✶ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼ð ✤➛✉ ✈➲ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ✤↕✐ sè tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷ ✺ ✺ ✶✳✷✳ ✣↕✐ sè ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸✳ ✣↕✐ sè tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ✈➔ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✷✷ ✷✳✶✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✳ ✳ ✳ ✳ ỵ ss số tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✷✳✹✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜→♥ ❦➼♥❤ ♣❤ê tr♦♥❣ ✤↕✐ sè F LM ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✷✾ ✳ ✳ ✳ ✳ ✸✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ é ỵ t❤✉②➳t ✈➲ ✤↕✐ sè ❇❛♥❛❝❤ ❧➔ ❧➽♥❤ ✈ü❝ q✉❛♥ trå♥❣ ❝õ❛ ❚♦→♥ ❣✐↔✐ t➼❝❤✱ ♥â ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤ð✐ ✤➛✉ ỳ t trữợ ỡ ❜↔♥ ❤♦➔♥ t❤✐➺♥ ❝→❝❤ ✤➙② ❦❤♦↔♥❣ ✈➔✐ t❤➟♣ ❦✛✳ ◆â ❝â r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ s➙✉ s➢❝ tr♦♥❣ ♥❤✐➲✉ ❝❤✉②➯♥ ♥❣➔♥❤ ❝õ❛ t♦→♥ ❤å❝✱ ✤➦❝ ❜✐➺t ❧➔ ù♥❣ ❞ö♥❣ tr♦♥❣ ự t ự số ỵ tt t♦→♥ tû✱✳✳✳✣↕✐ sè ❇❛♥❛❝❤ ❝â ❝➜✉ tró❝ ✤↕✐ sè ✈➔ ❣✐↔✐ t➼❝❤ ❧➔ tê♥❣ q✉→t ✈➔ ❣➛♥ ❣ô✐ ♥❤➜t ✤è✐ ✈ỵ✐ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✳ ▼ët sü ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ✤↕✐ sè ❇❛♥❛❝❤ ❧➔ ❝→❝ ✤↕✐ sè ❋r❡❝❤❡t✳ ✣↕✐ sè ❋r❡❝❤❡t ①✉➜t ❤✐➺♥ ✈➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ✻✵ t trữợ õ ữủ ❝ù✉ ❦❤→ t❤➜✉ ✤→♦ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ s➙✉ s➢❝ tr♦♥❣ ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ ❣✐↔✐ t➼❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❋r❡❝❤❡t ✭①❡♠ ❬✼❪✮✳ ❈→❝ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ✤↕✐ sè ❋r❡❝❤❡t ❧➔ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ✤↕✐ sè tỉ♣ỉ✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❧➔ ❦❤â ❦❤➠♥ ❤ì♥ ♥❤✐➲✉ s♦ ✈ỵ✐ ❝→❝ ❧ỵ♣ ✤↕✐ sè tr➯♥✳ ◆❤➡♠ t➻♠ ❤✐➸✉ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ợ số tổổ õ t ữỡ ✭❧♦❝❛❧❧② ♠✉❧t✐♣❧✐❝❛t✐✈❡✮✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ s❛✉ ❝❤♦ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤ ❧➔✿ ❱➲ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ët ❧ỵ♣ ✤↕✐ sè tỉ♣ỉ✳ ❉ü❛ ✈➔♦ ❝❤õ ②➳✉ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✻✱ ✸✱ ✹✱ ✺❪✱ ❝❤ó♥❣ tỉ✐ tr ỳ ỵ tt ✤↕✐ sè tæ♣æ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ♠ët sè t➼♥❤ ❝❤➜t ✤➣ ❜✐➳t tø ✤↕✐ sè ❇❛♥❛❝❤ ❝❤♦ ❧ỵ♣ ✤↕✐ sè ♥➔②✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ư ✈➔ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥✱ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝✳ ◆❣♦➔✐ ✈✐➺❝ tr➻♥❤ ❜➔② ❝â ❤➺ t❤è♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ✤➣ ✸ tr➻♥❤ ❜➔② tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➲ ①✉➜t ♠ët sè ❦➳t q✉↔ ✈➔ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ❦➳t q✉↔✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ▼ð ✤➛✉ ✈➲ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ✤↕✐ sè tỉ♣ỉ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ð sð ♥❤➡♠ ❝❤✉➞♥ ❜à ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ✈➲ s❛✉❀ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè ❇❛♥❛❝❤❀ ❦❤→✐ ♥✐➺♠✱ ❝→❝ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè tỉ♣ỉ✱ ✤↕✐ sè ❋r❡❝❤❡t✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ✈➔ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ❈❤÷ì♥❣ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ✈➔ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝✳ ✣➙② ❧➔ ❝→❝ ❧ỵ♣ ✤↕✐ sè tỉ♣ỉ ❝â ❝➜✉ tró❝ ✧❣➛♥ ✈ỵ✐✧ ✤↕✐ sè ❇❛♥❛❝❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ♥❣♦➔✐ ✈✐➺❝ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❆♥s❛r✐✱ ❩♦❤r✐ ✈➔ ❏❛❜❜❛r✐✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➲ ①✉➜t ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤ê✱ ❜→♥ ❦➼♥❤ ♣❤ê✱✳✳✳ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ✤↕✐ sè ❇❛♥❛❝❤✳ ◆❣♦➔✐ r❛ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ①➙② ❞ü♥❣ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ❦➳t q✉↔✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ tr÷í♥❣ ✣↕✐ ữợ sỹ ữợ t ❑✐➲✉ P❤÷ì♥❣ ❈❤✐✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ t❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ◗✉❛ ✤➙②✱ t→❝ ❣✐↔ ❣û✐ ❧á♥❣ ❝↔♠ ì♥ ✤➳♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ❚❍P❚ ▼↕❝ ✣➽♥❤ ❈❤✐✱ ◗✉➟♥ ✻✱ ỗ t t ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❈✉è✐ ũ ỡ ỗ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✷✶ ●✐↔✐ t➼❝❤ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ✹ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ú tổ rt ữủ ỳ ỵ õ ❣â♣ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✺ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❱➠♥ ❇➻♥❤ ✺ ❈❍×❒◆● ✶ ▼Ð ✣❺❯ ❱➋ ✣❸■ ❙➮ ❇❆◆❆❈❍ ❱⑨ ✣❸■ ❙➮ ❚➷P➷ ❈❤÷ì♥❣ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tæ♣æ✱ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ✤↕✐ sè ❋r❡❝❤❡t✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ▼ư❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ✱ ❦❤ỉ♥❣ ❣✐❛♥ ❋r❡❝❤❡t✳ ◆❤ú♥❣ ♥ë✐ ❞✉♥❣ ♥➔② ✤÷đ❝ tê♥❣ ❤đ♣ ✈➔ tr➼❝❤ r❛ ❝❤õ ②➳✉ tø ❬✷❪✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝ị♥❣ ✈ỵ✐ ♠ët tỉ♣ỉ tr➯♥ ✤â s t ổ ữợ ❧➔ ❧✐➯♥ tư❝✳ ❚➟♣ ❝♦♥ U tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì X ữủ U U ợ ♠å✐ α ∈ K ✈➔ |α| 1❀ t➟♣ U ✤÷đ❝ út ợ x X tỗ t↕✐ δ > s❛♦ ❝❤♦ αx ∈ U ✈ỵ✐ || ỵ r ổ tỡ tổổ ổ tỗ t ỡ s U ỗ t út ợ U U tỗ t V U s V + V ⊂ U✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❝♦♥ U ổ tỡ X ữủ ỗ ✈ỵ✐ ♠å✐ x, y ∈ U ✱ ✈ỵ✐ ♠å✐ λ 1✱ t❤➻ λx + (1 − λ)y ∈ U ✳ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥ tr÷í♥❣ R✳ ❍➔♠ : E → R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✻ ✶✮ x 0✱ ✈ỵ✐ ♠å✐ x ∈ E ✈➔ x = ⇔ x = 0❀ ✷✮ λx = |λ| x ✱ ✈ỵ✐ ♠å✐ λ ∈ R ✈➔ ✈ỵ✐ ♠å✐ x ∈ E ❀ ✸✮ x + y x + y , ✈ỵ✐ ♠å✐ x, y ∈ E ✳ ❑❤✐ ✤â (E, ) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ d(x, y) = x−y , ∀x, y ∈ E ✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ E ✤➛② ✤õ ✈ỵ✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥✳ ✣è✐ ✈ỵ✐ tỉ♣ỉ s✐♥❤ ❜ð✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ❝→❝ t ổ ữợ tr E ❧✐➯♥ tư❝✳ ❉♦ ✤â✱ ♠é✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈ỵ✐ Bn = {x ∈ E : x < }, n = 1, 2, ỡ s ỗ n t ỗ E ổ tỡ tổổ ữủ ỗ ữỡ ♥➳✉ ♥â ❝â ♠ët ❝ì sð ❧➙♥ ❝➟♥ U ❝õ❛ ỗ t ỗ sỷ X ổ ỗ ữỡ õ ∈ X ❝â ❝ì sð ❧➙♥ ❝➟♥ U t❤♦↔ ♠➣♥✿ ✶✮ U, V ∈ U t❤➻ ❝â W ∈ U s❛♦ ❝❤♦ W ⊂ U ∩ V ❀ ✷✮ αU ∈ U ✈ỵ✐ ♠å✐ α ∈ K, α = ✈➔ ✈ỵ✐ ♠å✐ U ∈ U; ✸✮ ▼å✐ U ∈ U ỗ út ỡ ỳ ổ ❣✐❛♥ t✉②➳♥ t➼♥❤ tæ♣æ X ❝â ❤å ❝→❝ t➟♣ ❝♦♥ U t❤♦↔ ♠➣♥ ✶✮✱ ✷✮ ✈➔ ✸✮ t❤➻ ♥â ❧➔ ổ ỗ ữỡ ổ tỡ E õ U ỗ t ỗ út t tr E tỗ t tổổ ②➳✉ ♥❤➜t s❛♦ ❝❤♦ ❤❛✐ ♣❤➨♣ t♦→♥ tr➯♥ E ❧✐➯♥ tử E tr t ổ ỗ ữỡ ❍ì♥ ♥ú❛✱ ❝ì sð ❧➙♥ ❝➟♥ ❝õ❛ tr♦♥❣ E ❧➔ ❤å ❝→❝ t➟♣ n Vi , ε > 0, Vi ∈ U, U =ε i=1 i n, n N tổổ ỗ ✤à❛ ♣❤÷ì♥❣ τ tr➯♥ X ♥❤➟♥ U ❧➔♠ ❝ì sð ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ ∈ X t❤➻ tæ♣æ ♥➔② ❧➔ ❍❛✉s❞♦r❢❢ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ εU = U ∈U;ε>0 ❙❛✉ ✤➙② t❛ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦➳t q✉↔ ❝èt sỹ tổổ ỗ ữỡ t❤æ♥❣ q✉❛ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥✳ ✣➛✉ t✐➯♥ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ♥û❛ ❝❤✉➞♥✳ ✶✳✶✳✾ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳ ❍➔♠ p ①→❝ ✤à♥❤ tr➯♥ X ✈➔ ♥❤➟♥ ❣✐→ trà t❤ü❝ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ tr➯♥ X ♥➳✉ ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ ✈ỵ✐ ♠å✐ λ ∈ K t❛ ❝â N1 ✮ p(x) 0; N2 ✮ p(x + y) p(x) + p(y); N3 ✮ p(λx) = |λ|p(x) ❘ã r➔♥❣✱ ♥û❛ ❝❤✉➞♥ p tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X ❧➔ ❝❤✉➞♥ tr➯♥ X ♥➳✉ p(x) = s✉② r❛ x = 0✳ ✶✳✶✳✶✵ ▼➺♥❤ ✤➲✳ ◆➳✉ p ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì X t❤➻ ✈ỵ✐ ♠å✐ α > ❝→❝ t➟♣ A = {x ∈ X : p(x) < α} ✈➔ B = {x ∈ X : p(x) α} ❧➔ ỗ út t sỷ P ❧➔ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X ✳ ❑❤✐ ✤â✱ ❦➳t ❤ñ♣ ❝→❝ ▼➺♥❤ ✤➲ ✶✳✶✳✼ ✈➔ t õ r X tỗ t ởt tỉ♣ỉ ②➳✉ ♥❤➜t s❛♦ ❝❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈➔ ❝→❝ p ∈ P ❧✐➯♥ tư❝✳ ❍ì♥ ♥ú❛✱ X ổ ỗ ữỡ ỡ s t t ỗ õ ❞↕♥❣ U = {x ∈ X : sup pi (x) < ε, i = 1, , n}, tr♦♥❣ ✤â ε > 0✱ pi ∈ P ✱ n ∈ N∗ ✳ ✽ ✶✳✶✳✶✷ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ t➟♣ ỗ út ổ tỡ tổổ X ❍➔♠ t❤ü❝ ❦❤ỉ♥❣ ➙♠ µA : X → R+ ❝❤♦ ❜ð✐ µA (x) = inf{t > : x ∈ tA} ợ x X ữủ s t ủ A ỵ A t ỗ út ổ ✈❡❝tì tỉ♣ỉ X t❤➻ µA := p ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ X ✳ ❍ì♥ ♥ú❛ {x ∈ X : p(x) < 1} ⊂ A ⊂ {x ∈ X : p(x) 1} t X ổ ỗ ✤à❛ ♣❤÷ì♥❣ t❤➻ X ❝â ♠ët ❝ì sð ❧➙♥ ❝➟♥ ỗ t ỗ út õ ỡ s tữỡ ự ợ ❝→❝ ♥û❛ ❝❤✉➞♥ ❧➔ ❝→❝ ♣❤✐➳♠ ❤➔♠ ▼✐♥❦♦✇s❦✐ t÷ì♥❣ ù♥❣✳ ❑➳t ❤đ♣ ✈ỵ✐ ◆❤➟♥ ①➨t ✶✳✶✳✶✶ s✉② r❛ r➡♥❣ ♠é✐ tổổ ỗ ữỡ t ữủ ♠ët ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ ✈➔ ♥❣÷đ❝ ❧↕✐✳ ✶✳✶✳✶✺ ◆❤➟♥ ①➨t✳ ●✐↔ sû P ❧➔ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ s✐♥❤ r tổổ ỗ ữỡ tr X õ X ❧➔ ❍❛✉s❞♦r❢❢ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p(x) = ✈ỵ✐ ♠å✐ p ∈ P ❦➨♦ t❤❡♦ x = 0✳ ổ ỗ ữỡ ♠➯tr✐❝ ❣å✐ ❧➔ F ✲❦❤æ♥❣ ❣✐❛♥✱ ♥➳✉ ♥â ✤➛② ✤õ t ổ rt ỵ s r ổ ỗ ữỡ tr ỵ X ổ sr ỗ ữỡ X ữủ ✤à♥❤ ❜ð✐ ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ♥û❛ ❝❤✉➞♥ t❤➻ X tr tự tr X tỗ t ởt tr s r tổổ trũ ợ tổổ ỗ ữỡ õ t ứ ỵ tr t❛ ❝â✱ ❞➣② {xk } ❤ë✐ tư tỵ✐ x t❤❡♦ ♠➯tr✐❝ d ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ pn (x − xk ) ợ ộ n ữỡ tỹ {xk } ❧➔ ❞➣② ✾ ❈❛✉❝❤② ✤è✐ ✈ỵ✐ ♠➯tr✐❝ d ❦❤✐ ✈➔ ❝❤✐ ❦❤✐ ♥â ❧➔ ❞➣② ❈❛✉❝❤② ✤è✐ ✈ỵ✐ ♠é✐ ♥û❛ ❝❤✉➞♥ pn ✳ ✶✳✶✳✶✾ ❱➼ ❞ö✳ ●✐↔ sû C∞ := {x = {zn } : zn ∈ C, n 1} ợ ổ ữợ tổ tữớ t tø♥❣ sè ❤↕♥❣✳ ❳➨t ❤å Q = {pn } ❧➔ ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ♥û❛ ❝❤✉➞♥ tr➯♥ C∞ ①→❝ ✤à♥❤ ❜ð✐ pn (x) = |zn |; x = {zn }, n = 1, 2, ❑❤✐ ✤â C∞ ❧➔ ❦❤æ♥❣ ỗ ữỡ ỷ ữủ C tr ợ tr ❜ð✐ ∞ d(x, y) = n=1 |xn − yn | 2n + |xn − yn | ✈ỵ✐ ♠å✐ x, y ∈ C∞ ✳ ❚✉② ♥❤✐➯♥✱ C∞ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❚❤➟t ✈➟②✱ ♥➳✉ ♥❣÷đ❝ ❧↕✐ t õ ổ õ tỗ t↕✐ ❝❤✉➞♥ tr➯♥ C∞ s❛♦ ❝❤♦ tæ♣æ s✐♥❤ r❛ ❜ð✐ ❝❤✉➞♥ trị♥❣ ✈ỵ✐ tỉ♣ỉ s✐♥❤ r❛ ❜ð✐ {pn }✳ ❳➨t B(0, 1) = {x ∈ C∞ : x < 1}✳ õ tỗ t V = {x C : pi (x) = |xi | < δ, i ∈ I} tr♦♥❣ ✤â I ❧➔ t➟♣ ❤ú✉ ❤↕♥ s❛♦ ❝❤♦ V ⊂ B(0, 1)✳ ▲➜② x0 = {x0n } ∈ C∞ s❛♦ ❝❤♦ x0n = ♥➳✉ n ∈ I ✈➔ x0n = ✈ỵ✐ n ∈ / I ✳ ❑❤✐ ✤â✱ x0 = ✈➔ s✉② r❛ x0 = r > 0✳ ❱ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ k ❞♦ ❝→❝❤ ①→❝ ✤à♥❤ ❝õ❛ x0 ✈➔ V t❛ ❝â kx0 ∈ V ✳ ❉♦ ✤â kx0 ∈ B(0, 1) ✈ỵ✐ ♠å✐ k ✳ ❙✉② r❛ kx0 = kr < ✈ỵ✐ ♠å✐ k ✳ ❚❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ✶✳✶✳✷✵ ❱➼ ❞ư✳ ●å✐ C(R) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tư❝ tr➯♥ R✳ ❱ỵ✐ ♠é✐ n = 1, 2, ✤➦t pn (f ) = sup{|f (x)| : x ∈ [−n, n]}, ✶✵ ✈ỵ✐ ♠å✐ f ∈ C(R)✳ ❑❤✐ ✤â✱ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ pn ❧➔ ❝→❝ ♥û❛ ❝❤✉➞♥ tr➯♥ C(R)✳ ❉♦ ✤â✱ C(R) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ỗ ữỡ s ỷ {pn }✳ ❍ì♥ ♥û❛✱ C(R) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❋r❡❝❤❡t ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ ∞ d(f, g) = n=1 pn (f − g) , 2n + pn (f − g) ✈ỵ✐ ♠å✐ f, g ∈ C(R)✳ ✶✳✷✳ ✣↕✐ sè ❇❛♥❛❝❤ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ♠ð ✤➛✉ ❝õ❛ ✣↕✐ sè ❇❛♥❛❝❤✳ ❈→❝ ❦➳t q✉↔ ✤÷đ❝ tr➻❝❤ r❛ tø ❬✶❪✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ▼ët ✤↕✐ sè ♣❤ù❝ A ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì A tr➯♥ tr÷í♥❣ C ❝ị♥❣ ✈ỵ✐ ♠ët ♣❤➨♣ ♥❤➙♥ tr♦♥❣ tr➯♥ A t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ ✶✮ x(yz) = (xy)z, ∀x, y, z ∈ A❀ ✷✮ x(y + z) = xy + xz, (x + y)z = xz + yz, ∀x, y, z ∈ A ✸✮ (αx)y = α(xy) = x(αy), ∀x, y ∈ A, ∀α ∈ C ✣↕✐ sè ♣❤ù❝ A ✤÷đ❝ ❣å✐ ❧➔ ❣✐❛♦ ❤♦→♥ ♥➳✉ ♣❤➨♣ ♥❤➙♥ tr➯♥ A ❧➔ ❣✐❛♦ ❤♦→♥✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ I ❝õ❛ ✤↕✐ sè ự A ữủ ợ x ∈ A ✈➔ y ∈ I t❤➻ xy ∈ I ✳ ▼ët ✐❞❡❛❧ I ✤÷đ❝ ❣å✐ ❧➔ ✐❞❡❛❧ t❤ü❝ sü ♥➳✉ I = ✈➔ I = A✳ ▼ët ✐❞❡❛❧ t❤ü❝ sü ✤÷đ❝ ❣å✐ ❧➔ ✐❞❡❛❧ ❝ü❝ ✤↕✐ ♥➳✉ ♥â ❦❤æ♥❣ ❝❤ù❛ tr♦♥❣ ❜➜t ❦ý ✐❞❡❛❧ t❤ü❝ sü ♥➔♦✳ ợ ộ tỹ sỹ I số tữỡ A/I ✤÷đ❝ ①➙② ❞ü♥❣ ❜ð✐ ❝→❝ ♣❤➨♣ t♦→♥ (x + I) + (y + I) = x + y + I, α(x + I) = αx + I, (x + I)(y + I) = xy + I ❚r♦♥❣ ✤↕✐ sè ♥❣÷í✐ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣✿ A/I ❧➔ ♠ët tr÷í♥❣ ✭♠å✐ ♣❤➛♥ tû ❦❤→❝ ❦❤æ♥❣ ✤➲✉ ❦❤↔ ♥❣❤à❝❤✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ I ❧➔ ✐❞❡❛❧ ❝ü❝ ✤↕✐✱ ✈➔ ♠é✐ ✐❞❡❛❧ t❤ü❝ sü ❧✉æ♥ ❝❤ù❛ tr♦♥❣ ♠ët ✐❞❡❛❧ ❝ü❝ ✤↕✐ ♥➔♦ ✤â✳ ✷✺ ✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tæ♣æ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ▼ư❝ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ ❧✉ỉ♥ ❣✐↔ t❤✐➳t A ❝â ✤ì♥ e ỵ s rở ỵ ỵ A ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ x ∈ A✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ b > 1✱ bn xn → tr♦♥❣ A t❤➻ e − x ❦❤↔ ♥❣❤à❝❤ ✈➔ ∞ −1 (e − x) xn =e+ n=1 ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû sn = n xk ✳ ❚ø ❣✐↔ t❤✐➳t✱ t❛ ❝â k=1 bn (sn − sn−1 ) = bn xn → ❉♦ A ❧➔ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ♥➯♥ (sn ) ❧➔ ❞➣② ❈❛✉❝❤②✳ ◆❤í ❣✐↔ t❤✐➳t A ❧➔ ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ♥➯♥ (sn ) ❤ë✐ tư tỵ✐ y ∈ A✱ tù❝ ❧➔ y = lim sn = n→∞ ❚❛ ❝â ∞ xk ✳ k=1 n (xk − xk+1 ) = e − x + (x − xn+1 ) = e − xn+1 (e − x)(e + sn ) = (e − x) + k=1 ❱➻ ❝❤✉é✐ ∞ xk ❤ë✐ tö ♥➯♥ xn+1 → 0✳ ❉♦ ✤♦✱ tø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ♣❤➨♣ k=1 ♥❤➙♥ tr♦♥❣ A t❛ ♥❤➟♥ ✤÷đ❝ (e − x)(e + y) = (e + y)(e − x) = e ❚ù❝ ❧➔ e − x ❦❤↔ ♥❣❤à❝❤✳ ❚❛ ♥❤➟♥ ✤÷đ❝ ♥❣❛② ❤➺ q✉↔ s❛✉✳ ✷✳✷✳✷ ❍➺ q✉↔✳ ✭❬✺❪✮ ❈❤♦ A ❧➔ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ x ∈ A✳ ❦❤✐ ✤â ✈ỵ✐ ♠é✐ b > 1✱ bn (e − x)n → tr♦♥❣ A t❤➻ x ❦❤↔ ♥❣❤à❝❤✳ ✷✻ ✷✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✺❪✮ ✣↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ A ✤÷đ❝ ❣å✐ ❧➔ ✤↕✐ số õ t ữỡ tt tỗ t↕✐ ❧➙♥ ❝➟♥ U0 ❝õ❛ tr♦♥❣ A s❛♦ ❝❤♦ V tỗ t k0 s ❝❤♦ U0k ⊂ V ✈ỵ✐ ♠å✐ k k0 ✳ ✣↕✐ số õ t ữỡ ữủ ỵ F LM ✳ ✷✳✷✳✹ ❱➼ ❞ö✳ ▼å✐ ✤↕✐ sè ❇❛♥❛❝❤ ❧➔ ❝â t➼♥❤ ♥❤➙♥ ✤à❛ ♣❤÷ì♥❣✳ ❚❤➟t ✈➙②✱ ❧➜② U0 ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ ❜→♥ ❦➼♥❤ ✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ U0 t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤à♥❤ ♥❣❤➽❛ ♥â✐ tr➯♥✳ ▼➺♥❤ ✤➲ s❛✉ ❧➔ rở ỵ ố ợ số F LM ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ✳ ✷✳✷✳✺ ▼➺♥❤ ✤➲✳ ✭❬✺❪✮ ❈❤♦ A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ✳ ❑❤✐ ✤â✱ t➟♣ ❝→❝ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ G(A) ❝õ❛ A ❧➔ t➟♣ ♠ð ✈➔ G(A) ❧➔ ♠ët ♥❤â♠ ✤è✐ ✈ỵ✐ ♣❤➨♣ t♦→♥ ♥❤➙♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr ữủ G(A) õ ợ t ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ G(A) ♠ð✳ ●å✐ U0 ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ tr♦♥❣ A s❛♦ ❝❤♦ ♠å✐ ❧➙♥ ❝➟♥ V tỗ t k0 s U0k V ✈ỵ✐ ♠å✐ k k0 ✳ ●✐↔ sû u ∈ G(A)✱ b > U tũ ỵ ❝õ❛ 0✳ ❈❤å♥ ❧➙♥ ❝➟♥ ❝➙♥ V0 ❝õ❛ s❛♦ ❝❤♦ u−1 V0 ⊂ b−1 U0 ❱ỵ✐ ♠å✐ v ∈ u − V0 t❛ ❝â u − v ∈ V0 ✈➔ ✈➻ t❤➳ e − u−1 v ∈ u−1 V0 ❉♦ ✤â b(e − u−1 v) ∈ bu−1 V0 ⊂ U0 ❱➻ ✈➟②✱ bn (e − u−1 v)n ∈ U0n ⊂ U ✷✼ ✈ỵ✐ ♠å✐ n ✤õ ❧ỵ♥✳ ❉♦ ✤â✱ bn (e − u−1 v)n → 0✳ ❱➻ ✈➟②✱ →♣ ❞ư♥❣ ❍➺ q✉↔ ✷✳✷✳✷ t❛ ♥❤➟♥ ✤÷đ❝ u−1 v ∈ G(A)✳ ❙✉② r❛ v = u(u−1 v) ∈ G(A)✳ ❱➻ u − V0 ❧➔ ❧➙♥ ❝➟♥ ♠ð ❝õ❛ u ♥➡♠ tr♦♥❣ G(A) ♥➯♥ G(A) ❧➔ ♠ð✳ ❚❛ ữủ t q q trồ s ỵ ✭❬✺❪✮ ◆➳✉ A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ✤ì♥ ✈à e ✈➔ a ∈ A t❤➻ ♣❤ê ❝õ❛ a ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû a ∈ A✳ ❳➨t →♥❤ ①↕ φ : C → A ①→❝ ✤à♥❤ ❜ð✐ φ(λ) = λe − a✱ ợ C tr ữủ ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ❉♦ ✤â✱ σ(a) = C \ φ−1 G(A) ❧➔ t➟♣ ✤â♥❣ ❝õ❛ C✳ ❚❛ ❝❤➾ r❛ σ(a) ❜à ❝❤➦♥✳ ●✐↔ sû U0 ❧➔ ❧➟♥ ❝➟♥ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✷✳✺✳ ❑❤✐ ✤â✱ ❝❤å♥ α ∈ C∗ s❛♦ ❝❤♦ a ∈ αU0 ✳ ❑❤✐ ✤â (α−1 a)n → ❦❤✐ b n n → ∞✳ ●✐↔ sû |λ| > ✈➔ |λ| > b > 1✳ ❚❛ ❝â → 0✳ ❙✉② r❛ λ bn a αλ n → a ❱➻ ✈➟②✱ t❤❡♦ ✣à♥❤ ỵ t õ e G(A) s r αλ ∈ / σ(a)✳ ❇➙② αλ β ❣✐í✱ ♥➳✉ β ∈ σ(a)✳ ✣➦t λ = ✳ ❚❛ ♥❤➟♥ ✤÷đ❝ |λ| 1✱ tù❝ ❧➔ |β| |α|✳ α ❉♦ ✤â σ(a) ❜à ❝❤➦♥✳ ◆❤÷ ✈➟②✱ σ(a) ❧➔ t➟♣ ✤â♥❣ ✈➔ ❜à ❝❤➦♥ ❝õ❛ C✱ ✈➻ t❤➳ ♥â ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❑➳t q✉↔ s rở ỵ ỗ ự tr số F LM ✤➛② ✤õ ❧➔ ❧✐➯♥ tö❝✳ ❦❤↔ ♠➯tr✐❝ ✷✽ ❈❤ù♥❣ ♠✐♥❤✳ : A C ỗ ự ✈➔ b > 1✳ ●✐↔ sû x ∈ A ✈➔ bn xn ❤ë✐ tư tỵ✐ 0✳ ❑❤✐ ✤â✱ ✤➦t sn = n xk ✱ n = 1, 2, ❚❛ ❝â (sn ) ❧➔ k=1 ❞➣② ❈❛✉❝❤②✳ ❱➻ ✈➟②✱ (sn ) ❤ë✐ tư tỵ✐ y ∈ A ✈➔ ∞ xn y= n=1 ❚❛ ❝â y − xy = lim sn − x lim sn = lim (sn − xsn ) = lim (x − xn+1 ) = x n→∞ n→∞ n→∞ n→∞ ❉♦ ✤â φ(x) = φ(y − xy) = φ(y) − φ(x)φ(y) x ❙✉② r❛ φ(x) = 1✳ ◆➳✉ |φ(x)| > t❤➻ ❝❤å♥ u = ✳ ❚❛ ❝â φ(x) n n b u =b n xn φ(x) n = φ(x) n n nb x → ❱➻ ✈➟②✱ φ(u) = 1✳ ▼➙✉ t❤✉➝♥ ✈ỵ✐ φ(u) = 1✳ ❉♦ ✤â✱ ♥➳✉ bn xn → t❤➻ |φ(x)| < ❉♦ φ ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ φ ❧✐➯♥ tö❝ t↕✐ 0✳ ●✐↔ sû (xn ) ❧➔ ❞➣② ❤ë✐ tư tỵ✐ tr A õ ợ > tỗ t↕✐ n ✤õ ❧ỵ♥ s❛♦ ❝❤♦ b−1 εxn ∈ U0 ✱ tr♦♥❣ ✤â U0 ❧➔ ❧➙♥ ❝➟♥ s❛♦ ❝❤♦ ✈ỵ✐ U tỗ t số tỹ ♥❤✐➯♥ k s❛♦ ❝❤♦ U k ⊂ U0 ✳ ●✐↔ sỷ V tũ ỵ k0 ❧➔ sè tü ♥❤✐➯♥ s❛♦ ❝❤♦ U0k ⊂ V ✈ỵ✐ k k0 ✳ ❑❤✐ ✤â bk (ε−k xkn ) ∈ V ✱ ✈➔ ✈➻ t❤➳ lim bk (ε−k xkn ) = 0✳ ❱➻ ✈➟②✱ k→∞ |φ(ε−1 xn )| < 1, tù❝ ❧➔ |φ(xn )| < ε ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❉♦ ✤â lim φ(xn ) = 0✳ ❉♦ ✤â✱ tử n ỵ A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ✤ì♥ ✈à e✳ ●✐↔ sû φ : A → C ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ s❛♦ ❝❤♦ φ(e) = ✈➔ ker φ ⊂ A \ G(A)✳ ❑❤✐ ✤â tử ự ỵ r ợ b > ✈➔ x ∈ A s❛♦ ❝❤♦ bn xn → t❤➻ e − x ∈ G(A)✳ ❙✉② r❛ e − x ∈ / ker φ✱ tù❝ ❧➔ φ(e − x) = 0✳ ❉♦ φ t✉②➳♥ t➼♥❤✱ t❛ ❝â φ(x) = φ(e) = ❚✐➳♣ tư❝ ❧➟♣ ❧✉➟♥ ♥❤÷ tr ự ỵ t ữủ tử ỵ ss số tổổ ỡ tr ự ỵ ❈❧❡s❛s♦♥✲❩❡❧❛③❦♦ ❝❤♦ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝✳ ✣➙② sỹ rở ỵ sè ❇❛♥❛❝❤ A ✈➔ a ∈ A✳ ❑❤✐ ✤â✱ t❛ ✤➣ ❜✐➳t ∞ E(a) = n=1 an n! ❧✉æ♥ ①→❝ ✤à♥❤✱ ❜ð✐ ✈➻ ❝❤✉é✐ ð ✈➳ ♣❤↔✐ ❧➔ ❤ë✐ tö t✉②➺t ✤è✐✳ ✣è✐ ✈ỵ✐ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ tr E(a) õ t ổ tỗ t ợ ộ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ A t❛ ✤➦t E(A) = {a A : E(a) tỗ t} ợ ộ a ∈ E(A)✱ t❛ ✤➦t ea = e + E(a)✱ tr♦♥❣ ✤â e ❧➔ ✤ì♥ ✈à ❝õ❛ A✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉ ✷✳✸✳✶ ▼➺♥❤ ✤➲✳ ✭❬✺❪✮ ❈❤♦ A ❧➔ ✤↕✐ sè ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ✤ì♥ ✈à e✳ ◆➳✉ a, −a ∈ E(A) t❤➻ ea ∈ G(A) ự t t t ũ ỵ ❤✐➺✉ x0 = e ✈ỵ✐ ♠é✐ x ∈ A✳ ❑❤✐ ✤â✱ tø ❣✐↔ t❤✐➳t a, −a ∈ E(a) t❛ ❝â ∞ e=e =e a+(−a) = n=0 ∞ = (a + (−a))n n! n! n Cnk ak (−a)n−k n=0 k=0 ∞ n = n=0 k=0 n ∞ = k=0 n=0 n ∞ = k=0 n=k n ∞ = = k Cn (−1)n−k an n! k Cn (−1)n−k an n! k Cn (−1)n−k an n! 1 (−a)n−k ak k! (n − k)! k=0 n=k ∞ k ∞ a k=0 ∞ = k! (−a)n−k (n − k)! k! (−a)n n! n=k ∞ ak n=0 k=0 a −a =e e ❉♦ ✤â ea ∈ G(A) ✈➔ (ea )1 = ea ỵ s x E(A) ỵ A ❧➔ ✤↕✐ sè ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ✤ì♥ ✈à e ✈➔ x ∈ A✳ ❑❤✐ ✤â✱ ♥➳✉ {xn : n = 0, 1, 2, } ❜à ❝❤➦♥ t❤➻ λx ∈ E(A) ✈ỵ✐ ♠å✐ λ ∈ C✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {xn : n = 0, 1, 2, } ❧➔ t➟♣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ bn λn (λx)n n b > tø → ✈ỵ✐ ♠å✐ λ ∈ C✱ s✉② r❛ b → 0✳ ❉♦ õ n! n! (x)n ỵ t❛ ♥❤➟♥ ✤÷đ❝ ❝❤✉é✐ ❤ë✐ tư tr♦♥❣ A✱ tù❝ ❧➔ ex n=1 n! ỵ ❈❤♦ A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ✤ì♥ ✈à e ✈➔ ϕ : A → C ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❦❤→❝ ❦❤æ♥❣ tr➯♥ A✳ ❑❤✐ ✤â✱ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✶✮ ϕ(e) = ✈➔ ker ϕ ⊂ A \ G(A)❀ ✭✷✮ ϕ(a) ∈ σ(a) ✈ỵ✐ ♠å✐ a ∈ A❀ ✭✸✮ ỗ ự ự (1) (2) ●✐↔ sû ϕ(e) = ✈➔ ker ϕ ⊂ A \ G(A)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ a ∈ A t❛ ❝â ♥➳✉ ϕ(a) ∈ / σ(a) t❤➻ ϕ(a)e − a ∈ G(A)✳ ❙✉② r❛ ϕ(a)e − a ∈ / ker ϕ✳ ❱➻ ✈➟②✱ ϕ ϕ(a)e − a = 0✳ ❚✉② ♥❤✐➯♥✱ ❞♦ ϕ t✉②➳♥ t➼♥❤ ✈➔ ϕ(e) = t❛ ❝â ϕ ϕ(a)e − a = ϕ(a)ϕ(e) − ϕ(a) = 0✳ ❚❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ❱➟② (1) ⇒ (2)✳ ❇➙② ❣✐í✱ ❣✐↔ sû ϕ(a) ∈ σ(a) ✈ỵ✐ ♠å✐ a ∈ A✳ ❑❤✐ ✤â✱ ♥➳✉ ϕ(e) = λ t❤➻ λ ∈ σ(e)✱ tù❝ ❧➔ λe − e ∈ / G(A)✳ ◆➳✉ λ = t❤➻ e ✳ ❉♦ ✤â λ = 1, tù❝ ❧➔ φ(e) = 1✳ ❇➙② λe − e ∈ G(A) ✈➔ (λe − e)−1 = λ−1 ❣✐í✱ ❣✐↔ sû ker ϕ A \ G(A)✳ ❑❤✐ ✤â✱ tỗ t b ker s b G(A)✳ ❱➻ b ∈ ker ϕ ♥➯♥ ϕ(b) = 0✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ ✭✷✮ t❛ ❝â ∈ σ(b)✳ ❙✉② r❛ 0e − b = −b ∈ / G(A)✱ ✈➻ t❤➳ b ∈ / G(A)✳ ❚❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ❱➟② (2) ⇒ (1)✳ ❚❛ ✤➣ ❜✐➳t (3) ⇒ (1)✳ ❱➜♥ ✤➲ ❝á♥ ❧↕✐ ❧➔ ❝❤ù♥❣ ♠✐♥❤ (1) ⇒ (3)✳ ❱ỵ✐ x ∈ A ✈➔ b > ♥➳✉ bn xn → t❤➻ ✈ỵ✐ ♠å✐ λ ∈ C s❛♦ ❝❤♦ |λ| > t❛ ❝â λ−n bn xn → 0, tù❝ ❧➔ bn (λ−1 x)n → õ t ỵ t õ e λ−1 x ∈ G(A)✳ ❙✉② r❛ ϕ(x) = λ ◆❤÷ ✈➟②✱ ♥➳✉ bn xn → t❤➻ |ϕ(x)| 1✳ ❉♦ ✤â✱ ϕ ❜à ❝❤➦♥ tr➯♥ V := b−1 U0 ✱ tr♦♥❣ ✤â U0 ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✤↕✐ sè F LM ✈➔ ✈➻ t❤➳ ϕ ❧✐➯♥ tö❝✳ ✸✷ ●✐↔ sû b > ✈➔ a ∈ V ✳ ✣➦t x = ak ✈ỵ✐ k ∈ N✳ ❚ø (bk )n xn = bkn akn → s✉② r❛ |ϕ(x)| 1✱ tù❝ ❧➔ |ϕ(ak )| ✈ỵ✐ ♠é✐ k ✳ ❱ỵ✐ ♠é✐ a ∈ V ✱ ①➨t →♥❤ ①↕ F : C → C ①→❝ ✤à♥❤ ❜ð✐ N z n an n! za F (z) = ϕ(e ) = ϕ e + lim N →∞ k=1 N = ϕ(e) + lim N →∞ ∞ =1+ k=1 z n ϕ(an ) n! n=1 z n ϕ(an ) n! z n ϕ(an ) ❱➻ ✈ỵ✐ ♠å✐ n ♥➯♥ ❝❤✉é✐ ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✈➻ n! t❤➳ F (z) ❧➔ ❤➔♠ ♥❣✉②➯♥ ✭❝❤➾♥❤ ❤➻♥❤ tr➯♥ t♦➔♥ ❜ë ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✮✳ ∞ n=1 |ϕ(an )| ❍ì♥ ♥ú❛✱ ∞ |F (z)| n=0 |z|n = e|z| n! ❚ø eza ∈ G(A) ✈ỵ✐ ♠å✐ z s✉② r❛ F (z) = ợ z tỗ t ∈ C s❛♦ ❝❤♦ ∞ F (z) = e αz =1+ n=1 ❑❤❛✐ tr✐➸♥ eαz αn z n n! t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ ✈➔ s♦ s→♥❤ ❤❛✐ ✈➳ t❛ ♥❤➟♥ ✤÷đ❝ ϕ(a) = α ✈➔ ϕ(a2 ) = α2 ợ ộ x A tỗ t λ > s❛♦ ❝❤♦ a = λx ∈ V ✳ ❱➻ t❤➳ 2 ϕ(λ x ) = ϕ(λx) ❱➻ ϕ t✉②➳♥ t➼♥❤ t❛ ♥❤➟♥ ✤÷đ❝ ϕ(x2 ) = ϕ(x) ✈ỵ✐ ♠å✐ x ∈ A✳ ❇➙② ❣✐í✱ t❛ ❝❤➾ r❛ ϕ(xy) = ϕ(x)ϕ(y) ✈ỵ✐ ♠å✐ x, y ∈ A✳ ❚❤➟t ✈➟②✱ tø ϕ (x + y)2 = ϕ(x + y) ✸✸ s✉② r❛ ϕ(xy + yx) = 2ϕ(x)ϕ(y) ✈ỵ✐ ♠å✐ x, y ∈ A✳ ❱➻ ✈➟②✱ ♥➳✉ A ❣✐→♦ ❤♦→♥ t❤➻ ϕ(xy) = ϕ(x)ϕ(y) ❈❤ù♥❣ t tú qt ổ ỗ ự t tỗ t a, b A s ❝❤♦ ϕ(ab) = ϕ(a)ϕ(b)✳ ✣➦t x = ϕ(a)e − a✳ ❚❛ ❝â ϕ(x) = 0✳ ❈❤å♥ y t❤❡♦ b ♣❤ò ❤ñ♣ t❛ ❝â ϕ(xy) = 1✳ ❑❤✐ ✤â✱ tø ϕ(xy + yx) = 2ϕ(x)ϕ(y) ✈➔ t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ ϕ s✉② r❛ ϕ(yx) = −1✳ ✣➦t z = yxy ✳ ❚❛ ❝â = 2ϕ(x)ϕ(z) = ϕ(xz + zx) = ϕ(xyxy + yxyx) = ϕ[(xy + yx)2 − (xy)2 − (yx)2 ] = [ϕ(xy + yx)]2 − [ϕ(xy)]2 − [ϕ(yx)]2 = 4ϕ(x)2 ϕ(y)2 − − = −2 ❚❛ ữủ sỹ t ỗ ♣❤ù❝✳ ✷✳✹✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜→♥ ❦➼♥❤ ♣❤ê tr♦♥❣ ✤↕✐ sè F LM ▼ö❝ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜→♥ ❦➼♥❤ ♣❤ê tr♦♥❣ ✤↕✐ sè ❋▲▼✳ ✷✳✹✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✻❪✮ ❈❤♦ A ❧➔ ✤↕✐ sè tỉ♣ỉ ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♠➯tr✐❝ dA✳ ❚❛ ❣å✐ ♠➯tr✐❝ tr➯♥ A ❧➔ ♥û❛ ♥❤➙♥ ♥➳✉ dA (0, xy) dA (0, x)dA (0, y) ✈ỵ✐ ♠å✐ x, y ∈ A✳ ❑❤✐ ✤â dA ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ✤♦ tr➯♥ A✳ ❚❛ ỵ dA (0, x) DA (x) õ t❤➸ ❣✐↔ t❤✐➳t ♠➯tr✐❝ dA ❧➔ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ tà♥❤ t✐➳♥✱ tù❝ ❧➔ dA (x + z, y + z) = dA (x, y) ✈ỵ✐ ♠å✐ x, y, z ∈ A ✭①❡♠ ❬✽❪✮✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉✳ ✸✹ ✷✳✹✳✷ ▼➺♥❤ ✤➲✳ ●✐↔ sû A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♣❤➨♣ ✤♦ ♥û❛ ♥❤➙♥ dA ✳ ❑❤✐ ✤â✱ ♥➳✉ x ∈ A s❛♦ ❝❤♦ DA (x) < t❤➻ e − x ❦❤↔ ♥❣❤à❝❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x ∈ A ✈➔ DA (x) = dA (x, 0) < 1✳ ▲➜② b > s❛♦ ❝❤♦ dA (0, bx) < 1✳ ✣✐➲✉ ♥➔②✱ ❧✉æ♥ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❜ð✐ ❞➣② (1 + )x → x ❦❤✐ n n → ∞✳ ❉♦ ✤â dA (0, (1 + )x) n dA (0, x) + dA (x, (1 + )x) < n ♥➳✉ n ✤õ ❧ỵ♥✳ ❑❤✐ ✤â✱ tø t➼♥❤ ♥û❛ ♥❤➙♥ ❝õ❛ ♠➯tr✐❝ dA t❛ s✉② r❛ n dA (0, bn xn ) dA (0, bx) ❱➻ ✈➟②✱ dA (0, bn xn ) → ❦❤✐ n → ∞✱ tù❝ ❧➔ bn xn → 0✳ ❉♦ ✤â✱ →♣ ❞ö♥❣ ❍➺ q✉↔ ✷✳✷✳✷ t❛ õ e x G(A) ỵ s t q ỵ sû A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♣❤➨♣ ✤♦ ♥û❛ ♥❤➙♥ dA ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ A ρ(x) = lim n→∞ DA (xn ) n ❚❛ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ s❛✉✿ ✷✳✹✳✹ ▼➺♥❤ ✤➲✳ ●✐↔ sû A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♣❤➨♣ ✤♦ ♥û❛ ♥❤➙♥ dA ✳ ❑❤✐ ✤â ρ(xy) = ρ(yx) ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø (xy)n = x(yx)n−1 y s✉② r❛ DA ((xy)n ) = dA (0, (xy)n ) dA (0, x)dA (0, (yx)n−1 )dA (0, y) = DA (x)DA ((yx)n−1 )DA (y) ✸✺ ❙✉② r❛ DA ((xy)n ) n n−1 n DA (y) n DA ((yx)n−1 ) n − 1 DA (x) n ❈❤♦ n → ∞ t❛ ♥❤➟♥ ✤÷đ❝ ρ(xy) ❚÷ì♥❣ tü✱ t❛ ❝â ρ(yx) ρ(yx) ρ(xy)✳ ❱➻ ✈➟② ρ(xy) = ρ(yx)✳ ✷✳✹✳✺ ▼➺♥❤ ✤➲✳ ●✐↔ sû A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♣❤➨♣ ✤♦ ♥û❛ ♥❤➙♥ dA ✳ ◆➳✉ dA (0, x2 ) = dA (0, x) ✈ỵ✐ ♠å✐ x ∈ A t❤➻ ρ(x) = DA (x)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t t❛ ❝â DA (x4 ) = dA (0, x4 ) = dA (0, x2 ) = dA (0, x) 4 = DA (x) ❇➡♥❣ q✉② ♥↕♣ t❛ ❝â 2k DA (x ) = DA (x) 2k ✈ỵ✐ ♠å✐ k ∈ N∗ ✳ ❉♦ ✤â✱ ρ(x) = lim n→∞ = lim k→∞ = lim k→∞ DA (xn ) n k DA (x2 ) 2k DA (x) 2k 2k = DA (x) ✷✳✹✳✻ ▼➺♥❤ ✤➲✳ ●✐↔ sû A ❧➔ ✤↕✐ sè F LM ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ♣❤➨♣ ✤♦ ♥û❛ ♥❤➙♥ dA ✳ ◆➳✉ x, y ∈ A ✈➔ xy = yx t❤➻ ρ(xy) ρ(x)ρ(y), ρ(x + y) ρ(x) + ρ(y) ✸✻ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø xy = yx s✉② r❛ (xy)n = xn y n ✈ỵ✐ ♠å✐ n✳ ❑❤✐ ✤â✱ tø t➼♥❤ ♥û❛ ♥❤➙♥ ❝õ❛ dA t❛ ❝â ρ(xy) = lim n→∞ = lim n→∞ DA ((xy)n ) n DA (xn )DA (y n ) n lim n→∞ DA (xn ) n lim n→∞ DA (y n ) n = ρ(x)ρ(y) x x ✈➔ b = t❛ ❝â α β ●✐↔ sû ρ(x) < α ✈➔ ρ(y) < β ✳ ❑❤✐ ✤â✱ ✈ỵ✐ a = ρ(a) < 1, ρ(b) < 1✳ ❚ø ρ(x) = lim n→∞ DA (xn ) n ✈ỵ✐ ♠é✐ x ∈ A s✉② r❛ tỗ t số tỹ N s n n max{DA (a2 ), DA (b2 )} < 1, ✈ỵ✐ ♠å✐ n ❱ỵ✐ ♠é✐ n N ✤➦t γn = max n DA (a2 n k n DA (x + y)2 2n = DA −k ).D (bk ) A 2n n C2kn x2 −k k y N ❚❛ ❝â 2n k=0 2n C2kn α2 = DA n n −k k −k k β a b 2n k=0 2n n n C2kn α2 −k β k DA (a2 −k )DA (bk ) 2n k=0 1/2n (α + β).γn ✸✼ ❇ð✐ ✈➻ (γn ) ❧➔ ❞➣② ❣✐↔♠ t❛ ❝â ρ(x + y) = lim n→∞ n DA (x + y)2 2n n (α + β) lim sup γn2 n→∞ n (α + β) lim sup γN2 = α + β n→∞ ❉♦ α > (x), > (y) tũ ỵ t ✤÷đ❝ ρ(x+y) ρ(x)+ρ(y) ✸✽ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ s❛✉✿ ✶✮ ❚r➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tổổ ổ ỗ ữỡ ổ rt ✤↕✐ sè ❇❛♥❛❝❤✳ ✷✮ ❚r➻♥❤ ❜➔② ❤➺ t❤è♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ ♠ð ✤➛✉ ✈➲ ✤↕✐ sè tæ♣æ✳ ✸✮ ❚r➻♥❤ ❜➔② ♥❤ú♥❣ ♥❤ú♥❣ ❦➳t q✉↔ ✈➲ ♥❤â♠ ❝→❝ ♣❤➛♥ tû ❦❤↔ ỗ ự trú ỵ ss ✤è✐ ✈ỵ✐ ❧ỵ♣ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥✱ ✤↕✐ sè tỉ♣ỉ ❝ì ❜↔♥ ❦❤↔ ♠➯tr✐❝✳ ✹✮ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ ❝→❝ t➔✐ ❧✐➺✉ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤ ❤♦➦❝ ❝❤ù♥❣ ♠✐♥❤ ✈➠♥ t➢t ♥❤÷✿ ▼➺♥❤ ✤➲ ✶✳✸✳✹✱ ❇ê ✤➲ ✶✳✸✳✺✱ ◆❤➟♥ ①➨t ✷✳✶✳✸✱ ▼➺♥❤ ✤➲ ✷✳✶✳✹✱ ỵ ỵ ỵ t ởt số t q t➼♥❤ ❝❤➜t ❝õ❛ ❜→♥ ❦➼♥❤ ♣❤ê ❝õ❛ ♣❤➛♥ tû tr♦♥❣ ✤↕✐ sè tæ♣æ ❋▲▼ t❤➸ ❤✐➺♥ ð ❝→❝ ▼➺♥❤ ✤➲ ✷✳✹✳✷✱ ▼➺♥❤ ✤➲ ✷✳✹✳✹✱ ▼➺♥❤ ✤➲ ✷✳✹✳✺✱ ▼➺♥❤ ✤➲ ✷✳✹✳✻✳ ✻✮ ✣÷❛ r❛ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ❦➳t q✉↔ ♥❤÷✿ ❱➼ ❞ư ✶✳✸✳✶✵✱ ❱➼ ❞ư ✷✳✶✳✺✳ ✸✾ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉ ✭✷✵✶✵✮✱ ◆❤➟♣ ♠æ♥ ✤↕✐ sè ✤➲✉✱ ◆❤➔ ①✉➜t ❜↔♥ ✣❍❙P✲❍➔ ♥ë✐✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✈➔ ▲➯ ▼➟✉ ❍↔✐ ✭✷✵✵✷✮✱ ❈ì sð ỵ tt t ◆❳❇ ●✐→♦ ❉ö❝✳ ❬✸❪ ❆♥s❛r✐ P✳ ❊ ✭✶✾✾✵✮ ❆ ❝❧❛ss ♦❢ ❢❛❝t♦r❛❜❧❡ t♦♣♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛s Pr♦❝✳ ❊❞✐♥❜✉r❣❤ ▼❛t❤✳ ❙♦❝✳ ✭✷✮ ✸✸ ✭✶✾✾✵✮✱ ♥♦✳ ✶✱ ✺✸✲✺✾✳ ❬✹❪ ❆♥s❛r✐ P✳ ❊ ✭✷✵✵✶✮ ❚♦♣✐❝s ♦♥ ❢✉♥❞❛♠❡♥t❛❧ t♦♣♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛s✱ ❍♦♥❛♠ ▼❛t❤✳ ❏✳ ✷✸ ✱ ♥♦✳ ✶✱ ✺✾✲✻✻✳ ❬✺❪ ❆♥s❛r✐ P✳ ❊ ✭✷✵✶✵✮ ❚❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧s ♦♥ ❢✉♥❞❛♠❡♥t❛❧ ❧♦❝❛❧❧② ♠✉❧t✐♣❧✐❝❛t✐✈❡ t♦♣♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛s✱ ❚✉r❦✐s❤ ❏✳ ▼❛t❤✳ ✸✹ ✭✷✵✶✵✮✱ ♥♦✳ ✸✱ ✸✽✺✲✸✾✶✳ ❬✻❪ ❩♦❤r✐ ❆✳ ❛♥❞ ❏❛❜❜❛r✐ ❆ ✭✷✵✶✷✮✱●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s t♦ ❢✉♥❞❛♠❡♥t❛❧ ❧♦❝❛❧❧② ♠✉❧t✐♣❧✐❝❛t✐✈❡ t♦♣♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛s✱ ❚✉r❦✐s❤ ❏✳ ▼❛t❤✳ ✸✻✱ ✹✹✺✲✹✺✶✳ ❬✼❪ ●♦❧❞♠❛♥♥✱ ❍✳✱ ✭✶✾✾✵✮✱ ❯♥✐❢♦r♠ ❋r❡❝❤❡t ❛❧❣❡❜r❛s✱ ◆♦rt❤✲❍♦❧❧❛♥❞ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✱ ✶✻✷✳ ◆♦rt❤✲❍♦❧❧❛♥❞ P✉❜❧✐s❤✐♥❣ ❈♦✳✱ ❆♠st❡r✲ ❞❛♠✳ ❬✽❪ ❘✉❞✐♥✱ ❲✳✱ ✭✶✾✾✶✮ ❋✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❙❡r✐❡s ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✳ ▼❝●r❛✇✲❍✐❧❧✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✳ ... ♣❤➨♣ t♦→♥ ♥❤➙♥ A × A → A✱ (a, b) → ab ❧✐➯♥ tư❝✳ ✷✮ ✣↕✐ sè tỉ♣ỉ ✤÷đ❝ ❣å✐ ❧➔ số ỗ ữỡ õ ỡ s ỗ t ỗ số ỗ ữỡ ữủ số t rt tổổ ỗ ữỡ õ ữủ s ♠ët ♠➯tr✐❝✳ ✣↕✐ sè t✐➲♥ ❋r❡❝❤❡t ❧➔ ✤↕✐ sè ❋r❡❝❤❡t... ♥❤✐➲✉ ❝❤✉②➯♥ ♥❣➔♥❤ ❝õ❛ t♦→♥ ❤å❝✱ ✤➦❝ ❜✐➺t ❧➔ ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ●✐↔✐ t➼❝❤ ♣❤ù❝✱ ✣↕✐ số ỵ tt t tỷ số õ tró❝ ✤↕✐ sè ✈➔ ❣✐↔✐ t➼❝❤ ❧➔ tê♥❣ q✉→t ✈➔ ❣➛♥ ❣ơ✐ ♥❤➜t ✤è✐ ✈ỵ✐ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✳ ▼ët... ✤➦❝ trữ ỗ ự sỷ ởt ỗ ự tr số ♣❤ù❝ A ❝â ✤ì♥ ✈à e✳ ❑❤✐ ✤â ✶✮ ϕ(e) = 1❀ ✷✮ ϕ(x) = ♥➳✉ x ❧➔ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤✳ ✶✳✸✳✶✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶❪✮❈❤♦ A ❧➔ ♠ët số tổổ Pờ x A ữủ ỵ ❤✐➺✉ ❧➔ σ(x)✱ ❧➔