Header Page of 54 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI HOÀNG DUY THẮNG ĐỐI NGẪU CỦA KHUNG KẾT HỢP LUẬN VĂN THẠC SĨ TOÁN HỌC HÀ NỘI, 2018 Footer Page of 54 Header Page of 54 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI HOÀNG DUY THẮNG ĐỐI NGẪU CỦA KHUNG KẾT HỢP Chun ngành: Tốn Giải tích Mã số: 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC Người hướng dẫn khoa học: TS Nguyễn Quỳnh Nga HÀ NỘI, 2018 Footer Page of 54 HÀ NỘI, 2012 Header Page of 54 ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ ❚❙✳ ý ữớ ổ ữợ t t t ữợ tổ õ t ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ ỗ ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❍♦➔♥❣ ❉✉② ❚❤➢♥❣ Footer Page of 54 Header Page of 54 ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ữợ sỹ ữợ ❚❙✳ ◆❣✉②➵♥ ◗✉ý♥❤ ◆❣❛✱ ❧✉➟♥ ✈➠♥ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐✿✏✣è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤đ♣ ✑ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ sü ♥❤➟♥ t❤ù❝ ✈➔ t➻♠ ❤✐➸✉ ❝õ❛ ❜↔♥ t❤➙♥ t→❝ ❣✐↔✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❍♦➔♥❣ ❉✉② ❚❤➢♥❣ Footer Page of 54 Header Page of 54 ✷ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✸ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✻ ✶✳✶ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✻ ✶✳✷ ❑❤✉♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ ✷✹ ✷✳✶ ❑❤✉♥❣ ❦➳t ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸ ❚♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❝➦♣ ❞➣② ❦➳♣ ❤ñ♣ ❇❡ss❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ Footer Page of 54 ✹✹ ✹✺ Header Page of 54 ✸ ▼ð ✤➛✉ ✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐ ❚r♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ♥❤➜t ❧➔ ❦❤→✐ ♥✐➺♠ ❝ì sð✱ ♥❤í ✤â ♠é✐ ✈❡❝tì tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝â t❤➸ ✈✐➳t ♥❤÷ tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð✳ ❚✉② ♥❤✐➯♥ ✤✐➲✉ ❦✐➺♥ ✤➸ trð t❤➔♥❤ ❝ì sð ❧➔ ❦❤→ ❝❤➦t✱ ❦❤ỉ♥❣ ❝❤♦ ♣❤➨♣ sü ♣❤ư t❤✉ë❝ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð✳ ✣✐➲✉ ♥➔② ❧➔♠ ❝❤♦ ❦❤â t➻♠ ❤♦➦❝ t❤➟♠ ❝❤➼ ❧➔ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ❝→❝ ❝ì sð t❤ä❛ ♠➣♥ ♠ët sè s ỵ ú t❛ ✤✐ t➻♠ ♠ët ❝ỉ♥❣ ❝ư ❦❤→❝ ❧✐♥❤ ❤♦↕t ❤ì♥ ✈➔ ❦❤✉♥❣ ❝❤➼♥❤ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ♥❤÷ ✈➟②✳ ❑❤✉♥❣ ❝❤♦ ♣❤➨♣ ❝❤ó♥❣ t❛ ❜✐➸✉ ❞✐➵♥ ♠é✐ ♣❤➛♥ tû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤÷ ♠ët tê ❤đ♣ t✉②➳♥ t➼♥❤ ✭✈ỉ ❤↕♥✮ ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❦❤✉♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ✤á✐ ❤ä✐ t➼♥❤ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû ❦❤✉♥❣✳ ữủ ợ t 1952 ❉✉❢❢✐♥ ✈➔ ❆✳ ❈✳ ❙❝❤❛❡❢❢❡r ❬✻❪ tr♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ộ rr ổ ỏ ỗ t ❦❤æ♥❣ ♥❤➟♥ r❛ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ t 30 trữợ ổ tr t✐➳♣ t❤❡♦ ①✉➜t ❤✐➺♥✳ ❱➔♦ ♥➠♠ 1980 ❘✳ ❨♦✉♥❣ ✤➣ ✈✐➳t ❝✉è♥ s→❝❤ ❝â ♥❤ú♥❣ ❦➳t q✉↔ ❝ì ❜↔♥ ✈➲ ❦❤✉♥❣✱ ❧↕✐ tr♦♥❣ ♥❣ú ❝↔♥❤ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r ❦❤æ♥❣ ✤✐➲✉ ❤á❛✳ ❚✉② ♥❤✐➯♥ ♣❤↔✐ ✤➳♥ ♥➠♠ 1986, s❛✉ ❜➔✐ ❜→♦ ❝õ❛ ■✳ ❉❛✉❜❡❝❤✐❡s✱ ❆✳ ●r♦ss♠❛♥♥ ✈➔ ❨✳ ▼❡②❡r [5] t❤➻ ỵ tt ợ ữủ q t➙♠ rë♥❣ r➣✐✳ ❑❤✉♥❣ ❝â Footer Page of 54 Header Page of 54 ✹ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr ỷ ỵ t ỵ tt t ❞ú ❧✐➺✉✱✳ ✳ ✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❝→❝❤ ✤➸ ①➙② ỹ trữợ t ỹ ỳ t ♣❤➛♥ ✤à❛ ♣❤÷ì♥❣ s❛✉ ✤â s➩ ①➙② ❞ü♥❣ ❦❤✉♥❣ t♦➔♥ ❝ư❝ tø ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥ ♥➔②✳ ▼ët ÷✉ ✤✐➸♠ ❝õ❛ þ t÷ð♥❣ ♥➔② ❧➔ ❣✐ó♣ ❝❤♦ ✈✐➺❝ ①➙② ❞ü♥❣ ❝→❝ ❦❤✉♥❣ ❝â ♥❤ú♥❣ ù♥❣ ❞ư♥❣ ✤➦❝ ❜✐➺t ❞➵ ❞➔♥❣ ❤ì♥✳ ❚ø ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❦❤✉♥❣ ✈➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♥â✱ ❈❛s❛③③❛ ✈➔ ❑✉t②♥✐♦❦ [2] ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤✉♥❣ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ✭❋r❛♠❡ ♦❢ s✉❜s♣❛❝❡s✮✳ ▼ët t➯♥ ❣å✐ ❦❤→❝ ❝õ❛ ❦❤✉♥❣ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❧➔ ❦❤✉♥❣ ❦➳t ❤đ♣ ✭❋✉s✐♦♥ ❢r❛♠❡✮✳ ❑❤✉♥❣ ❦➳t ❤đ♣ ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ tê♥❣ q✉→t ❤â❛ ❝õ❛ ❦❤✉♥❣✳ ❑❤✉♥❣ ❦➳t ❤ñ♣ ❧➔ ♠ët ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ✤➸ ❣✐↔✐ q✉②➳t ♥❤ú♥❣ ❜➔✐ t♦→♥ ỷ ỵ t t tờ ủ ỳ ❧✐➺✉✳ ❑❤✉♥❣ ❦➳t ❤ñ♣ ❝❤♦ ♣❤➨♣ ♣❤➙♥ t➼❝❤ t➼♥ ❤✐➺✉ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝→❝ ♣❤➨♣ ❝❤✐➳✉ tr➯♥ ♠ët ❤å ổ ỗ ợ ố t➻♠ ❤✐➸✉ s➙✉ s➢❝ ❤ì♥ ✈➲ ❦❤✉♥❣ ❦➳t ❤đ♣✱ ✤➦t t ố õ ữủ sỹ ỗ ỵ ữợ ý tổ qt ✤à♥❤ ❝❤å♥ ✏✣è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✑ ❧➔♠ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❝❛♦ ❤å❝ ❝õ❛ ♠➻♥❤✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ tê♥❣ q✉❛♥ ✈➲ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤đ♣✳ ✸✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉ • ◆➢♠ ✈ú♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t♦→♥ tû t✉②➳♥ t tr ổ rt ỵ tt tờ qt tr ổ rt ữủ ởt sè ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ ♠ët sè ❦➳t q✉↔ ✈➲ ❝→❝ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ t♦→♥ tû ❦❤✉♥❣ Footer Page of 54 Header Page of 54 ✺ ❝õ❛ ❝➦♣ ❞➣② ❦➳t ❤đ♣ ❇❡ss❡❧✳ ✹✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ • ❑❤✉♥❣ ❦➳t ❤đ♣✳ • t tr ữợ q✉❛♥ ✤➳♥ ❦❤✉♥❣ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❙û ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠ ✤➸ t✐➳♣ ❝➟♥ ✈➜♥ ✤➲✳ ❚❤✉ t❤➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜➔✐ ❜→♦ ♠ỵ✐ tr♦♥❣ ✈➔ ữợ tỵ✐✳ ✻✳ ✣â♥❣ ❣â♣ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ✈➲ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ Footer Page of 54 Header Page of 54 ✻ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ s➩ ♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤✉♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ❞ü❛ tr➯♥ ❝→❝ t➔✐ ❧✐➺✉ [4], [8] ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ sû ❞ư♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❣✐↔ t❤✐➳t ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✶ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❚♦→♥ tû t✉②➳♥ t➼♥❤ T tø ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt K ❧➔ ❧✐➯♥ tö❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❜à ❝❤➦♥✱ tỗ t số c > s ❝❤♦ T(x) ≤ c x , ✈ỵ✐ ♠å✐ x ∈ H ❑➼ ❤✐➺✉ L (H, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø H ✈➔♦ K ❑❤✐ H = K t❤➻ L (H, K) ✤÷đ❝ ❦➼ ❤✐➺✉ ✤ì♥ ❣✐↔♥ ❧➔ L (H) ❈❤✉➞♥ ❝õ❛ T ∈ L(H, K) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❤➡♥❣ sè c ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ T(x) ≤ c x , ợ x H õ ởt tữỡ ữỡ T = sup { T (x) : x ∈ H, x ≤ 1} = sup { T (x) : x ∈ H, x = 1} Footer Page of 54 Header Page 10 of 54 ✼ ●✐↔ sû H, L, K ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ◆➳✉ T L (H, K) t tỗ t t ởt tỷ T ∗ ∈ L (K, H) s❛♦ ❝❤♦ ▼➺♥❤ ✤➲ ✶✳✶✳✶✳ ∈ T ∗ (x), y = x, T (y) , x ∈ K, y ∈ H ❍ì♥ ♥ú❛✱ ✐✮ (aS + bT )∗ = aS ∗ + bT ∗ ✐✐✮ (RS)∗ = S ∗R∗ ✐✐✐✮ (T ∗ )∗ = T ✐✈✮ I ∗ = I, tr♦♥❣ ✤â I ❧➔ t tỷ ỗ t tở L(H) T ♥❣❤à❝❤ t❤➻ T ∗ ❝ô♥❣ ❦❤↔ ♥❣❤à❝❤ ✈➔ (T −1)∗ = (T ∗)−1, tr♦♥❣ ✤â S, T ∈ L (H, K) , R ∈ L (K, L) ✈➔ a, b ∈ C ❚♦→♥ tû T ∗ ð ▼➺♥❤ ✤➲ 1.1.1 ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ ❤đ♣ ❝õ❛ t♦→♥ tû T ▼➺♥❤ ✤➲ ✶✳✶✳✷✳ ●✐↔ sû T ∈ L(H, K) ✈➔ S ∈ L (K, L) ❑❤✐ ✤â ✐✮ T (x) ≤ T ✐✐✮ ST ≤ S ✐✐✐✮ T = T∗ ✐✈✮ TT∗ = T x T ❚♦→♥ tû T ∈ L(H) ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû tü ❧✐➯♥ ❤đ♣ ♥➳✉ T = T ∗ , ❧➔ ✉♥✐t❛ ♥➳✉ T ∗ T = T T ∗ = I T ✤÷đ❝ ❣å✐ ❧➔ ❞÷ì♥❣ (❦➼ ❤✐➺✉ T ≥ 0) ♥➳✉ T (x) , x ≥ ✈ỵ✐ ♠å✐ x ∈ H T, K ∈ L(H), T ≥ K ♥➳✉ T − K ú ỵ r ợ ộ T L(H) t❤➻ T ∗ T (x) , x = T (x) , T (x) ≥ ✈ỵ✐ ♠å✐ x ∈ H ❉♦ ✤â T ∗ T ❧➔ ❞÷ì♥❣✳ Footer Page 10 of 54 Header Page 34 of 54 ✸✶ ∗ TW,υ (fk − fl ) ♥➯♥ fk − fl → ❦❤✐ ❧➔ ❞➣② ❈❛✉❝❤②✳ ❉♦ ✤â k, l → ∞✳ ❉♦ ✤â {fk } {fk } ❤ë✐ tö ✤➳♥ f ♥➔♦ ✤â t❤✉ë❝ H✳ ❱➻ ✈➟② ∗ ∗ ∗ TW,υ (fk ) → TW,υ (f ) = g ✳ ❉♦ ✤â Range TW,υ ✤â♥❣✳ ∗ ❇➙② ❣✐í t❛ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐ ❧➔ TW,υ ❧➔ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤✱ ❜à ❝❤➦♥✱ ✤ì♥ →♥❤ ✈➔ ❝â ♠✐➲♥ ❣✐→ trà õ (2) ữỡ ữỡ ợ (3) TW, t♦➔♥ ①→❝ ✤à♥❤ ❜à ❝❤➦♥ ✈➔ t♦➔♥ →♥❤✳ ❚❤❡♦ ✣à♥❤ ỵ 1.1.1(ii), tỗ t A > s ❝❤♦ TW,υ (f ) ≥ A f ✱ ✈ỵ✐ ♠å✐ f ∈ H✳ ❚❤❡♦ (2.3), υi πWi (f ) ≥A f ✈ỵ✐ ♠å✐ f ∈ H i∈I TW, tỗ t B > s❛♦ ❝❤♦ ❚❤❡♦ (2.3) t❛ s✉② r❛ υi2 πWi (f ) ≤B ∗ TW,υ (f ) ≤ √ B f ✳ f 2✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛ {(Wi , vi )}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣ H ụ tữỡ tỹ ữ ố ợ t s r tỗ t ởt t tỷ ❦➳t ✈ỵ✐ ♠é✐ ❦❤✉♥❣ ❦➳t ❤đ♣ t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t t÷ì♥❣ tü ♥❤÷ ❝❤ó♥❣ t❛ s➩ t❤➜② tr♦♥❣ ♠➺♥❤ ✤➲ t✐➳♣ t❤❡♦✳ ❱➼ ❞ư ❝❤ó♥❣ t❛ ❝ơ♥❣ ♥❤➟♥ ✤÷đ❝ ♠ët ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✸✳ ❈❤♦ {(Wi , vi )}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣ ❝õ❛ H ❑❤✐ ✤â t♦→♥ tû ❦❤✉♥❣ SW,υ ❝õ❛ {(Wi , vi )}i∈I ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ∗ (f ) = TW,υ {υi πwi (f )}i∈I = SW,υ (f ) = TW,υ TW,υ υi πWi (f ) i∈I ❚÷ì♥❣ tü ♥❤÷ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❝→❝ ▼➺♥❤ ✤➲ 1.2.2 ✈➔ ✣à♥❤ ỵ 1.2.2 t õ t q s ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈❤♦ {(Wi, vi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣ ❝õ❛ H ✈ỵ✐ ❝→❝ ❝➟♥ ❦❤✉♥❣ A ✈➔ B ✳ ❑❤✐ ✤â t♦→♥ tû ❦❤✉♥❣ SW,υ ❝õ❛ {(Wi, vi)}i∈I ❧➔ t♦→♥ tû ❦❤↔ ♥❣❤à❝❤✱ tü ❧✐➯♥ ❤đ♣ ❞÷ì♥❣ tr➯♥ H ✈ỵ✐ AI ≤ SW,υ ≤ BI ◆❣♦➔✐ r❛ ❝❤ó♥❣ t❛ ❝â ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝ −1 f= υi2 SW,υ (πW (f )) ✈ỵ✐ ♠å✐ f ∈ H ▼➺♥❤ ✤➲ ✷✳✶✳✷✳ i i∈I Footer Page 34 of 54 Header Page 35 of 54 ✸✷ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ❜➜t ❦➻ f ∈ H, ❝❤ó♥❣ t❛ ❝â υi2 πWi (f ) , f SW,υ (f ), f = i∈I υi2 πWi (f ) , f = i∈I = i∈I υi2 πW (f ) , f i υi2 πWi f, πWi f = i∈I υi2 = πWi (f ) i∈I ❚ø ✤â s✉② r❛ r➡♥❣ SW,υ ❧➔ ♠ët t♦→♥ tû ❞÷ì♥❣✳ ❱➻ ✈➟② SW,υ ❧➔ tü ❧✐➯♥ ❤đ♣✳ ❈❤ó♥❣ t❛ t➼♥❤ t♦→♥ t❤➯♠ Af, f = A υi2 ≤ f πWi (f ) i∈I = SW,υ (f ), f ≤ Bf, f ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② r➡♥❣ AI ≤ SW,υ ≤ BI ✈➔ ✈➻ ✈➟② SW,υ ❧➔ ♠ët t♦→♥ tû ❦❤↔ ♥❣❤à❝❤ tr➯♥ H ❈✉è✐ ❝ò♥❣ t❛ ❝â ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝ s✉② trü❝ t✐➳♣ tø −1 f = SW,υ (SW,υ (f )) = i∈I −1 υi2 SW,υ (πWi (f )) ✈ỵ✐ ♠å✐ f ∈ H ❍➺ t❤ù❝ ♥➔② ❝❤➾ r❛ r➡♥❣ ❤å ❝→❝ t♦→♥ tû −1 υi2 SW,υ πW i i∈I ❧➔ ♠ët ❦❤❛✐ tr✐➸♥ ❝õ❛ t tỷ ỗ t rst tt r➡♥❣ ♠ët ❤å ❝→❝ t♦→♥ tû ❜à ❝❤➦♥ {Ti }i∈I ữủ ởt tr t tỷ ỗ ♥❤➜t tr➯♥ H ♥➳✉ ✈ỵ✐ ♠å✐ f ∈ H t❛ ❝â f= Ti f i∈I Footer Page 35 of 54 Header Page 36 of 54 ✸✸ ✭ ✈➔ ❝❤✉é✐ ❤ë✐ tư ♠ët ❝→❝❤ ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ✈ỵ✐ ♠å✐ f ∈ H ✮✳ ❍å −1 SW,v Wi , vi i∈I ✤÷đ❝ ❣å✐ ❧➔ ❦❤✉♥❣ ❦➳t ❤đ♣ ✤è✐ ♥❣➝✉✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦❤✉♥❣ ❦➳t ❤ñ♣ ✤è✐ ♥❣➝✉ ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ ❈❛s❛③③❛ ✈➔ ❑✉t②♥✐♦❦ [2] ✤➣ sû ❞ö♥❣ ❦➳t q✉↔ s❛✉✿ ❈❤♦ {(Wi, υi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣ ✈➔ ❝❤♦ T : H → H ❧➔ t♦→♥ tû ❦❤↔ ♥❣❤à❝❤ tr➯♥ H✳ ❑❤✐ ✤â {(T Wi , υi )}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ▼➺♥❤ ✤➲ ✷✳✶✳✸✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ♥➔②✱ ❝→❝ t→❝ ❣✐↔ ✤➣ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝✿ πT W = T πW T −1 ✭✷✳✹✮ ◆❤➟♥ t õ ởt ố ợ ữỡ tr (2.4) ✤â ❧➔ ✈➳ ♣❤↔✐ ❧✉æ♥ ❧✉æ♥ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❧➯♥ T W ✱ ✭p ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ tø H ❧➯♥ K ♥➳✉ p2 = p✮✱ ♥❤÷♥❣ ♥â ❦❤æ♥❣ ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ ✭p ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ ♥➳✉ p2 = p, p = p∗ ) trø ❦❤✐ T ∗ T W ⊂ W ✭ ①❡♠ ▼ö❝ 2.2✮✱ ✣➦❝ ❜✐➺t✱ ✤✐➲✉ ♥➔② ❝â t❤➸ ①↔② r❛ ♥➳✉ T ❧➔ ♠ët t♦→♥ tû ✉♥✐t❛✳ ❚r♦♥❣ ▼ö❝ 2.2 ❝❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ (2.4) ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ❚→❝ ❣✐↔ ●❛✈r✉t❛[7] ✤➣ ❝❤ù♥❣ ♠✐♥❤ t❤❡♦ ❝→❝❤ ❦❤→❝ r➡♥❣ ▼➺♥❤ ✤➲ 2.1.3 ✈➝♥ ✤ó♥❣✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ♠ư❝ 2.2 ✤÷đ❝ tr➼❝❤ ❞➝♥ tø [7] ✷✳✷ ✣è✐ ♥❣➝✉ t ủ ổ tự (2.4) tữỡ ữỡ ợ πT W T = T π W ✣➛✉ t✐➯♥✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ s❛✉ ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ t♦→♥ tû✳ Footer Page 36 of 54 Header Page 37 of 54 ✸✹ ❈❤♦ T ∈ L(H) ✈➔ W ⊂ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ▼➺♥❤ ✤➲ ✷✳✷✳✶✳ ✭✐✮ πT W T = T πW ; ✭✐✐✮ T ∗ T W ⊂ W ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ⇒ ✭✐✐✮✳ ▲➜② h ∈ W ⊥✳ ❚❛ ❝â πT W T h = T πW h = ♥➯♥ T h ∈ (T W )⊥ = (T W )⊥ ◆❤÷♥❣ T h, T v = 0, ✈ỵ✐ ♠å✐ v ∈ W ⇔ h, T ∗ T v = 0, ✈ỵ✐ ♠å✐ v ∈ W ⇔ h ∈ (T ∗ T W )⊥ ❚ø ✤â W ⊥ ⊂ (T ∗ T W )⊥ ❤❛② T ∗ T W ⊂ W ✭✐✐✮ ⇒ ✭✐✮✳ ◆➳✉ v ∈ W t❤➻ πT W T v = T v ✈➔ T πW v = T v ◆➳✉ h ∈ W ⊥ t❤➻ T πW h = T = 0, ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t✱ t❛ ❝â h ∈ (T ∗ T W )⊥ ✳ ◆❤÷ ð tr➯♥✱ t❛ ❝â T h ∈ (T W )⊥ ✱ ❝❤♦ ♥➯♥ πT W T h = ❱ỵ✐ ♠å✐ f ∈ H, t❛ ❝â f = f1 + f2 tr♦♥❣ ✤â f1 ∈ W, f2 ∈ W⊥ ❚❛ ❝â πT W T f = πT W T f1 + πT W T f2 = T πW f1 + T W f2 = T W f ỗ t ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H✱ t♦→♥ tû ❦❤↔ ♥❣❤à❝❤ T L(H) ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ W ❝õ❛ H s❛♦ ❝❤♦ ❍➺ q✉↔ ✷✳✷✳✶✳ πT W T = T π W ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❧➜② H = R2, W = {(x, 0) | x ∈ R} ✈➔ T : R2 → R2 , T (x, y) = (x + y, y) Footer Page 37 of 54 ∈ Header Page 38 of 54 ✸✺ ✈ỵ✐ ♠å✐ (x, y) ∈ R2 ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ T ❧➔ T ∗ (x, y) = (x, x + y) ✈ỵ✐ ♠å✐ (x, y) ∈ R2 ❚❤➟② ✈➟②✱ ✈ỵ✐ ♠å✐ (x, y), (u, v) ∈ R2 t❛ ❝â T ∗ (x, y), (u, v) = (x, y), T (u, v) = (x, y), ((u + v), v) = x(u + v) + yv = xu + (x + y)v = (x, x + y), (u, v) ❚ø ✤â T ∗ (x, y) = (x, x + y) ❚❛ ❝â T ∗ T (x, 0) = T ∗ (x, 0) = (x, x) ∈ / W ✈ỵ✐ ♠å✐ x = ❚ø ▼➺♥❤ ✤➲ 2.2.1 s✉② r❛ πT W T = T πW ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ♠ö❝ ♥➔②✱ t❛ ❝➛♥ ❜ê ✤➲ s❛✉✿ ❇ê ✤➲ ✷✳✷✳✶✳ ✤â t❛ ❝â ❈❤♦ T ∈ L(H) ✈➔ W ⊂H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣✳ ❑❤✐ π W T ∗ = πW T ∗ πT W ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f ∈ H✱ t❤➻ f = πT W f + g, g ∈ (T W )⊥ = (T W )⊥ ❚→❝ ✤ë♥❣ T ∗ ❧➯♥ ❝↔ ❤❛✐ ✈➳ t❛ ✤÷đ❝ T ∗ f = T ∗ πT W f + T g ữ ợ v W, t õ T ∗ g, v = g, T v = 0, ❝❤♦ ♥➯♥ T ∗ g ∈ W ⊥ ✳ ❚ø ✤â s✉② r❛ πW T ∗ f = πW T ∗ πT W f + πW T ∗ g = πW T ∗ πT W f Footer Page 38 of 54 Header Page 39 of 54 ✸✻ ❈❤♦ {(Wi, vi)}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ C, D✳ ◆➳✉ T ∈ L(H) ❧➔ t♦→♥ tû ❦❤↔ ♥❣❤à❝❤✱ t❤➻ {(T Wi , vi )}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ C ✈➔ D T ∗ T ∗−1 ∗ ∗−1 T T ✣à♥❤ ❧➼ ✷✳✷✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ 2.2.1 t❛ ❝â πWi T ∗ f = πWi T ∗ πT Wi f ≤ πWi T ∗ πT Wi f ≤ T ∗ πT Wi f ❝❤♦ ♥➯♥ C T ∗f vi2 πWi T ∗ f ≤ ≤ T∗ i∈I vi2 πT Wi f i∈I ❱➻ T ∗ ❦❤↔ ♥❣❤à❝❤ ♥➯♥ f = T ∗−1 T ∗ f ≤ T ∗−1 T ∗ f ❤❛② T ∗f ≥ ❉♦ ✤â vi2 πT Wi f i∈I ≥ f T ∗−1 T∗ C T ∗−1 f ▼➦t ❦❤→❝✱ tø ❇ê ✤➲ 2.2.1✱ t❤❛② T −1 ❝❤♦ T ✱ t❛ t❤✉ ✤÷đ❝ πT Wi = πT Wi T ∗−1 πWi T ∗ , ❝❤♦ ♥➯♥ πT Wi f ≤ T ∗−1 · πWi T ∗ f ❉♦ ✤â vi2 πT Wi f i∈I Footer Page 39 of 54 ≤ T ∗−1 vi2 πWi T ∗ f i∈I Header Page 40 of 54 ✸✼ ≤ T ∗−1 D T ∗ f ≤ D · T ∗−1 2 T∗ f ❑❤✉♥❣ ❦➳t ❤ñ♣ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤đ♣ {(Wi, vi)}i∈I ✈ỵ✐ ❝→❝ ❝➟♥ C, D ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ ❍➺ q✉↔ ✷✳✷✳✷✳ C SW,v −1 SW,v ✈➔ D SW,v −1 SW,v tr♦♥❣ ✤â SW,v ❧➔ t♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ {Wi, vi}i∈I −1 ự T = SW,v tr ỵ 2.2.1 ❈❤♦ {(Wi, vi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ C, D ✈➔ U ❧➔ t♦→♥ tû ✉♥✐t❛ tr➯♥ H ❑❤✐ ✤â {(U Wi, vi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ C, D ✈➔ t♦→♥ tû ❦❤✉♥❣ U SW,v U ∗ ❍➺ q✉↔ ✷✳✷✳✸✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ỵ 2.2.1 ợ T = U t❛ s✉② r❛ {(U Wi , vi )}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ ❝➟♥ C, D✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ t♦→♥ tû ❦❤✉♥❣ {(U Wi , vi )}i∈I ❧➔ USW,v U ∗ t❛ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ 2.2.1 vi2 U πWi U ∗ (f ) = U SW,v U ∗ f, vi2 πU Wi (f ) = SU W,v = iI f H, iI tr õ SUW,v ỵ ❤✐➺✉ ❧➔ t♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ {(UWi , vi )}i∈I ❇➙② ❣✐í ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝ ♥❤í ❦❤✉♥❣ ❦➳t ❤đ♣ ✤è✐ ♥❣➝✉✳ ❚❤❡♦ ❇ê ✤➲ 2.2.1✱ t❛ ❝â −1 −1 −1 πWi SW,v = πWi SW,v πSW,v Wi , Footer Page 40 of 54 Header Page 41 of 54 ✸✽ ▲➜② ❧✐➯♥ ❤đ♣ ❤❛✐ ✈➳ t❛ ✤÷đ❝ −1 ∗ −1 −1 (πWi SW,v ) = πWi SW,v πSW,v Wi ∗ −1 ∗ −1 ❤❛② (SW,v ) (πWi )∗ = πSW,v Wi −1 SW,v ∗ ∗ (πWi )∗ −1 ∗ −1 ∗ ❉♦ (SW,v ) = (SW,v )−1 = SW,v −1 ✈➔ (πWi )∗ = πWi ✈➔ πSW,v Wi ∗ −1 = πSW,v Wi ♥➯♥ −1 −1 −1 SW,v πWi = πSW,v Wi SW,v πWi ❉♦ ✤â ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝ ❝â ❞↕♥❣ −1 −1 vi2 πSW,v Wi SW,v πWi (f ), f= f ∈ H ✭✷✳✺✮ i∈I ✣✐➲✉ ♥➔② ❞➝♥ tỵ✐ ✤à♥❤ ♥❣❤➽❛ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ {(Vi , vi )}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤đ♣ ✈ỵ✐ SV,v ❧➔ t♦→♥ tû ❦❤✉♥❣✳ ❳➨t ❞➣② ❦➳t ❤ñ♣ ❇❡ss❡❧ {(Wi , wi )}i∈I ✳ ❚❛ ♥â✐ r➡♥❣ {{(Wi , wi )}i∈I ❧➔ ✤è✐ ♥❣➝✉ t❤❛② ♣❤✐➯♥ ❝õ❛ {(Vi , vi )}i∈I ♥➳✉ t❛ ❝â −1 vi wi πWi SV,v πVi (f ), f= ✭✷✳✻✮ i∈I ✈ỵ✐ ♠å✐ f ∈ H ❚❤❡♦ ❤➺ t❤ù❝ (2.5) t❛ ❝â ❦❤✉♥❣ ❦➳t ❤ñ♣ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ ❧➔ ♠ët ❦❤✉♥❣ ✤è✐ ♥❣➝✉ t❤❛② ♣❤✐➯♥✳ ❚❛ ❝ô♥❣ ❝â ❦➳t q✉↔ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❦➳t ❤ñ♣✳ ✣è✐ ♥❣➝✉ t❤❛② ♣❤✐➯♥ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣ ❧➔ ♠ët ❦❤✉♥❣ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ (2.6) t❛ ❝â f −1 vi wi SV,v πVi (f ), πWi (f ) = i∈I Footer Page 41 of 54 Header Page 42 of 54 ✸✾ −1 vi wi SV,v πVi (f ) πWi (f ) ≤ i∈I 1/2 −1 vi2 SV,v πVi (f ) ≤ wi2 πWi (f ) i∈I −1 ≤ SV,v 1/2 i∈I 1/2 √ wi2 πWi (f ) D f , i∈I tr♦♥❣ ✤â D ❧➔ ❝➟♥ tr➯♥ ❝õ❛ ❦❤✉♥❣ {(Vi , vi )}i∈I ❚ø ✤â s✉② r❛ −1 SV,v D f ≤ i∈I wi2 πWi (f ) ❑➳t ❤đ♣ ✈ỵ✐ {(Wi , wi )}i∈I ❧➔ ❞➣② ❦➳t ❤đ♣ ❇❡ss❡❧✱ t❛ s✉② r❛ {(Vi , vi )}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ✷✳✸ ❚♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❝➦♣ ❞➣② ❦➳♣ ủ ss r ữợ ú tổ t ❞➣② ❦➳t ❤đ♣ ❇❡ss❡❧ {(Vi , vi )}i∈I ✈ỵ✐ ❝➟♥ ❇❡ss❡❧ D1 ✈➔ {(Wi , wi )}i∈I ✈ỵ✐ ❝➟♥ ❇❡ss❡❧ D2 ✳ ❚❛ ①➨t t♦→♥ tû SV W f := f ∈ H vi wi πVi πWi f, i∈I ❚❤❡♦ ❇ê ✤➲ 2.1.1✱ ❝❤✉é✐ ❤ë✐ tư ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥✳ ❚❛ ❝ô♥❣ ❝â SV W f, g = ✭✷✳✼✮ vi wi πWi f, πVi g , i∈I ✈ỵ✐ ♠å✐ f, g ∈ H ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❛ ❝â 1/2 vi2 | SV W f, g | ≤ πV i g 1/2 wi2 · i∈I π Wi f i∈I ❚ø (2.8) s✉② r❛ | SV W f, g | ≤ Footer Page 42 of 54 D1 D2 g f , ✭✷✳✽✮ Header Page 43 of 54 ✹✵ ❝❤♦ ♥➯♥ SV W ❧➔ t♦→♥ tû ❜à ❝❤➦♥ ✈➔ SV W ≤ D1 D2 ❚ø (2.8) t❛ ❝ô♥❣ ❝â 1/2 SV W f ≤ wi2 πWi f D1 ✭✷✳✾✮ i∈I ✈➔ 1/2 SV∗ W g ≤ vi2 πVi g D2 ✭✷✳✶✵✮ i∈I ◆❣♦➔✐ r❛✱ tø (2.7) t❛ ❝â vi wi f, πWi πVi g = f, SW V g , SV W f, g = i∈I ❝❤♦ ♥➯♥ SV∗ W = SW V ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ SV W ❜à ữợ ()K L(H) s {Ti }iI tr t tỷ ỗ t tr õ ✣à♥❤ ❧➼ ✷✳✸✳✶✳ Ti = vi wi KπVi πWi , i ∈ I ◆➳✉ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ①↔② r❛ t❤➻ {(Wi, wi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈❤ù♥❣ SV W ữợ t tỗ t K L(H) s KSV W = IH ✳ ❚ø ✤â f= vi wi KπVi πWi f, i∈I Footer Page 43 of 54 Header Page 44 of 54 ✹✶ tù❝ ❧➔ Ti ❧➔ ♠ët ❦❤❛✐ tr t tỷ ỗ t ✭✐✐✮ ✤ó♥❣ t❤➻ ✈ỵ✐ f ∈ H t❛ ❝â f= vi wi KπVi πWi f i∈I ❚ø ✤â f =K vi wi πVi πWi f , i∈I ❞♦ ✈➟② IH = KSV W ✳ ❚ø ✤â IH = KSV W ≤ K SV W ❱➻ ✈➟② SV W K SV W ữợ SV W ữợ A t (2.9) A f ≤ SV W f ≤ wi πW i f ) D1 ( i∈I ♥➯♥ wi πWi f 2 ≥ A2 f i∈I ❑➳t ❤đ♣ ✈ỵ✐ {(Wi , wi )}i∈I ❧➔ ♠ët ❞➣② ❦➳t ❤ñ♣ ❇❡ss❡❧ s✉② r❛ {(Wi , wi )}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ SV W ❦❤↔ ♥❣❤à❝❤❀ ✭✐✐✮ (∃)K ∈ L(H) ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ ❍➺ q✉↔ ✷✳✸✳✶✳ Ti = vi wi KπVi πWi ❧➔ ♠ët ❦❤❛✐ tr✐➸♥ ❝õ❛ t♦→♥ tỷ ỗ t ởt tr r❛✱ t❤➻ {(Vi , vi )}i∈I , {(Wi , wi )}i∈I ❧➔ ❝→❝ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ⇒ (ii) ●✐↔ sû SV W ❦❤↔ ♥❣❤à❝❤✳ ❑❤✐ ✤â SV W ữợ t ỵ 2.3.1 tỗ t↕✐ K ∈ L(H) s❛♦ ❝❤♦ {Ti }i∈I ❧➔ ❦❤❛✐ tr t tỷ ỗ t ụ õ ♥❣❤➽❛ ❧➔ IH = KSV W ❚ø Footer Page 44 of 54 Header Page 45 of 54 ✹✷ ✤â ❞♦ SV W ❦❤↔ ♥❣❤à❝❤ ♥➯♥ K ❝ô♥❣ ❦❤↔ (ii) (i) sỷ tỗ t K L(H) ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ {Ti }i∈I ❧➔ ❦❤❛✐ tr✐➸♥ t tỷ ỗ t õ IH = KSW V ❚ø ✤â SV W ❦❤↔ ♥❣❤à❝❤✳ ◆➳✉ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ①↔② r❛ t❤➻ t❤❡♦ ✣à♥❤ ỵ 2.3.1 {(Wi , wi )}iI ởt ❦➳t ❤ñ♣✳ ❉♦ SV W = SWV ✈➔ SVW ❦❤↔ ♥❣❤à❝❤ ♥➯♥ SWV ❝ô♥❣ ❦❤↔ ♥❣❤à❝❤✳ ❚ø ✤â {(Vi , vi )}i∈I ❝ơ♥❣ ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤đ♣✳ ❈❤♦ {(Wi, wi)}i∈I ❧➔ ♠ët ❞➣② ❦➳t ❤ñ♣ ❇❡ss❡❧✳ ❑❤✐ ✤â {(Wi , wi )}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣ ❦❤✐ ✈➔ ❝❤➾ tỗ t {(Vi , vi )}iI t ủ ss s SV W ữợ q✉↔ ✷✳✸✳✷✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ {(Wi, wi)}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣ t❤➻ ✤➦t Vi = Wi , vi = wi , i ∈ I ❑❤✐ ✤â SW ❝❤➼♥❤ ❧➔ SW,w ữợ ợ ữủ t sỷ ỵ 2.3.1 f sỷ tỗ t < 1, > s vi wi πVi πWi (f ) ≤ λ1 f + λ2 i∈I vi wi πVi πWi (f ) , i∈I ✈ỵ✐ ❜➜t ❦ý f ∈ H ❑❤✐ ✤â {(Wi, wi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣ ✈➔ − λ1 + λ2 f D1 wi2 πWi f , ≤ f ∈ H i∈I ❈❤ù♥❣ ♠✐♥❤✳ ◆❤÷ trữợ t ỵ SV W f = vi wi πVi πWi (f ) i∈I ❚❛ ❝â f − SV W f ≤ λ1 f + λ2 SV W f ❉♦ f − SV W f ≥ | f − SV W f | , Footer Page 45 of 54 Header Page 46 of 54 ✹✸ ♥➯♥ λ1 f + λ2 SV W f ≥ f − SV W (f ) , ❝❤♦ ♥➯♥ − λ1 f + λ2 ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❚ø (2.9) t❛ t❤✉ ✤÷đ❝ SV W f ≥ ❚ø ✣à♥❤ ỵ 2.3.1 {(Wi , wi )}iI wi2 Wi f i∈I ≥ D1 − λ1 + λ2 f ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ λ2 = 0✱ t❛ ❝â ❦➳t q✉↔ ♠↕♥❤ ❤ì♥ ♥❤÷ s q sỷ tỗ t [0, 1) s❛♦ ❝❤♦ f− vi wi πVi πWi (f ) ≤ λ f , f ∈ H i∈I ❑❤✐ ✤â {(Wi, wi)}i∈I ✈➔ {(Vi, vi)}i∈I ❧➔ ❝→❝ ❦❤✉♥❣ ❦➳t ủ ữợ ữủ s ú wi2 W i f i∈I vi2 πVi f i∈I (1 − λ)2 f 2, ≥ D1 (1 − λ)2 f 2, ≥ D2 ✈ỵ✐ ❜➜t ❦ý f ∈ H ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ✈ỵ✐ f ∈ H f − SW V (f ) = (IH − SV W )∗ (f ) ≤ (IH − SV W )∗ = IH − SW V ❤❛② f − f ≤λ f , vi wi πWi πVi (f ) ≤ λ f i∈I ⑩♣ ❞ö♥❣ ỵ 2.3.1 t ữủ ự ♠✐♥❤✳ Footer Page 46 of 54 f Header Page 47 of 54 ✹✹ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣✱ ❝❤✐ t✐➳t ❝→❝ ✈➜♥ ✤➲ s❛✉✿ tự ỗ t tỷ t t➼♥❤ ❜à ❝❤➦♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❦❤✉♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✲❑❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ t♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❞➣② ❦➳t ❤ñ♣ ❇❡ss❡❧✳ Footer Page 47 of 54 Header Page 48 of 54 ✹✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼✳ ❙✳ ❆s❣❛r✐✱ ❆✳ ❑❤♦sr❛✈✐ ✭✷✵✵✺✮✱ ✏❋r❛♠❡s ❛♥❞ ❜❛s❡s ♦❢ s✉❜s♣❛❝❡s ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ❱♦❧✳ ✸✵✽✱ ◆♦✳✷✱ ✺✹✶✲✺✺✸✳ ❬✷❪ P✳●✳ ❈❛s❛③③❛✱ ●✳ ❑✳ ❑✉t②♥✐♦❦ ✭✷✵✵✹✮✱ ✏❋r❛♠❡s ♦❢ s✉❜s♣❛❝❡s ✑✱ ❈♦♥✲ t❡♠♣✳ ▼❛t❤✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱❱♦❧✳ ✸✹✺✱ ✽✼✲✶✶✸✳ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ❬✸❪ P✳●✳ ❈❛s❛③③❛✱ ●✳ ❑✳ ❑✉t②♥✐♦❦ ✱ ❙✳ ▲✐✭✷✵✵✽✮✱ ✏❋✉s✐♦♥ ❢r❛♠❡s ❛♥❞ ❞✐str✐❜✉t❡❞ ♣r♦❝❡ss✐♥❣ ✑✱ ❆♣♣❧✳ ❈♦♠♣✉t✳ ❍❛r♠♦♥✳ ❆♥❛❧✳ ✱ ❱♦❧✳ ✷✺✱ ✶✶✹✲✶✸✷✳ ❬✹❪ ❖✳ ❈❤r✐st❡♥s❡♥ ✭✷✵✵✸✮✱ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❢r❛♠❡s ❛♥❞ ❘✐❡s③ ❜❛s❡s ✱ ❇✐r❦❤☎❛✉s❡r✱ ❇♦st♦♥✳ ❬✺❪ ■✳ ❉❛✉❜❡❝❤✐❡s✱ ❆✳ ●r♦ss♠❛♥ ❛♥❞ ❨✳ ▼❡②❡r ✭✶✾✽✻✮✱ ✏P❛✐♥❧❡ss ♥♦♥♦rt❤♦❣♦♥❛❧ ❡①♣❛♥s✐♦♥s✑✱ ❏✳ ▼❛t❤✳ P❤②s✳✱❱♦❧✳ ✼✷✱ ✶✷✼✶ ✲ ✶✷✽✸✳ ❬✻❪ ❘✳ ❏✳ ❉✉❢❢✐♥ ❛♥❞ ❆✳ ❈✳ ❙❝❤❛❡❢❢❡r ✭✶✾✺✷✮✱ ✏❆ ❝❧❛ss ♦❢ ♥♦♥❤❛r♠♦♥✐❝ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱❱♦❧✳ ✼✷✱ ✸✹✶ ✲ ✸✻✻✳ ❬✼❪ P✳ ●❛✈r✉t❛ ✭✷✵✵✼✮✱ ✏❖♥ t❤❡ ❞✉❛❧✐t② ♦❢ ❢✉s✐♦♥ ❢r❛♠❡s✑✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱❱♦❧✳ ✸✸✸✱ ✽✼✶✲✽✼✾✳ ❬✽❪ ❘✳❱✳ ❑❛❞✐s♦♥ ❛♥❞ ❏✳ ❘✳ ❘✐♥❣r♦s❡ ✭✶✾✽✸✮✱ ❋✉♥❞❛♠❡♥t❛❧s ♦❢ t❤❡ t❤❡♦r② ♦❢ ♦♣❡r❛t♦r ❛❧❣❡❜r❛s✱ ❱♦❧✳ ✶✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ❋♦✉r✐❡r s❡r✐❡s✑✱ Footer Page 48 of 54 ...Header Page of 54 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI HOÀNG DUY THẮNG ĐỐI NGẪU CỦA KHUNG KẾT HỢP Chuyên ngành: Tốn Giải tích Mã số: 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC Người hướng