x) ❉♦ ✤â✱ x (∆) = σ (x) ❚ø ✤â✱ t❛ ❝â xˆ ∞ = sup |ˆ x (h)| = sup |h (x)| = sup |λ| = f (x) ≤ x ✳ h∈∆ h∈∆ λ∈σ(x) ✸✻ § ✷ P❍➆P ✣➮■ ❍ÑP ✸✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✭❦❤æ♥❣ ✤á✐ ❤ä✐ t➼♥❤ ❣✐❛♦ ❤♦→♥✮✳ ⑩♥❤ ①↕ x → x∗ tø A ✈➔♦ ❝❤➼♥❤ ♥â ✤÷đ❝ ❣å✐ ❧➔ P❤➨♣ ✤è✐ ❤ñ♣ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✶✳ (x + y)∗ = x∗ + y ∗ ✱ ∀x, y ∈ A✳ ✷✳ (λx)∗ = λ.x∗ ✱ ∀x ∈ A✳ ✸✳ (xy)∗ = x∗ y ∗ , ∀x, y ∈ A✳ ✹✳ x∗∗ = (x∗ )∗ = x ∀x ∈ A✳ P❤➛♥ tû x ∈ A ✤÷đ❝ ❣å✐ ❧➔ tü ❧✐➯♥ ❤ñ♣ ♥➳✉ x∗ = x✳ ✸✳✷✳✷ ỵ sỷ A số ợ ♣❤➨♣ ✤è✐ ❤ñ♣ ✈➔ x ∈ A✳ ❑❤✐ ✤â✱ t❛ ❝â ❛✳ ❈→❝ ♣❤➛♥ tû x + x∗✱ y (x − x∗) ✈➔ xx∗ tü ❧✐➯♥ ❤ñ♣✳ ❜✳ P❤➛♥ tû x ữủ t ữợ x = u + iv✱ ✈ỵ✐ u, v ❧➔ ❝→❝ ♣❤➛♥ tû tü ❧✐➯♥ ❤đ♣ tr♦♥❣ A✳ ❝✳ ✣ì♥ ✈à e ❧➔ tü ❧✐➯♥ ❤ñ♣✳ ❞✳ x ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x∗ ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A✳ ❍ì♥ ♥ú❛✱ (x∗)−1 = x−1 ∗✳ ❈❤ù♥❣ ♠✐♥❤✳ ❛✳ ✯ ❚❛ ❝â (x + x∗ )∗ = x∗ + (x∗ )∗ = x + x∗ ✳ ❙✉② r❛ x + x∗ ❧➔ ♣❤➛♥ tû tü ❧✐➯♥ ❤ñ♣✳ ✯ ❚❛ ❝â [i (x − x∗ )]∗ = −i(x − x∗ )∗ = −i [x∗ − (x∗ )∗ ] = −i (x∗ − x) = i (x − x∗ )✳ ❙✉② r❛ (x − x∗ ) ❧➔ ♣❤➛♥ tû tü ❧✐➯♥ ❤ñ♣✳ ✯ ❚❛ ❝â (xx∗ )∗ = (x∗ )∗ x∗ = xx∗ ✳ ✸✼ ❙✉② r❛ xx∗ ❧➔ ♣❤➛♥ tû tü ❧✐➯♥ ❤ñ♣✳ x + x∗ i (x − x∗ ) ❜✳ ●✐↔ sû x ∈ A✳ ❚❛ ✤➦t u = , v= ✳ 2 ❚ø ❦❤➥♥❣ ✤à♥❤ ✭❛✮ t❛ s✉② r❛ u, v ❧➔ ❝→❝ ♣❤➛♥ tû tü ❧✐➯♥ ❤ñ♣ ✈➔ tø ❝→❝❤ ✤➦t u, v t❛ s✉② r❛ x = u + iv ✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✤â✳ ●✐↔ sû x = u + iv ✱ ✈ỵ✐ u , v tü ❧✐➯♥ ❤ñ♣✳ ✣➦t w = v − v ✳ ❑❤✐ ✤â✱ ✈➻ u, v, u , v tü ❧✐➯♥ ❤ñ♣ ✈➔ w = v − v , iw = u − u ♥➯♥ ❝↔ ❤❛✐ ♣❤➛♥ tû w, iw ❧➔ tü ❧✐➯♥ ❤ñ♣✳ ❉♦ ✤â t❛ ❝â iw = (iw)∗ = −iw∗ = −iw✳ ❙✉② r❛ w = ⇒ u = u , v = v ✳ ❱➟② ❜✐➸✉ ❞✐➵♥ ✤â ❧➔ ❞✉② ♥❤➜t✳ ❝✳ ❱➻ e ❧➔ ✤ì♥ ✈à ♥➯♥ t❛ ❝â e∗ = e.e∗ ✳ ◆❤í ❦❤➥♥❣ ✤à♥❤ ✭❛✮ e.e∗ ❧➔ tü ❧✐➯♥ ❤ñ♣✳ ❙✉② r❛ e∗ tü ❧✐➯♥ ❤ñ♣✳ ❉♦ ✤â e ❧➔ tü ❧✐➯♥ ❤ñ♣✳ ❞✳ ●✐↔ sû x ∈ A✱ x ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ∃x−1 ∈ A s❛♦ ❝❤♦ xx−1 = x−1 x = e ⇔ xx−1 ∗ ∗ = x−1 x ∗ ⇔ x−1 x∗ = x∗ x−1 = e∗ = e ∗ =e ❇ð✐ ✈➟②✱ x∗ ❦❤↔ ♥❣❤à❝❤ (x )1 = x1 ỵ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✈➔ ♥û❛ ✤ì♥ t❤➻ ♠é✐ ♣❤➨♣ ✤è✐ ❤đ♣ tr➯♥ A ❧➔ ❧✐➯♥ tư❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x → x∗ ❧➔ ♣❤➨♣ ✤è✐ ❤đ♣ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ♥û❛ ✤ì♥ A✳ ❱➻ →♥❤ ①↕ x → x∗ ❧➔ →♥❤ ①↕ tø ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ A ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ A ♥➯♥ ự õ tử ỵ ỗ t õ t ự r {xn } ∈ A ✱ ♠➔ xn → x ✈➔ xn ∗ y t y = x rữợ t t õ t r h ỗ ự tũ ỵ số A ✤➦t ϕ (x) = h (x∗ )✱ ∀x ∈ A✱ t❤➻ ❝→❝ ✤✐➲✉ ❦✐➺♥ (1), (2), (3) tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ t s r ởt ỗ ự ❱➻ t❤➳ ϕ ❧✐➯♥ tö❝✳ ✯ h = ⇒ ϕ = 0✳ ✯ ϕ (x + y) = h [(x + y)∗ ] = h (x∗ + y ∗ ) = h (x∗ ) + h (y ∗ ) = h (x∗ ) + h (y ∗ ) = ϕ (x) + ϕ (y) ✯ ϕ (λx) = h [(λx)∗ ] = h λx∗ = λh (x∗ ) = λh (x∗ ) = λϕ (x) ✯ ϕ (xy) = h [(xy)∗ ] = h (x∗ y ∗ ) = h (x∗ ) h (y ∗ ) = h (x∗ )h (y ∗ ) = ϕ (x) ϕ (y) ❱➻ xn → x ✱ xn ∗ → y ✈➔ ϕ ❧✐➯♥ tö❝ ♥➯♥ t❛ ❝â h (x∗ ) = ϕ (x) = lim ϕ (xn ) = lim h (x∗ ) = h (y), ∀h ∈ ∆ n→∞ n→∞ ∗ ∗ ⇔ h (x ) = h (y) , ∀h ∈ ∆ ⇔ y − x ∈ rad A = {0} ⇔ y = x∗ ✸✳✷✳✹ ✣à♥❤ ♥❣❤➽❛✳ ✣↕✐ sè ❇❛♥❛❝❤ A ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤đ♣ x → x∗ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ xx∗ = x ✱ ∀x ∈ A ✤÷đ❝ ❣å✐ ❧➔ B ∗ ✸✳✷✳✺ ◆❤➟♥ ①➨t✳ ◆➳✉ A ❧➔ B∗ ✲ ✤↕✐ sè✱ t❤➻ ✸✾ ✲ ✤↕✐ sè✳ x∗ = x , ∀x ∈ A✳ § ✸ ❇■➊◆ ❙❍■▲❖❱ ❇➢t ✤➛✉ tø ❜➔✐ ♥➔②✱ ❦❤✐ ♥â✐ ✤➳♥ ✤↕✐ sè ❇❛♥❛❝❤ A t❤➻ t❛ ❤✐➸✉ r➡♥❣ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à✳ ✸✳✸✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❝♦♥ E ❝õ❛ MA ✤÷đ❝ ❣å✐ ❧➔ ❜✐➯♥ ❝õ❛ A ♥➳✉ ♠å✐ ❤➔♠ f ∈ A, f ✤↕t ❝ü❝ ✤↕✐ tr➯♥ E ✳ ✸✳✸✳✷ ❇ê ✤➲✳ ●✐↔ sû f1, f2, , fn ∈ A ✈➔ V ❧➔ t➟♣ ♠ð ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ V = ψ : fj (ψ) < 1, ≤ j ≤ n ⊂ MA ✳ ❑❤✐ ✤â✱ ❤♦➦❝ V ❣✐❛♦ ✈ỵ✐ ♠å✐ ❜✐➯♥ ❝õ❛ A✱ ❤♦➦❝ E\V ❧➔ ❜✐➯♥ ✤â♥❣ ❝õ❛ A ♥➳✉ E õ A ự sỷ tỗ t↕✐ ❜✐➳♥ ✤â♥❣ E ❝õ❛ A ♠➔ E\V ❦❤æ♥❣ ❧➔ ❜✐➯♥ ✤â♥❣ ❝õ❛ A✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ V ❣✐❛♦ ✈ỵ✐ ♠å✐ ❜✐➯♥ ❝õ❛ A✳ ❚❤➟t ✈➟②✱ ✯ ▲➜② f ∈ A s❛♦ ❝❤♦ f = 1, f MA E\V < 1✳ ❉♦ E\V ❦❤æ♥❣ ❧➔ ❜✐➯♥ ❝õ❛ A tỗ t f A s f ổ ✤↕t ❝ü❝ ✤↕✐ tr➯♥ E\V ✳ ❑❤✐ ✤â✱ max f (z) = M > z∈MA f ✱ t❛ ❝â g MA = 1✳ M ❦❤æ♥❣ ✤↕t ❝ü❝ ✤↕✐ tr➯♥ E\V ✈➔ E\V ❧➔ t➟♣ ❝♦♠♣❛❝t ✈➔ ❝❤ù❛ ❈❤å♥ g = ❱➻ f tr♦♥❣ MA ♥➯♥ f (z) < M, ∀z ∈ E\V ⇒ f E\V ❙✉② r❛ ✹✵