❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ✣➱ ❚❍➚ ❍❯➏ ▼❐❚ ❙➮ C ∗✲✣❸■ ❙➮ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▼❆ ❚❘❾◆ ✣❆ ❚❍Ù❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✶✾ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ✣➱ ❚❍➚ ❍❯➏ ▼❐❚ ❙➮ C ∗✲✣❸■ ❙➮ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▼❆ ❚❘❾◆ ✣❆ ❚❍Ù❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ✈➔ ❧➼ t❤✉②➳t số số ữớ ữợ ❞➝♥✿ ❚❙✳ ▲➊ ❚❍❆◆❍ ❍■➌❯ ✐ ▼ö❝ ❧ö❝ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇↔♥❣ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✐ ✐✐ ✐✐✐ ✲✣❸■ ❙➮ ❚✃◆● ◗❯⑩❚ ❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ✶ C∗ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ✣↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ✣↕✐ sè ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺ ∗✲✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✻ C ∗ ✲✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷ ▼❐❚ ❙➮ C ∗✲✣❸■ ❙➮ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▼❆ ❚❘❾◆ ✣❆ ❚❍Ù❈ ✸✻ ✷✳✶ ❚➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ t❤ü❝ ✈➔ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷ ❚➟♣ ❝→❝ ✤❛ t❤ù❝ t❤ü❝ ✈➔ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸ ❚➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✶ ✐✐ ỵ R rữớ số tỹ C ❚r÷í♥❣ sè ♣❤ù❝✳ K ✿ ❚r÷í♥❣ sè t❤ü❝ ❤♦➦❝ tr÷í♥❣ sè ♣❤ù❝✳ Kn×n ✿ ❚➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n tr➯♥ tr÷í♥❣ K✳ K[x] ✿ ❚➟♣ ❝→❝ ✤❛ t❤ù❝ n ❜✐➳♥ tr➯♥ tr÷í♥❣ K✳ C[z, z] ✿ ❚➟♣ ❝→❝ tự ộ t tr trữớ C Msìt ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝ ❝ï s × t tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ K[x]✳ XT ✿ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ X ✳ XH ✿ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❧✐➯♥ ❤ñ♣ ❝õ❛ ♠❛ tr➟♥ X ✳ L(E, K) ✿ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ①→❝ ✤à♥❤ tr➯♥ E ✳ C0 (R) ✿ ❚➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ ✈➔ tr✐➺t t✐➯✉ t↕✐ ✈ỉ ❝ị♥❣✳ B(H) ✿ ❚➟♣ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tr➯♥ H✳ σ(a) ✿ P❤ê ❝õ❛ ♣❤➛♥ tû a✳ r(a) ✿ ❇→♥ ❦➼♥❤ ♣❤ê ❝õ❛ ♣❤➛♥ tû a✳ ✐✐✐ ▼ð ✤➛✉ C ∗ ✲✤↕✐ sè✱ ♠➦❝ ❞ị tø ❧➙✉ ✤÷đ❝ ①❡♠ ❧➔ ♠ët ♥❤→♥❤ ❝õ❛ ●✐↔✐ t➼❝❤ ❤➔♠ ❞♦ ♥â t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ♥❤÷♥❣ ♥â ❧➔ ♠ët sü ❦➳t ❤đ♣ ❝õ❛ ❝➜✉ tró❝ ❦❤ỉ♥❣ ❣✐❛♥ ợ trú số ữ ổ ✈❡❝tì✱ ✈➔♥❤✱ ♠ỉ✤✉♥ ❝ơ♥❣ ♥❤÷ ∗✲✤↕✐ sè✳ ✣➦❝ ❜✐➺t✱ ❝→❝ C ∗ ✲✤↕✐ sè ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝â ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ✈ỵ✐ t➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉ỉ♥❣ ✈➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ tr➯♥ ♠ët tr÷í♥❣✳ ◆❣♦➔✐ r❛✱ ❝→❝ C ∗ ✲✤↕✐ sè ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝✱ ❝❤➥♥❣ ❤↕♥ tr♦♥❣ ❈ì ❧÷đ♥❣ tû✱ C ∗ ✲✤↕✐ sè ✤÷đ❝ ❞ị♥❣ ✤➸ ♠ỉ ❤➻♥❤ ❤â❛ ✤↕✐ sè ❝→❝ q st t ỵ r ỳ ❣➛♥ ✤➙②✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ q✉❛♥ t➙♠ ♥❤✐➲✉ ❤ì♥ ✤➳♥ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠❛ tr➟♥ ✤❛ t❤ù❝ ❞♦ ❝â ♥❤✐➲✉ ✈➜♥ ✤➲ t♦→♥ ❤å❝ ❝➛♥ ♠ët sü ❤✐➸✉ ❜✐➳t ♥❤➜t ✤à♥❤ ✈➲ ♠❛ tr➟♥ ✤❛ t❤ù❝✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ t➻♠ ❤✐➸✉ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ C ∗ ✲✤↕✐ sè✳ ❚ø ✤â✱ ❝❤ó♥❣ tỉ✐ ❦✐➸♠ tr❛ sü t❤ä❛ ♠➣♥ ♠ët sè ❝➜✉ tró❝ ❧✐➯♥ q✉❛♥ ✤➳♥ C ∗ ✲✤↕✐ sè ❝❤♦ ♠ët sè t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝✳ ✣✐➲✉ ♥➔② s➩ ❣✐ó♣ ➼❝❤ ❝❤♦ ❝→❝ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ♥➔②✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ C ∗✲✤↕✐ sè tê♥❣ q✉→t ✈➔ ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ ✐✈ ❦❤ỉ♥❣ ❣✐❛♥ ✉♥✐t❛✱ ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤✱ ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët ✤↕✐ sè✱ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ P❤➛♥ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❜❛ ❝➜✉ tró❝ ✤↕✐ sè✿ ∗✲✤↕✐ sè✱ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ C ∗ ✲✤↕✐ sè✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè C ∗✲✤↕✐ sè ❧✐➯♥ q✉❛♥ ✤➳♥ ♠❛ tr➟♥ ✤❛ t❤ù❝ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❦✐➸♠ tr❛ ❝→❝ t➟♣ ♥➯✉ tr♦♥❣ ❝→❝ ✈➼ ❞ư ð ❈❤÷ì♥❣ ✶✱ t➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ ✭t❤ü❝ ✈➔ ♣❤ù❝✮✱ t➟♣ ❝→❝ ✤❛ t❤ù❝ ✭t❤ü❝ ✈➔ ♣❤ù❝✮✱ t➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝✱ ❝â t❤ä❛ ♠➣♥ ❜❛ ❝➜✉ tró❝ ♥➯✉ tr➯♥✱ tr♦♥❣ ✤â ❝â ❝➜✉ tró❝ C ∗ ✲✤↕✐ sè✱ ❤❛② ❦❤ỉ♥❣✳ ❈→❝ ❦➳t q ữủ t ự ữợ ỵ ữủ t ữợ sỹ ữợ ú ù t t ❝õ❛ t❤➛② ❣✐→♦ ❚❙✳ ▲➯ ❚❤❛♥❤ ❍✐➳✉✱ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥✳ ❚ỉ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ◗✉❛ ✤➙②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ỡ qỵ rữớ ỡ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ũ qỵ ổ ợ ❤å❝ ✣↕✐ sè ✈➔ ❧➼ t❤✉②➳t sè ❦❤â❛ ✷✵ ✤➣ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ◆❤➙♥ ✤➙②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü ❤é trñ ✈➲ ♠å✐ ♠➦t tø ♣❤➼❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ỗ ỡ ỡ tổ ổ t ❚r÷í♥❣ ❉ü ❜à ✤↕✐ ❤å❝ ❞➙♥ të❝ tr✉♥❣ ÷ì♥❣ ◆❤❛ ❚r❛♥❣ ✤➣ ❧✉ỉ♥ ❤é trđ✱ ❣✐ó♣ ✤ï ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ò ữủ tỹ ợ sỹ ộ ỹ ố ❣➢♥❣ ❝õ❛ ❜↔♥ t❤➙♥✱ ✈ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ỳ õ ỵ qỵ t ổ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ◗✉② ◆❤ì♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ ✣é ❚❤à ❍✉➺ ✶ ❈❤÷ì♥❣ ✶ ✲✣❸■ ❙➮ ❚✃◆● ◗❯⑩❚ ❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ C∗ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ ❦❤æ♥❣ ❣✐❛♥ ✉♥✐t❛✱ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤✱ ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët ✤↕✐ sè✱ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✸❪ ✈➔ ❬✹❪✳ ❚ø ✤â ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ❣➛♥ ✈ỵ✐ C ∗ ✲✤↕✐ sè ♥❤÷✿ ✤↕✐ sè✱ ∗✲✤↕✐ sè✱ ✤↕✐ sè ❇❛♥❤❛❝❤❀ ✈➔ ❝❤➼♥❤ C ∗ ✲✤↕✐ sè tê♥❣ q✉→t✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❝❤õ ②➳✉ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✸❪✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ K ❧➔ ♠ët tr÷í♥❣ ✭K = R ❤♦➦❝ K = C✮✳ ▼ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ✭❝á♥ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✮ ♠ët t➟♣ ❤đ♣ ❦❤→❝ ré♥❣ ❝ị♥❣ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥ X tr➯♥ trữớ K + : X ì X −→ X ·: K×X −→ X (λ, x) −→ λx (x, y) −→ x + y t❤ä❛ ♠➣♥ t→♠ t✐➯♥ ✤➲ s❛✉ ♠➔ t❛ ❣å✐ ❧➔ t→♠ t✐➯♥ ✤➲ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✿ (i) x + y = y + x ✈ỵ✐ ♠å✐ x, y ∈ X ❀ (ii) x + (y + z) = (x + y) + z ợ x, y, z X (iii) ỗ t↕✐ ∈ X s❛♦ ❝❤♦ x + = + x = x ✈ỵ✐ ♠å✐ x ∈ X (iv) ợ x X tỗ t x ∈ X s❛♦ ❝❤♦ x + (−x) = 0❀ (v) (à)x = (àx) ợ , K ♠å✐ x ∈ X ❀ (vi) (λ + µ)x = x + àx ợ , K ♠å✐ x ∈ X ❀ (vii) λ(x + y) = λx + λy ✈ỵ✐ ♠å✐ λ ∈ K ✈➔ ♠å✐ x, y ∈ X ❀ (viii) 1.x = x ✈ỵ✐ ♠å✐ x ∈ X, tr♦♥❣ ✤â ✶ ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ K✳ ❑❤✐ ✤â✱ t❛ ❝ơ♥❣ ♥â✐ X ❧➔ ♠ët K✲❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈❤♦ H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ K✳ ▼ët t➼❝❤ ✈ỉ ữợ tr H ởt , : H × H −→ K (x, y) −→ x, y t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ (i) x, x ≥ ✈ỵ✐ ♠å✐ x ∈ H; ✈➔ x, x = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0❀ (ii) x, y = y, x ✈ỵ✐ ♠å✐ x, y ∈ H ❀ (iii) x + y, z = x, z + y, z ✈ỵ✐ ♠å✐ x, y, z ∈ H ❀ (iv) λx, y = λ x, y ✈ỵ✐ ♠å✐ x, y ∈ H ✈➔ λ ∈ K✳ ❑❤æ♥❣ tỡ H ũ ợ ởt t ổ ữợ tr ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt H ởt ỵ (H, , ) H ❝❤♦ ❣å♥✳ ❚❛ ♥â✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ✉♥✐t❛ ✭t✳÷✳✱ ❊✉❝❧✐❞ ✮ ♥➳✉ H ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✸ tr➯♥ C tữ tr R ỵ Km×n ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ❝ï m × n tr➯♥ tr÷í♥❣ K✳ ❑❤✐ ✤â a) Km×n ❧➔ ♠ët K✲❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝ị♥❣ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥✿ ♣❤➨♣ ❝ë♥❣ ❤❛✐ ♠❛ tr➟♥ ✈➔ ♣❤➨♣ ♥❤➙♥ ♠ët ♣❤➛♥ tû trữớ K ợ ởt tr õ r Rmìn ❧➔ ♠ët R✲❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ✈➔ Cm×n ❧➔ ♠ët C✲❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì mn ❝❤✐➲✉✳ ❍ì♥ ♥ú❛✱ Cm×n ❝ơ♥❣ ❧➔ ♠ët R✲❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì 2mn ❝❤✐➲✉✳ b) ❳➨t →♥❤ ①↕ • : Rm×n × Rm×n −→ R (X, Y ) −→ X • Y := T r(X T Y ) tr♦♥❣ ✤â X T ❧➔ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ X ✳ ❑❤✐ ✤â✱ →♥❤ ①↕ ✏ •✑ ❧➔ ♠ët t➼❝❤ ổ ữợ tr Rmìn õ Rmìn ởt ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❊✉❝❧✐❞✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ X = [xij ], Y = [yij ] ∈ Rm×n ✱ t❛ ❝â (i) T r(X T X) = m i=1 n j=1 xij ≥ ✈➔ m i=1 n j=1 xij = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xij = 0, ✈ỵ✐ ♠å✐ ≤ i ≤ m, ≤ j ≤ n✳ ❉♦ ✤â✱ X • X ≥ ợ X Rmìn X ã X = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X = 0✳ (ii) X • Y = T r(X T Y ) = T r(X T Y )T = T r(Y T X) = Y • X ✳ (iii) (X + Y) • Z = T r(X + Y )T Z = T r(X T Z + Y T Z) = T r(X T Z) + T r(Y T Z) = X • Z + Y • Z (iv) (λX)•Y = T r((λX)T Y ) = T r(λX T Y ) = λT r(X T Y ) = (X ãY ), ợ ∈ R✳ ✣➦❝ ❜✐➺t✱ ❦❤✐ m = n t❛ ❝â X • Y = T r(X T Y ) = n i=1 n j=1 xij yij ✳ ❑❤✐ m = n = t t ổ ữợ tr tr t t ổ ữợ tổ tữớ tr Rn tữ Rm Rnìn ũ ợ A = ✤↕✐ sè ❇❛♥❛❝❤ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ T r(AT A) ✭❝❤✉➞♥ ❋r♦❜❡♥✐✉s✮ ❧➔ ♠ët C ∗ ✲✤↕✐ sè✳ Rnìn ũ ợ A = T r(AT A) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛✲ ♥❛❝❤✳ ✣➸ ❦✐➸♠ tr❛ Rn×n ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✱ t❛ ❝➛♥ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ AB ≤ A B ợ A, B Rnìn t sû A = [aij ], B = n k=1 aik bkj ✳ ❚❛ ❝â n n n n n n n aik bkj aik b2kj ≤ i=1 j=1 i=1 j=1 k=1 k=1 k=1 n n n n n n n n 2 aik bkj = aik b2kj i=1 k=1 j=1 k=1 i=1 k=1 j=1 k=1 [bij ] ✈➔ AB = [uij ] ∈ Rn×n tr♦♥❣ ✤â✱ uij = AB = = = A B ❚ø ✤â s✉② r❛ AB ≤ A B ợ A, B Rnìn ❱➟② Rn×n ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ ❚❛ s➩ ❝❤➾ r Rnìn ợ ổ tọ ❦✐➺♥ A∗ A = A ✈➔ ❞♦ ✤â ♥â ❦❤æ♥❣ ❧➔ ♠ët C ∗ ✲✤↕✐ sè✳ ❚❤➟t ✈➟②✱ ❧➜② ♠❛ tr➟♥ A= ❚❛ ❝â 1 , AT A = A∗ = AT = 2 √ √ ❑❤✐ ✤â✱ A = ✈➔ AT A = 28 < = A ✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ Cnìn ởt số ợ ♣❤➨♣ ✤è✐ ❤ñ♣ A∗ = AH tr♦♥❣ ✤â T AH = A ❧➔ ❝❤✉②➸♥ ✈à ❧✐➯♥ ❤ñ♣ ❝õ❛ ♠❛ tr➟♥ A ∈ Cn×n ✳ ✸✽ ✭✐✐✮ ✣↕✐ sè Cn×n ❝ị♥❣ ợ rs số ữ ổ ự ♠✐♥❤✳ C ∗ ✲✤↕✐ T r(AAH ) A = ❧➔ ♠ët sè✳ ✭✐✮ ❳❡♠ ❱➼ ❞ư ✶✳✺✳✷❜✮✳ ✭✐✐✮ Cn×n ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❋r♦❜❡♥✐✉s A = T r(AAH ) ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤ ✭①❡♠ ❱➼ ❞ư ✶✳✹✳✷✳✷❛✮✮✳ ❚✉② ♥❤✐➯♥✱ Cn×n ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ ♥➔②✳ ❚❤➟t ✈➟②✱ ①➨t ♠❛ tr➟♥ i ∈ C2×2 A= ❑❤✐ ✤â −i −2i , AH A = AH = 2i √ √ ❚❛ ❝â✱ A = ✈➔ AH A = 28 < = A ✳ ❇➙② ❣✐í✱ t❛ t số Knìn ợ à ❣å♥ ❝❤ó♥❣ tỉ✐ ❣ë♣ ❤❛✐ tr÷í♥❣ ❤đ♣ t❤➔♥❤ ♠ët ♠➺♥❤ ✤➲✳ ▼➺♥❤ ✤➲ ✷✳✶✳✹✳ ❚r➯♥ Kn×n t❛ ①➨t ❝❤✉➞♥ n A |aij | = max 1≤j≤n tr♦♥❣ ✤â A = [aij ] ∈ Kn×n i=1 ❑❤✐ ✤â ✭✐✮ ✭✐✐✮ (Kn×n , ) ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ (Kn×n , ) ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ ❈❤ù♥❣ ♠✐♥❤✳ ❧➔ ♠ët ∗✲✤↕✐ sè ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤đ♣ A∗ = AH ♥❤÷♥❣ sè ✳ ✭✐✮ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✶✷ ✈➔ ❱➼ ❞ư ✶✳✷✳✸ t❛ ❝â (Kn×n , ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ AB ≤ A B ✱ ✈ỵ✐ ✸✾ ♠å✐ A = [aij ], B = [bij ] ∈ Kn×n ✳ ❚❤➟t ✈➟②✱ t❛ ❝â n AB n 1≤j≤n aik bkj | | = max i=1 k=1 ✈➔ n n n i=1 i=1 k=1 n k=1 n |aik ||bkj | ≤ aik bkj | ≤ | n n ≤ max k |aik | |bkj | k=1 n |bkj |, ✈ỵ✐ ♠å✐ j = 1, 2, , n |aik | max j i=1 i=1 k=1 ❉♦ ✤â✱ n AB n | = max j i=1 n aik bkj | ≤ max k k=1 n |aik | max j i=1 |bkj | = A B k=1 ✭✐✐✮ ▲➜② ♠❛ tr➟♥ ∈ K2×2 A= ❑❤✐ ✤â 2 1 , AH A = AH = AT = ❚❛ ❝â✱ A = max{2; 2} = ✈➔ AH A = max{4; 6} = > = A 21 ✳ ❉♦ ✤â✱ (Kn×n , ) ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè✳ ◆❤➟♥ ①➨t ✷✳✶✳✺✳ ❇➡♥❣ ❝→❝❤ ✤ê✐ ✈à tr➼ ❣✐ú❛ ❝→❝ ❤➔♥❣ ✈➔ ❝→❝ ❝ët ❝õ❛ ♠❛ n tr➟♥ A t ụ t ữủ t q tữỡ tỹ ợ ❝❤✉➞♥ A tr♦♥❣ ✤â ∞ = max 1≤i≤n j=1 |aij |✱ A = [aij ] ∈ Kn×n ✳ ▼➺♥❤ ✤➲ ✷✳✶✳✻✳ ❈❤♦ ♠❛ tr➟♥ A = [aij ] ∈ Kn×n✳ õ số Knìn ũ ợ ổ ❧➔ A = max |aij | C ∗ ✲✤↕✐ 1≤i,j≤n sè✳ ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ❉♦ ✤â✱ ♥â ✹✵ ự ỵ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❑❤✐ ✤â✱ B(H) ❧➔ ♠ët C ∗ ✲✤↕✐ sè✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤➣ ❜✐➳t B(H) ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ữ tr ợ ộ A B(H)✱ ❦➼ ❤✐➺✉ A∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ A✳ ❑❤✐ ✤â✱ tø ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû ❧✐➯♥ ❤đ♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt s✉② r❛ →♥❤ ①↕ A −→ A∗ , ∀A ∈ B(H) ❧➔ ♠ët ♣❤➨♣ ✤è✐ ❤đ♣ tr➯♥ B(H)✳ ❍ì♥ ♥ú❛✱ B(H) ❧➔ ♠ët C ∗ ✲✤↕✐ sè✳ ❚❤➟t ✈➟②✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✶✺ t❛ ❝â A = A∗ ✱ ✈ỵ✐ ♠å✐ A ∈ B(H)✳ ❉♦ ✤â✱ A∗ A ≤ A∗ A = A ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ h ∈ H✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③✱ t❛ ❝â A∗ (h) = A∗ (h), A∗ (h) = AA∗ (h), h ≤ AA∗ h ❚ø ✤â s✉② r❛ A = A∗ AA∗ , ∀A ∈ B(H)✳ ❱➟② AA∗ ≤ = A∗ A = A ✱ ✈➔ ❞♦ ✤â B(H) ❧➔ ♠ët C ∗ ✲✤↕✐ sè✳ ◆❤➟♥ ①➨t ✷✳✶✳✽✳ ✭✐✮ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ Rn×n, Cn×n ✈➔ R[x] ❧➔ ♥❤ú♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❉♦ ✤â✱ B(Rn×n ), B(Cn×n ), B(R[x]) ❧➔ ❝→❝ C ∗ ✲✤↕✐ sè✳ ✭✐✐✮ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ♥➳✉ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤➻ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ♥❤÷♥❣ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè✳ ❍➺ q✉↔ ✷✳✶✳✾✳ ❱ỵ✐ ♠é✐ A ∈ Kn×n t❛ ①➨t p✲❝❤✉➞♥✱ ≤ p < +∞ A p = sup Ax p : x ∈ Kn , x = x p = max{ Ax p : x ∈ Kn , x p = 1} B(X) ✹✶ ❑❤✐ ✤â✱ (i) (Kn×n , p ) (ii) (Kn×n , p ) ❈❤ù♥❣ ♠✐♥❤✳ AB p ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ ❧➔ ♠ët C ∗ ✲✤↕✐ sè ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤đ♣ A∗ = AH ✳ ✭✐✮ ❚❛ ❝â = sup ≤ sup ≤ sup ABx p : x = = sup x p ABx p : Bx = sup Bx p ABx p : Bx = sup Bx p ABx p Bx p : Bx = Bx p x p Bx p : Bx = x p Bx p :x=0 x p = A p B p ✭✐✐✮ ❉♦ Kn×n = B(Kn×n ) ♥➯♥ tø ❱➼ ❞ư ✶✳✻✳✹✳✸✮ ✈➔ ◆❤➟♥ ①➨t ✷✳✶✳✽ t❛ s✉② r❛ Kn×n ❧➔ ♠ët C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ p ✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ✷✳✷ ❚➟♣ ❝→❝ ✤❛ t❤ù❝ t❤ü❝ ✈➔ ♣❤ù❝ ❳➨t ✤↕✐ sè ✤❛ t❤ù❝ R[x]✳ ❱ỵ✐ ♠é✐ f (x) ∈ R[x]✱ ❣✐↔ sû f (x) = fα x α ✱ α∈Ωn,d ❦❤✐ ✤â t❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✳ ▼➺♥❤ ✤➲ ✷✳✷✳✶✳ (i) ✣↕✐ sè ✤❛ t❤ù❝ R[x] ❧➔ ♠ët ∗✲✤↕✐ sè ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤đ♣ t➛♠ t❤÷í♥❣ (ii) R[x] f ∗ (x) = f (x)✳ ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❝↔♠ s✐♥❤ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ❞♦ ✤â ❈❤ù♥❣ ♠✐♥❤✳ R[x] ❦❤æ♥❣ ❧➔ f = → − f = C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ ❝↔♠ s✐♥❤✳ → − → − f • f ❦❤ỉ♥❣ ❧➔ ✭✐✮ ❍✐➸♥ ♥❤✐➯♥✳ ✭✐✐✮ R[x] ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✭①❡♠ ❱➼ ❞ư ✶✳✷✳✸✳✷✮✮✳ ❱ỵ✐ ♠å✐ f, g ∈ R[x] t❛ ❝➛♥ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ f g ≤ f g ✳ ❚❤➟t ✈➟②✱ ❧➜② ✹✷ √ f = 1+x1 +2x2 ✈➔ g = x1 +x2 ∈ R[x1 , x2 ]✳ ❑❤✐ ✤â✱ f = + + 22 = √ √ 6, g = ✈➔ f g = x1 + x2 + 3x1 x2 + x21 + 2x22 ✳ √ ❚❛ ❝â✱ f g = > 12 = f g ✳ ❉♦ ✤â✱ R[x] ✈ỵ✐ ❝❤✉➞♥ ✤➣ ❝❤♦ ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ R[x] ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ ❝↔♠ s✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❱ỵ✐ ♠é✐ f (x) = f p → − = f fα xα ∈ R[x]✱ p n p p✲❝❤✉➞♥✱ p ≥ ①➨t α∈Ωn,d |fαi |p = i=1 ❑❤✐ ✤â (i) p = 1, R[x] ❱ỵ✐ (ii) p ≥ 2, R[x] ❱ỵ✐ (iii) (R[x], p ) ❈❤ù♥❣ ♠✐♥❤✳ ❝❤✉➞♥ p ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ p ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ sè ✈ỵ✐ ♠å✐ p ≥ 1✳ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✳✶✷ ✈➔ ❱➼ ❞ö ✶✳✷✳✸✳✷✮ s✉② r❛ R[x] ❝ị♥❣ ✈ỵ✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✭✐✮ ❑❤✐ p = 1✱ ✈ỵ✐ ♠å✐ f, g ∈ R[x]✱ ❣✐↔ sû f = fα xα ✈➔ g = α∈Ωn,d gβ xβ t❛ ❝â f g = (fα gβ ) xγ ✳ ❑❤✐ ✤â✱ γ∈Ωn,2d β∈Ωn,d α+β=γ fg | = γ fα gβ | ≤ α+β=γ |fα | = α ❱➟② R[x] ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ |fα gβ | = γ α+β=γ |fα |.|gβ | γ α+β=γ |gβ | = f g β ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ ✭✐✐✮ ▲➜② f = g = 1+x1 ∈ R[x1 ]✳ ❱ỵ✐ ♠å✐ p ≥ 2✱ t❛ ❝â f p = g p = √ p ✹✸ ✈➔ f g = + 2x1 + x21 ✳ ❑❤✐ ✤â✱ fg p = √ p + 2p > √ p ❉♦ ✤â✱ R[x] ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ = f p g p , ✈ỵ✐ ♠å✐ p ≥ p ✭p ≥ 2✮ ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ✭✐✐✐✮ ❑❤✐ p > 1✱ (R[x], p ) ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ♥➯♥ ♥â ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ sè✳ ❑❤✐ p = 1✱ t❛ ❝â 2 f |fα | = α α ≥ ❉♦ ✤â (R[x], |fα |.|fβ | α,β fα fβ | = f | γ |fα |2 + = = f ∗f α+β=γ p ) ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ❦❤✐ p = 1✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ❚r➯♥ R[x]✱ ✤➦t (i) ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ (ii) R[x] ❦❤æ♥❣ ❧➔ f = sup |fα |✳ ❑❤✐ ✤â✱ α∈Ωn,d R[x]✳ ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ✤➣ ❝❤♦ ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ❉♦ ✤â✱ C ∗ ✲✤↕✐ ❈❤ù♥❣ ♠✐♥❤✳ R[x] sè ✈ỵ✐ ❝❤✉➞♥ ♥➔②✳ ✭✐✮ ❱ỵ✐ ♠å✐ f, g ∈ R[x] ✈➔ ♠å✐ λ ∈ R t❛ ❝â (i1 ) f = sup |fα | ≥ ✈➔ f = ⇔ sup |fα | = ⇔ |fα | = 0✱ α∈Ωn,d α∈Ωn,d ✈ỵ✐ ♠å✐ α ∈ Ωn,d ❤❛② fα = ✈ỵ✐ ♠å✐ α ∈ Ωn,d ✳ ❉♦ ✤â✱ f = 0✳ (i2 ) λf = sup |λfα | = |λ| sup |fα | = |λ| f ✳ α∈Ωn,d α∈Ωn,d (i3 ) ❱ỵ✐ ♠å✐ α ∈ Ωn,d ✱ t❛ ❝â |fα + gα | ≤ |fα | + |gα | ≤ sup |fα | + sup |gα | = f + g ✳ α∈Ωn,d α∈Ωn,d ❉♦ ✤â f + g = sup |fα + gα | ≤ f + g ✳ α∈Ωn,d ✹✹ ✭✐✐✮ ▲➜② f = g = 1+x ∈ R[x]✳ ❚❛ ❝â f = g = ✈➔ f g = 1+2x+x2 ✳ ❑❤✐ ✤â f g = > = f g ✳ ❉♦ ✤â✱ R[x] ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ s✉♣ ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ❱➟② R[x] ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ ✤➣ ❝❤♦✳ ▼➺♥❤ ✤➲ ✷✳✷✳✹✳ (i) C[z] ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ f = sup |fγ |✳ γ∈Ωn,d (ii) C[z] ❧➔ ♠ët ∗✲✤↕✐ sè ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤đ♣ f ∗ (z) = fγ z γ tr♦♥❣ ✤â γ∈Ωn,d fγ z γ ∈ C[z]✳ f (z) = γ∈Ωn,d (iii) C[z] ❧➔ C ∗ ✲✤↕✐ ❦❤ỉ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✏s✉♣✑✳ ❉♦ ✤â✱ C[z] ❦❤æ♥❣ sè✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ C[z] ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈ỵ✐ ❝❤✉➞♥ f = sup |fγ |✳ ❱ỵ✐ ♠å✐ ❞➣② {fk (z)} ⊂ C[z], fk (z) −→ f (z) t❛ ♣❤↔✐ ❝❤ù♥❣ γ∈Ωn,d ♠✐♥❤ f (z) ∈ C[z]✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû fk (z) = (k) γ fγ z γ ✈➔ f (z) = (o) γ fγ z γ ∈ C[z]✱ ❦❤✐ ✤â✱ fk −→ f ⇔ fk − f −→ ⇔ sup |fγ(k) − fγ(o) | −→ γ (k) (o) (k) (o) ❚ø ✤â s✉② r❛ ≤ |fγ − fγ | ≤ sup |fγ − fγ | −→ ✈ỵ✐ ♠å✐ γ ✳ γ ❉♦ ✤â (k) fγ −→ (o) fγ ✈ỵ✐ ♠å✐ γ ❤❛② f (z) ∈ C[z]✳ ❱➟② C[z] ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✭✐✐✮ ❳❡♠ ❱➼ ❞ư ✶✳✺✳✷✳❞✮✳ ✭✐✐✐✮ ▲➜② f (z) = i − 2z + z + z ✈➔ g(z) = z − z ∈ C[z]✳ ❑❤✐ ✤â f = 2, g = ✈➔ f g = iz − 2z + (1 − i)z + 3z − z − z ✳ ❚❛ ❝â f g = > = f g ✳ ❉♦ ✤â C[z] ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✏s✉♣✑✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ C[z] ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❝❤✉➞♥ ✤➣ ❝❤♦✳ ✹✺ ▼➺♥❤ ✤➲ ✷✳✷✳✺✳ ❚r➯♥ C[z]✱ ①➨t p✲❝❤✉➞♥✱ p ≥ f p → − = f n p |fγi |p = p i=1 ❑❤✐ ✤â✱ p = 1, C[z] ✭✐✮ ❱ỵ✐ p ≥ 2, C[z] ✭✐✐✮ ❱ỵ✐ ✭✐✐✐✮ (C[z], p ) ❈❤ù♥❣ ♠✐♥❤✳ ❝❤✉➞♥ p ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ sè ✈ỵ✐ p ❦❤ỉ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ p ≥ 1✳ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✳✶✷ ✈➔ ▼➺♥❤ ✤➲ ✷✳✷✳✹ s✉② r❛ C[z] ❝ị♥❣ ✈ỵ✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✭✐✮ ❑❤✐ p = 1✱ ✈ỵ✐ ♠å✐ f, g ∈ C[z] ❣✐↔ sû f = fα z α ✈➔ g = α∈Ωn,d gβ z β t❛ ❝â f g = (fα gβ ) z γ ✳ ❑❤✐ ✤â✱ γ∈Ωn,2d β∈Ωn,d α+β=γ fg | = γ fα gβ | ≤ α+β=γ |fα | = α ❱➟② C[z] ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ |fα gβ | = γ α+β=γ γ |fα |.|gβ | α+β=γ |gβ | = f g β ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳ √ ✭✐✐✮ ▲➜② f = g = i+z1 ∈ C[z1 ]✳ ❱ỵ✐ ♠å✐ p ≥ 2✱ t❛ ❝â f p = g p = p √ √ ✈➔ f g = −1 + 2iz1 + z12 ✳ ❑❤✐ ✤â✱ f g p = p + 2p > p = f p g p ✈ỵ✐ ♠å✐ p ≥ 2✳ ❉♦ ✤â✱ C[z] ❝ị♥❣ ✈ỵ✐ ❝❤✉➞♥ p ✱ ✭p ≥ 2✮ ❦❤ỉ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ✭✐✐✐✮ ❑❤✐ p > 1✱ (C[z], p ) ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ♥➯♥ ♥â ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ sè✳ ✹✻ ❑❤✐ p = 1✱ t❛ ❝â f |fα | = |fα |2 + = α α = f2 ❱➟② (C[z], γ α,β fα fβ | | |fα |.|fβ | ≥ α+β=γ = f ∗f p ) ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ❦❤✐ p = 1✳ ❉♦ ✤â (C[z], p ) ❦❤æ♥❣ ❧➔ C ∗ ✲✤↕✐ số ợ p ỵ C[z, z] = fαβ z α z β : fαβ f (z, z) = α,β∈Ωn,d ∈C ✳ ▼➺♥❤ ✤➲ ✷✳✷✳✻✳ (i) C[z, z] ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✏s✉♣✑ f = sup |fαβ | α,β∈Ωn,d (ii) C[z, z] ❧➔ ♠ët ∗✲✤↕✐ sè ✈ỵ✐ ♣❤➨♣ ✤è✐ ❤ñ♣ f ∗ (z, z) = fαβ z α z β α,β∈Ωn,d tr♦♥❣ ✤â α β fαβ z z ∈ C[z, z]✳ f (z, z) = α,β∈Ωn,d (iii) C[z, z] ❦❤ỉ♥❣ ❧➔ ❦❤ỉ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✏s✉♣✑✳ ❉♦ ✤â✱ C ∗ ✲✤↕✐ ❈❤ù♥❣ ♠✐♥❤✳ C[z, z] sè✳ ▲➟♣ ❧✉➟♥ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ♥❤÷ ▼➺♥❤ ✤➲ ✷✳✷✳✹ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ ✭✐✮ ✈➔ ✭✐✐✮✳ ✭✐✐✐✮ ▲➜② f (z, z) = 2z + 3zz ✈➔ g(z, z) = zz + iz ✳ ❑❤✐ ✤â✱ f = 3✱ g = ✈➔ f g = 2zz + (3 + 2i)z z + 3iz z ✳ ❚❛ ❝â✱ fg = √ 13 > = f g ❱➟② C[z, z] ❦❤ỉ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ s✉♣✳ ❉♦ ✤â✱ C[z, z] ổ C số ỵ ❤✐➺✉ Ch [z, z] = {f (z, z) ∈ C[z, z] : f (z, z) ∈ R, ∀z ∈ Cn } = {f (z, z) ∈ C[z, z] : f (z, z) = f (z, z)} ▼➺♥❤ ✤➲ ✷✳✷✳✼✳ ❈❤♦ f (z, z) = fαβ z α z β ∈ C[z, z]✳ ❑❤✐ ✤â α,β∈Ωn,d f ∈ Ch [z, z] ⇔ fαβ = f βα , ∀α, β ∈ Ωn,d ⇔ ♠❛ tr➟♥ ❚❛ ❝â t❤➸ ✈✐➳t f (z, z) = foo + ❈❤ù♥❣ ♠✐♥❤✳ [fαβ ] ❧➔ ❍❡r♠✐t fαβ z α z β + fβα z β z α ✳ ❚ø ✤â s✉② r❛ f (z, z) ♥❤➟♥ ❣✐→ trà t❤ü❝ ✈ỵ✐ ♠å✐ z ∈ Cn ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (z, z) = f (z, z) ❤❛② fαβ = f βα ✱ ✈ỵ✐ ♠å✐ α, β ∈ Ωn,d ✳ ✣✐➲✉ ♥➔② ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠❛ tr➟♥ [fαβ ] ❧➔ ♠❛ tr➟♥ ❍❡r♠✐t✳ ▼➺♥❤ ✤➲ ✷✳✷✳✽✳ (i) Ch[z, z] ❧➔ ♠ët ✤↕✐ sè t❤ü❝ ❝♦♥ ❝õ❛ ✤↕✐ sè C[z, z]✳ ∗✲✤↕✐ (ii) Ch [z, z] ❧➔ ♠ët (iii) Ch [z, z] ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ợ số ợ ố ủ t tữớ f = f✳ f = sup |fαβ |✳ ❉♦ α,β∈Ωn,d ✤â Ch [z, z] ❦❤æ♥❣ ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ fαβ z α z β , g(z, z) = α,β∈Ωn,d fδθ z δ z θ ∈ Ch [z, z] t❛ δ,θ∈Ωn,d fαβ gδθ z γ z η ✳ ❱➻ f, g ∈ Ch [z, z] ♥➯♥ fαβ = γ,η sè✳ ✭✐✮ ❚❛ ❝â Ch [z, z] ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ C[z, z]✳ ❱ỵ✐ ♠å✐ f (z, z) = ❝â f g = C ∗ ✲✤↕✐ α+δ=γ, β+θ=η f βα ✈➔ gδθ = g θδ ✳ ❉♦ ✤â✱ fαβ gδθ = f βα g θδ = fβα gθδ ❚ø ✤â s✉② r❛ f g ∈ Ch [z, z]✳ ❱➟② Ch [z, z] ❧➔ ♠ët ✤↕✐ sè t❤ü❝ ❝♦♥ ❝õ❛ C[z, z]✳ ✭✐✐✮ ❘ã r➔♥❣ f → f ∗ = f ❧➔ ♠ët ♣❤➨♣ ✤è✐ ❤ñ♣ tr➯♥ Ch [z, z]✳ ✭✐✐✐✮ ▲➜② f (z, z) = z + zz + z ✈➔ g(z, z) = z + z ∈ Ch [z, z]✳ ❚❛ ❝â ✹✽ f = g = ✈➔ f g = z + 2z z + 2zz + z ✳ ❑❤✐ ✤â fg = > = f g ❱➟② Ch [z, z] ❦❤æ♥❣ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✤➣ ❝❤♦✳ ❉♦ ✤â Ch [z, z] ❦❤æ♥❣ ♣❤↔✐ C ∗ ✲✤↕✐ sè✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ✷✳✸ ❚➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝ ❈→❝ ✤❛ t❤ù❝ tr➯♥ ♠ët tr÷í♥❣ ✈➔ ❝→❝ ♠❛ tr➟♥ sè ❝ô♥❣ ❝â t❤➸ ①❡♠ ❧➔ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝ ✤➦❝ ❜✐➺t✳ ❱✐➺❝ ❦✐➸♠ tr❛ ❝→❝ t➟♣ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ð ❝→❝ ♠ư❝ ✷✳✶ ✈➔ ✷✳✷✳ Ð ✤➙②✱ ❝❤ó♥❣ tỉ✐ ❝❤➾ ①➨t t➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝ ❞↕♥❣ tê♥❣ q✉→t✳ ❳➨t ✤↕✐ sè ❝→❝ tr tự Msìs tr trữớ K (i) Msìs ởt số ợ ố ❤ñ♣ ❝❤♦ ❜ð✐ A∗ (x) = AH (x) ❝õ❛ ♠❛ tr➟♥ T AH (x) = A (x) ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❧✐➯♥ ❤đ♣ A(x) ∈ Ms×s ✳ (ii) (Ms×s , ❇❛♥❛❝❤ ❦❤✐ tr♦♥❣ ✤â p) ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤ ❦❤✐ p=1 ✈➔ ❦❤æ♥❣ ❧➔ ✤↕✐ sè p > 1✳ (iii) (Ms×s , ❈❤ù♥❣ ♠✐♥❤✳ p) ❦❤ỉ♥❣ ❧➔ C ∗ ✲✤↕✐ sè ✈ỵ✐ ♠å✐ p ≥ 1✳ ✭✐✮ ❳❡♠ ❱➼ ❞ö ✶✳✺✳✷✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✺ t❛ ❝â ✭✐✐✮ ✈➔ ✭✐✐✐✮✳ ❚❛ tâ♠ t➢t ❝→❝ ❦➳t q✉↔ tr ữỡ ữ ữợ tr ✤â ❝→❝ æ ❝â ❞➜✉ ✏ ✑ ♥❣❤➽❛ ❧➔ t➟♣ ð ❝ët ✤â ✏t❤ä❛ ♠➣♥✑ ❝➜✉ tró❝ ð ❞á♥❣ t÷ì♥❣ ù♥❣✳ ✣↕✐ sè ✣↕✐ sè ❇❛♥❛❝❤ ∗✲✤↕✐ sè C ∗ ✲✤↕✐ sè · √ √ max p √ √ √ (1 ≤ p < ∞) √ · √ √ · F · √ √ sup √ √ p=1 √ R[x], C[z] · √ p>1 √ p · √ √ sup C[z, z], Ch [z, z] ❇↔♥❣ ✷✳✷✿ ❇↔♥❣ tâ♠ t➢t ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ❝õ❛ ♠ët sè t➟♣ ✈ỵ✐ ❝→❝ ❝❤✉➞♥ ❦❤→❝ ♥❤❛✉✳ √ √ ✱ √ √ · · ∞ √ F √ · Rn×n , Cn×n (n > 1) √ √ p=1 √ · √ p>1 √ p Ms×s ✹✾ ✺✵ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤✐ t✐➳t✱ ❝â ❤➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè✿ ✤↕✐ sè✱ ✤↕✐ sè ❇❛♥❛❝❤✱ ∗✲✤↕✐ sè ✈➔ C ∗ ✲✤↕✐ sè✳ ❚r♦♥❣ ✤â✱ ❝→❝ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❤ì♥ s ợ t t ỗ õ ỵ ✶✳✻✳✻✱ ▼➺♥❤ ✤➲ ✶✳✻✳✽✱ ▼➺♥❤ ✤➲ ✶✳✻✳✾✱ ▼➺♥❤ ✤➲ ✶✳✻✳✶✵ ✭①❡♠ ❈❤÷ì♥❣ ✶✮✳ ✷✳ ❑✐➸♠ tr❛ sü t❤ä❛ ♠➣♥ ❝→❝ ❝➜✉ tró❝ ❧✐➯♥ q✉❛♥ ✤➳♥ C ∗ ✲✤↕✐ sè ♥➯✉ tr➯♥ ❝❤♦ ❝→❝ t➟♣✿ t➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ t❤ü❝ ✈➔ ♣❤ù❝✱ t➟♣ ❝→❝ ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ sè t❤ü❝ ✈➔ ♣❤ù❝✱ ✤➦❝ ❜✐➺t ❝â t➟♣ ❝→❝ ✤❛ t❤ù❝ ❤é♥ t↕♣✱ ✈➔ t➟♣ ❝→❝ ♠❛ tr➟♥ ✤❛ t❤ù❝✳ ❚r♦♥❣ ✤â✱ ù♥❣ ✈ỵ✐ ♠é✐ t➟♣ ❝❤ó♥❣ tỉ✐ ①➨t ♥❤✐➲✉ ❝❤✉➞♥ ❦❤→❝ ♥❤❛✉ ✭①❡♠ ❈❤÷ì♥❣ ✷✮✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② tâ♠ t➢t tr♦♥❣ ❇↔♥❣ ✷✳✷✳ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❑✳❘✳ ❉❛✈✐❞s♦♥✱ ❈✯✲❛❧❣❡❜r❛s ❜② ❡①❛♠♣❧❡✱ ◆❳❇ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✲ ✐❝❛❧ ❙♦❝✐❡t②✱ ✶✾✾✻✳ ❬✷❪ ❑✳ ❙❝❤☎ ✉❞❣❡♥✱ ◆♦♥❝♦♠♠✉t❛t✐✈❡ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✲s♦♠❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ❛♥❞ ❢✐rst ✐❞❡❛s✱ ◆❳❇ ❙♣r✐♥❣❡r ◆❡✇ ❨♦r❦✱ ✷✵✵✾✳ ❬✸❪ ▲➯ ❚❤❛♥❤ ❍✐➳✉✱ ❇➔✐ ❣✐↔♥❣ ❝❤✉②➯♥ ✤➲ ❚➼♥❤ t♦→♥ ♠❛ tr➟♥✱ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯✱ ❚r÷í♥❣ ✤↕✐ ❤å❝ ◗✉② ◆❤ì♥✱ ✷✵✶✽✳ ❬✹❪ ❚❤→✐ ❚❤✉➛♥ ◗✉❛♥❣✱ ❈ì sð ỵ tt t ố rữớ ◗✉② ◆❤ì♥✱ ✷✵✶✸✳ ❑❤♦❛ ❚♦→♥ ✈➔ ... ♠å✐ a, b, c ∈ A ✈➔ ♠å✐ λ ∈ K t❛ ❝â (i) [(a + I) + (b + I)] (c + I) = ((a + b) + I) (c + I) = (a + b )c + I = (ac + I) + (bc + I) = (a + I) (c + I) + (b + I) (c + I) (ii) (a + I)[(b + I) + (c + I)]... (A+B )C = AC + BC ✳ ✭✐✐✮ ✣➦t E = A(B + C) = [eij ] ∈ Kn×n ✳ ❚❛ ❝â n n eij = aik (bkj + ckj ) = k=1 n aik bkj + k=1 aik ckj k=1 ✈ỵ✐ ♠å✐ i, j t❤ä❛ ♠➣♥ ≤ i, j ≤ n✳ ❙✉② r❛ E = AB+AC ✳ ❱➟② A(B +C) =... C = [cij ] ∈ Kn×n ✈➔ ♠å✐ λ ∈ K✱ ✭✐✮ ●✐↔ sû D = (A + B )C = [dij ] ∈ Kn×n ✳ ❑❤✐ ✤â k=1 k=1 k=1 bik ckj aik ckj + (aik + bik )ckj = dij = n n n ✈ỵ✐ ♠å✐ i, j t❤ä❛ ♠➣♥ ≤ i, j ≤ n✳ ❉♦ ✤â D = AC +BC