Định lý hahn banach và đối ngẫu của một số không gian hàm

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❍❷■ ❍⑨ ✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❱⑨ ✣➮■ ◆●❼❯ ❈Õ❆ ▼❐❚ ❙➮ ❑❍➷◆● ●■❆◆ ❍⑨▼ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ❍⑨ ◆❐■✱ ✺✴✷✵✶✾ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❍❷■ ❍⑨ ✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❱⑨ ✣➮■ ◆●❼❯ ❈Õ❆ ▼❐❚ ❙➮ ❑❍➷◆● ●■❆◆ ❍⑨▼ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ❑❍➶❆ P ữợ ề ì ❍⑨ ◆❐■✱ ✺✴✷✵✶✾ ▲❮■ ❈❷▼ ❒◆ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥✱ ✈ỵ✐ sü ❝è ❣➢♥❣ t ụ ữ sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝æ ❝æ♥❣ t→❝ t↕✐ ❦❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐ ✷ ✈➔ ❝→❝ t❤➛② ❝æ ✤➣ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕②✱ tr✉②➲♥ ✤↕t ❝❤♦ ỳ tự qỵ ổ ụ ữ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ t❤í✐ ❣✐❛♥ ✈ø❛ q✉❛✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ❣✐→♦✱ t✐➳♥ s➽ ❇ị✐ ❑✐➯♥ ❈÷í♥❣✱ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ ❝ơ♥❣ ♥❤÷ ❝✉♥❣ ❝➜♣ ❝❤♦ ❡♠ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ♥➲♥ t↔♥❣ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❍↔✐ ữợ sỹ ữợ t ũ ữớ õ ữủ t ổ trũ ợ t ✤➲ t➔✐ ♥➔♦ ❦❤→❝✱❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❦❤â❛ ữủ ró ỗ ố ró r r ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❍↔✐ ❍➔ ✐ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✸ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✻ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❞➣② c0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❞➣② lp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❑❤æ♥❣ ❣✐❛♥ Lp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ỵ ố tr ổ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✷✶ ✷✳✷ ✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹ ▼ët sè ❤➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸ ✣è✐ ♥❣➝✉ tr♦♥❣ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ✸✶ ✸✳✶ ✣è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ lp , < p < ∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1 , ❦❤æ♥❣ ❣✐❛♥ l∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷✳✶✳ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷✳✷✳ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Lp (1 < p < ∞) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸ ✐✐ ❑➌❚ ▲❯❾◆ ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ▼Ð ✣❺❯ ✶✳ ỵ t t ởt ♥❣➔♥❤ ❝õ❛ ❣✐↔✐ t➼❝❤ t♦→♥ ❤å❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✤÷đ❝ tr❛♥❣ ❜à t❤➯♠ ♠ët ❝➜✉ tró❝ tỉ♣ỉ ♣❤ị ❤đ♣ ✈➔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ❣✐ú❛ ❝❤ó♥❣✳ ❈→❝ ❦➳t q✉↔ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ♥â t ữ ỵ tt ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ỵ tt t ỹ tr ữỡ t ỵ tt ỳ t t ỗ tứ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝õ❛ ❍✐❧❜❡rt✱ ❋r❡❞❤♦❧♠✱✳✳✳✱ ✤➳♥ ♥❛② ❣✐↔✐ t➼❝❤ ❤➔♠ t➼❝❤ ❧ơ② ✤÷đ❝ ♥❤ú♥❣ t❤➔♥❤ tü✉ q✉❛♥ trå♥❣ ✈➔ ♥â ✤➣ trð t❤➔♥❤ ❝❤✉➞♥ ♠ü❝ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ tự t r t ỵ ❧➔ ♠ët ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣✳ ◆â ❝❤♦ ♣❤➨♣ ♠ð rë♥❣ ❝õ❛ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❧➯♥ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ ✤â✱ ♥â ❝ơ♥❣ ❝❤ù♥❣ tä r➡♥❣ ❝â ✧✤õ✧ ❝→❝ ♣❤✐➳♠ ❤➔♠ ❧✐➯♥ tö❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ♠é✐ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✤➸ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❧➔ ❝â t❤➸✳ ◆â ✤÷đ❝ ✤➦t t➯♥ t❤❡♦ ❍❛♥s ❍❛❤♥ ✈➔ ❙t❡❢❛♥ ❇❛♥❛❝❤ ❧➔ ỳ ữớ ự ỵ ♥❤ú♥❣ ♥➠♠ ✶✾✷✵✳ ❱ỵ✐ ♠ư❝ t✐➯✉ ❧➔ ❤♦➔♥ t❤✐➺♥ ❦✐➳♥ tự t trữợ r trữớ ữủ sü ❣✐ó♣ ✹ ✤ï t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❇ị✐ ❑✐➯♥ ữớ tỹ õ tốt ợ ✤➲ t➔✐✿ ✏ ✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ ✈➔ ✤è✐ ♥❣➝✉ ❝õ❛ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✑ ✳ ✷✳ ▼ư❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ✤è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❜❛♥❛❝❤❀ ✤à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ t❤ü❝✱ ✤à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ ♣❤ù❝❀ ✤è✐ ♥❣➝✉ tr♦♥❣ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✿ ✲ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❧➼ t❤✉②➳t✳ ✲ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ✹✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥✿ ◆❣♦➔✐ ♣❤➛♥ ♠ư❝ ❧ư❝✱ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ ♣❤ö õ ỗ ữỡ ữỡ ởt số t❤ù❝ ❝❤✉➞♥ ❜à✧✳ ❈❤÷ì♥❣ ♥➔② s➩ ✤✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② c0 , lp , Lp ✳ ❈❤÷ì♥❣ ✷✿✧✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤✧✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ✈➲ ✤è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ổ ỵ tỹ ỵ ự ởt số q ❈❤÷ì♥❣ ✸✿✧✣è✐ ♥❣➝✉ tr♦♥❣ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✧✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ✈➲ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ∞ p , < p < ∞✱ ✤è✐ , ✱ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Lp , < p < ∞ ❑❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② tr➯♥ ❝ì sð ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤÷đ❝ ❧✐➺t ❦➯ tr♦♥❣ ♣❤➛♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✣â♥❣ ❣â♣ ❝õ❛ ❡♠ t❤➸ ❤✐➺♥ ð ❝❤é✱ ❝õ♥❣ ❝è ♥➢♠ ✈ú♥❣ ✤÷đ❝ ✤à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ ✈➔ ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ❧➼ ✺ ❍❛❤♥✲❇❛♥❛❝❤✳ ♥➢♠ ✤÷đ❝ t➼♥❤ ✤è✐ ♥❣➝✉ ❝õ❛ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠✳ ❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❦❤ỉ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❊♠ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ♣❤↔♥ ❜✐➺♥ tø q✉þ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì E tr➯♥ tr÷í♥❣ K ❧➔ ♠ët →♥❤ ①↕ f : E −→ [0, ∞) x −→ x t❤ä❛ t t ữợ t tự t❛♠ ❣✐→❝✮ x + y ≤ x + y ✭ ∀ ①✱② ∈ E ✮ ✭✐✐✮ αx = |α| x (∀α ∈ K, x ∈ E) ✭✐✐✐✮ x = ⇒ x = ✭∀x∈ E ✮ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E tr K ũ ợ x tr õ ữủ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✭tr➯♥ K✮ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✭E, ✮ ❚r÷í♥❣ K ❧➔ tr÷í♥❣ R ❤♦➦❝ tr÷í♥❣ C ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼ët ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ❧➔ ♠ët →♥❤ ①↕ p ✿ E → ❬ 0✱ ∞✮ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✭p✭x + y ✮ ≤ p✭x✮ + p✭y ✮ ✈ỵ✐ ∀ x✱ y ∈ ❊✮ ✈➔ t t t t ữỡ px = ||p(x) ợ x ∈ E ✈➔ α ∈ K✮✳ ✷✽ α(x0 ) = ✈➔ α(x) = ✈ỵ✐ ♠å✐ x ∈ M ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t d = dist(x0 , M ) = inf { x − x0 : x ∈ M } ❱➻ ▼ ✤â♥❣ ✈➔ x0 ∈ / M, d > 0✳ ❚✐➳♣ t❤❡♦ ✤➦t M1 = {x + λx0 : x ∈ M, λ ∈ K} ✈➔ ①→❝ ✤à♥❤ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr➯♥ M1 ❝❤♦ ❜ð✐ α : M1 → K α(x + λx0 ) = λ ❚❛ t❤➜② r➡♥❣ ✭λ = 0✮ x + λx0 = |λ| x + x0 ≥ |λ|d λ ✈➔ ❞♦ ✤â |α(x + λx0 )| = |λ| ≤ x + λx0 d ✭ ✤ó♥❣ ✈ỵ✐ ♠å✐ x + λx0 ∈ M1 ✱ ❝↔ ❦❤✐ λ = 0✮✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r➡♥❣ α ❧➔ ♠ët ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ M1 ✱ ❞♦ ✤â α ∈ M1∗ ✳ ❚❤❡♦ ✤à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤✱ t❛ ❝â t❤➸ ♠ð rë♥❣ α tỵ✐ ♠ët ♣❤➛♥ tû ❝õ❛ E ∗ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t tt ỵ E ởt ổ ổ t t tỗ t↕✐ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ α : E → K ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ K✱ ♣❤↔✐ ❝â ♠ët ❝ì sð ✤↕✐ sè ✭❤♦➦❝ ❝ì sð ❍❛♠❡❧✮✳ ❈❤♦ {ei : i ∈ I} ❧➔ ♠ët ❝ì sð ✈➔ ♥❤➢❝ ❧↕✐ r➡♥❣ ❦❤✐ ✤â ♠é✐ x ∈ E ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ t❤à ✭t❤❡♦ ♠ët ❝→❝❤ ❞✉② ♥❤➜t✮ ♥❤÷ ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥ x = xi1 ei1 + xi2 ei2 + + xin ein ❝õ❛ ❝→❝ ♣❤➛♥ tỷ ỡ s t ữợ ữ s xi ei x= i∈I tr♦♥❣ ✤â xi = ♥➳✉ i = ij ❝❤♦ ♠ët sè sè ❥✳ ●✐í t❛ ①➨t ❝→❝ ♣❤✐➳♠ ❤➔♠ ❤➺ sè αi : E −→ K x −→ xi ❑❤✐ ✤â ➼t ♥❤➜t ♠ët tr♦♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤ αi (i ∈ I) ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ ◆➳✉ t➜t ❝↔ αi ✤➲✉ ❧✐➯♥ tư❝✱ t❛ ❧➜② ♠ët ❞➣② ✈æ ❤↕♥ (in )n ❝õ❛ ❝→❝ ♣❤➛♥ tû ♣❤➙♥ ❜✐➺t ❝õ❛ I ✳ ✣➦t n xn = j=1 eij 2j eij ❉➵ t❤➜② r➡♥❣ (xn )n tr E tỗ t limn→∞ xn = x ∈ ✸✵ E ✳ ◆➳✉ ♠é✐ αij ❧➔ ❧✐➯♥ tö❝ αij (x) = lim αij (xn ) = n 2j eij ❧➔ ❦❤→❝ ✈ỵ✐ ✈æ ❤↕♥ ij ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈➻ x= αi (x)ei i∈I ❧➔ ♠ët tê♥❣ ❤ú✉ ❤↕♥ ✸✶ ❈❤÷ì♥❣ ✸ ✣è✐ ♥❣➝✉ tr♦♥❣ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ✸✳✶ ✣è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ lp, < p < ∞ ❱ỵ✐ < p < ∞✳ ✣è✐ ♥❣➝✉ ❝õ❛ lp ❧➔ lq ✱ tr♦♥❣ ✤â p + q = ❝â ♥❣❤➽❛ ❧➔ ♠å✐ ♣❤✐➳♥ ❤➔♠ t✉②➳♥ t tr lp õ t t ữợ ∞ f (x) = αk xk k=1 ✈ỵ✐ α = {αn } ∈ lq ✈➔ f = α q ✳ ▼➺♥❤ ✤➲ ✸✳✶✳ ◆➳✉ E = lp ✈➔ < p < ∞✱ ❦❤✐ ✤â ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ♥â 1 (lp )∗ = lq ( + = 1) p q ❈❤➼♥❤ ①→❝ ❤ì♥✱ lq ❧➔ ✤➥♥❣ ❝➜✉ ✈ỵ✐ (lp )∗ ❜ð✐ s♦♥❣ →♥❤ T : lq → (lp )∗ , ✸✷ tr♦♥❣ ✤â (bn )n ∈ lq lp −→ K T ((bn )n ) : ∞ (an )n −→ an bn n=1 ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ✤➛✉ t✐➯♥ t❛ t❤➜② r➡♥❣ →♥❤ ①↕ T ❧➔ ❝â ♥❣❤➽❛✳ ◆➳✉ t❛ ❝è ✤à♥❤ b = (bn )n ∈ lq ❦❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r ❝❤♦ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ ∞ an b n ≤ a p b q n=1 ✈ỵ✐ ♠é✐ a = (an )n ∈ lp ✳ ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② ❝❤✉é✐ (T (b))(a) ❧✉ỉ♥ ❤ë✐ tư ✈➔ ✈➻ t❤➳ T (b) : lp → K ❧➔ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤✳ ❉➵ t❤➜② r➡♥❣ ♥â ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r t❛ ✤â ❜➜t ✤➥♥❣ t❤ù❝ |T (b)(a)| ≤ a p b q ❝❤♦ t❤➜② T (b) ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ ✈➔ T (b) ❝â ❝❤✉➞♥ ❦❤æ♥❣ ✈÷đt q✉→ b q ✳ ❇ð✐ ✈➟② T (b) ♥➡♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ lp ✈➔ T (b) (lp )∗ ≤ b q✳ ❚✐➳♣ t❤❡♦ ❧➔ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤➥♥❣ t❤ù❝ ①↔② r❛ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝✳ ▼✉è♥ ✈➟② ❝❤ó♥❣ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ T (b) (lp )∗ ≥ b q ✳ ◆➳✉ b = ❦❤✐ ✤â t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❤♦➔♥ t♦➔♥ ✤ó♥❣✳ ❱➻ t❤➳✱ t❛ ❣✐↔ sû b = ✭♥❣❤➽❛ ❧➔ ❝â ♠ët sè n ✈ỵ✐ bn = 0✮✳ ❇➙② ❣✐í✱ ✈ỵ✐ < p < ∞✳ ◆➳✉ t❛ ❧➜② an = |bn |q (bn )−1 ✭❜➡♥❣ ✵ ♥➳✉ bn = 0✮✱ ❦❤✐ ✤â a = (an )n ∈ lp ✈➔ ∞ a p p |an | = n=1 p ∞ |bn | = n=1 ∞ 1/p qp−p 1/p q |bn | = n=1 = b q/p q ✸✸ ❜ð✐ ✈➻ (q − 1)p = q ✳ ❱➻ ∞ T (b)(a) = ∞ |bn |q = b qq an bn = n=1 n=1 ❉♦ ✤â ✭♥➳✉ b = 0✮ T (b) (lp )∗ b qq |T (b)(a)| ≥ = = b q/p a p b q ❚ø ✤â t❛ ❝â t❤➸ t❤➜② r➡♥❣ T (b) (lp )∗ q−q/p q = b q = b q ✳ ❱➻ ✈➟② T ❧➔ ♠ët →♥❤ ①↕ ❜↔♦ t♦➔♥ ❝❤✉➞♥ tø lq ✤➳♥ ♠ët t➟♣ ❝♦♥ ❝õ❛ (lp )∗ ✳ ❚❤ü❝ t➳ T ❝ô♥❣ ❧➔ t✉②➳♥ t➼♥❤ ✭ ✈➔ ✈➻ t❤➳ ♥â ❧➔ ♠ët ♣❤➨♣ t✉②➳♥ t➼♥❤ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ (lp )∗ ✮ ❜ð✐ ✈➻ ❦❤→ ❞➵ ❞➔♥❣ ✤➸ ❦✐➸♠ tr❛ r➡♥❣ ✈ỵ✐ b, ˜b ∈ lq ✈➔ λ ∈ K✱ t❛ ❝â T (λb + ˜b) = λT (b) + T (˜b)✳✭ →♣ ❞ư♥❣ ❝↔ ❤❛✐ ✈➳ ❝❤♦ tị② þ a ∈ lp t❤➻ t❤➜② ✤ó♥❣✮ ▲➜② α ∈ (lp )∗ ❦❤✐ ✤â ①→❝ ✤à♥❤ b = (bn )n ❜ð✐ bn = α(en )tr♦♥❣ ✤â en t❛ ❤✐➸✉ ❧➔ ❞➣② (0, 0, , 1, 0, ) ❝õ❛ t➜t ❝↔ sè ✵ trø sè ✶ ð ✈à tr➼ t❤ù ♥✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ b ∈ lq ✈➔ T (b) = ú ỵ t r ợ ❞➣② a = (a1 , a2 , a3 , , an , 00, ) ✈ỵ✐ ❤ú✉ ❤↕♥ ❜➜t ❦➻ ❝→❝ sè ❦❤→❝ ✵✱ a ∈ lp ✈➔ ❛ ❧➔ ♠ët tê ❤ñ♣ ❤ú✉ ❤↕♥ a = a1 e1 + a2 e2 + + an en ❝õ❛ ❝→❝ sè ej ✳ ❑❤✐ ✤â t❛ ❝â t❤➸ →♣ ❞ö♥❣ α ❝❤♦ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❞ị♥❣ t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ α t❤➜② r➡♥❣ α(a) = a1 b1 + a2 b2 + + an bn ✸✹ ▲➜② aj = |bj |q (bj )−1 ✈ỵ✐ j = 1, 2, , n ✭ ✈➔ aj = 0, j > n✮✳ ❚❛ ❝â t❤➸ t➼♥❤ ✤÷đ❝ ♠ët ❝→❝❤ ❞➵ ❞➔♥❣ ❤ì♥ n a p 1/p |bj |q = j=1 ✈➔ n α(a) = n |bj |q aj bj = j=1 j=1 ❇ð✐ ✈➟② α (lp )∗ n j=1 |bj |q |α(a)| = ≥ a p n j=1 |bj |q 1− p1 n p |bj |q = n 1/q |bj |q = j=1 j=1 ✭❝❤♦ ✤➳♥ ❦❤✐ ♠ët sè bj = tr♦♥❣ ♠✐➲♥ ≤ j ≤ n✮✳ ◆➳✉ ❣✐í t❛ ❝❤♦ n → ∞ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝✱ t❛ t➻♠ b ∈ lq ✭tù❝ ❧➔ b q ≤ α (lp )∗ ✮✳ ❱➻ ✈➟② ❣✐í t❛ ❝â t❤➸ ♥â✐ T (b) ∈ (lp )∗ ✈➔ ❞➵ t❤➜② r➡♥❣ T (b)(en ) = bn = α(en )✳ ❚❤❡♦ t➼♥❤ t✉②➳♥ t➼♥❤ ❝❤♦ ♠å✐ ❞➣② ❜➜t ❦➻ ❤ú✉ n ❤↕♥ ❝→❝ sè ❦❤→❝ ✵ a = aj ej ∈ lp t❛ ❝â T (b)(a) = α(a) j=1 ❈✉è✐ ❝ò♥❣✱ ❝→❝ ❞➣② ❤ú✉ ❤↕♥ ❝→❝ sè ❦❤→❝ ✵ ❧➔ trò ♠➟t tr♦♥❣ lp ✈➻ ✈ỵ✐ ❜➜t ❦➻ x = (xn )n ∈ lp ❦❤→ rã r➔♥❣ r➡♥❣ ∞ x − (x1 , x2 , , xn , 0, 0, ) p 1/p p |xj | = →0 j=n+1 ❦❤✐ n → ∞✳ ❱➻ T (b) ✈➔ α ❧➔ ❧✐➯♥ tö❝ tr➯♥ lp ✈➔ ❜➡♥❣ ♥❤❛✉ tr➯♥ ♠ët t➟♣ ❦➼♥✱ ♥â ♣❤↔✐ ❧➔ T (b) = α✳ ✸✺ ✸✳✷ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1, ❦❤æ♥❣ ❣✐❛♥ l∞ ✸✳✷✳✶✳ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1 ▼➺♥❤ ✤➲ ✸✳✷✳ ❑❤æ♥❣ ❣✐❛♥ (l1)∗ ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ l∞ ❤❛② ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1 ❧➔ l∞ ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤ü❝ ✈➟②✱ ✈ỵ✐ ♠é✐ ♣❤➛♥ tû u = (un ) ∈ l∞ ✱ ♣❤✐➳♥ ❤➔♠ ∞ un ξn (x = (ξn ) ∈ l1 ) fu (x) = n=1 ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ l1 ✱ ❜ð✐ ✈➻ ❝❤✉é✐ ð ✈➳ ♣❤↔✐ ❧➔ ❤ë✐ tö✱ t❤➟t ✈➟②✿ ∞ |fu (x)| ≤ ∞ |un |.|ξn | ≤ sup |un | n n=1 |ξn | = u ∞ x n=1 ❘ã r➔♥❣ fu ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤ù♥❣ tä r➡♥❣ fu ≤ u ✭✸✳✶✮ ∞ ◆❣÷đ❝ ❧↕✐✱ f (l1 ) tũ ỵ ợ x = (ξn ) ∈ l1 t❛ ✤➲✉ ❝â ∞ f (x) = f ( ∞ ξn en ) = ( n=1 ∞ f (en )ξn = ( n=1 un ξn ), n=1 tr♦♥❣ ✤â un = f (en ) ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ x✳ ❚ø ❜➜t ✤➡♥❣ t❤ù❝ un = |f (en )| ≤ f en = f , ✸✻ t❛ t❤➜② r➡♥❣ ❞➣② u = (un ) ❧➔ ❜à ❝❤➦♥✱ tù❝ ❧➔ u ∈ l∞ ✈➔ u ∞ = sup |un | ≤ f 1≤n