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 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 HÀ NỘI, 2012 ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ ❚❙✳ ◆❣✉②➵♥ ◗✉ý♥❤ ◆❣❛✱ ♥❣÷í✐ ❝ỉ ✤➣ ữợ t t t ữợ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ❝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✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ỗ ụ t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❍♦➔♥❣ ❉✉② ❚❤➢♥❣ ▲í✐ ❝❛♠ ổ ữợ sỹ ữợ ý ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ 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 ❚❤➢♥❣ ✷ ▼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❤❛♠ ❦❤↔♦ ✹✹ ✹✺ ✸ ▼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 ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð✳ ✣✐➲✉ ♥➔② ❧➔♠ ❝❤♦ ❦❤â t➻♠ ❤♦➦❝ t❤➟♠ ❝❤➼ ❧➔ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ❝→❝ ❝ì sð t❤ä❛ ♠➣♥ ởt số s ỵ ✤➸ ❝❤ó♥❣ 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❛ ❝→❝ ♣❤➛♥ tỷ ữủ ợ t 1952 ❘✳ ❏✳ ❉✉❢❢✐♥ ✈➔ ❆✳ ❈✳ ❙❝❤❛❡❢❢❡r ❬✻❪ tr♦♥❣ ❦❤✐ ự ộ rr ổ ỏ ỗ t ❤å❝ ✤➣ ❦❤æ♥❣ ♥❤➟♥ r❛ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ t 30 trữợ ❝æ♥❣ 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 ❤å❝ q✉❛♥ t➙♠ rë♥❣ r➣✐✳ ❑❤✉♥❣ ❝â ✹ ♥❤✐➲✉ ù♥❣ tr ỷ ỵ t ỵ tt t ♥➨♥ ❞ú ❧✐➺✉✱✳ ✳ ✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❝→❝❤ ✤➸ ỹ trữợ t ỹ ỳ t❤➔♥❤ ♣❤➛♥ ✤à❛ ♣❤÷ì♥❣ s❛✉ ✤â s➩ ①➙② ❞ü♥❣ ❦❤✉♥❣ t♦➔♥ ❝ư❝ tø ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥ ♥➔②✳ ▼ët ÷✉ ✤✐➸♠ ỵ tữ ú ỹ ❝→❝ ❦❤✉♥❣ ❝â ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t ❞➵ ❞➔♥❣ ❤ì♥✳ ❚ø ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❦❤✉♥❣ ✈➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♥â✱ ❈❛s❛③③❛ ✈➔ ❑✉t②♥✐♦❦ [2] ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤✉♥❣ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✭❋r❛♠❡ ♦❢ s✉❜s♣❛❝❡s✮✳ ▼ët t➯♥ ❣å✐ ❦❤→❝ ❝õ❛ ❦❤✉♥❣ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❧➔ ❦❤✉♥❣ ❦➳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 ❞ú ❧✐➺✉✳ ❑❤✉♥❣ ❦➳t ❤ñ♣ ❝❤♦ ♣❤➨♣ ♣❤➙♥ t➼❝❤ t➼♥ ❤✐➺✉ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝→❝ ♣❤➨♣ ❝❤✐➳✉ tr➯♥ ♠ët ổ ỗ ợ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ s➢❝ ❤ì♥ ✈➲ ❦❤✉♥❣ ❦➳t ❤đ♣✱ ✤➦t ❜✐➺t ❧➔ ❝→❝ ✤è✐ ♥❣➝✉ ❝õ❛ ♥â✱ ✤÷đ❝ sü ỗ ỵ ữợ ý tổ q✉②➳t ✤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 t tr ổ rt ỵ tt tờ qt tr ổ rt ữủ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ ♠ët sè ❦➳t q✉↔ ✈➲ ❝→❝ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤ñ♣✱ t♦→♥ tû ❦❤✉♥❣ ✺ ❝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 ❤đ♣✳ ✻ ❈❤÷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 ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✶ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡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 , ✈ỵ✐ ♠å✐ x ∈ H ◆â✐ ♠ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣ T = sup { T (x) : x ∈ H, x ≤ 1} = sup { T (x) : x ∈ H, x = 1} ✼ ●✐↔ 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♦→♥ tû ỗ t tở L(H) T 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➼♥❤ ❧✐➯♥ ❤ñ♣ ❝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♥❣✳ ✸✶ ∗ 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 ✤â♥❣✳ ❉♦ (2) ữỡ ữỡ ợ (3) TW, t t 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,υ 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 ❤đ♣ ❝õ❛ H ❈ơ♥❣ tữỡ tỹ ữ ố ợ t s r tỗ t ởt t tỷ t ợ ộ ❦❤✉♥❣ ❦➳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 ỵ 1.2.2 ❦❤✉♥❣ t❛ ❝â ❦➳t q✉↔ s❛✉ ❝❤♦ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈❤♦ {(Wi, vi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣ ❝õ❛ H ✈ỵ✐ ❝→❝ ❝➟♥ ❦❤✉♥❣ A ✈➔ B ✳ ❑❤✐ ✤â t♦→♥ tû ❦❤✉♥❣ SW,υ ❝õ❛ {(Wi, vi)}i∈I ❧➔ t♦→♥ tû tỹ ủ ữỡ tr H ợ 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 ✸✷ ❈❤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✐ ♣❤ö❝ 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 iI ởt tr t tỷ ỗ t ✭r❡s♦❧✉t✐♦♥ ♦❢ ✐❞❡♥t✐t②✮✳ ❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣ ♠ët ❤å ❝→❝ t♦→♥ tû ❜à ❝❤➦♥ {Ti }i∈I ✤÷đ❝ ❣å✐ ❧➔ ♠ët tr t tỷ ỗ t tr H ✈ỵ✐ ♠å✐ f ∈ H t❛ ❝â f= Ti f i∈I ✸✸ ✭ ✈➔ ❝❤✉é✐ ❤ë✐ tư ♠ët ❝→❝❤ ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ✈ỵ✐ ♠å✐ f ∈ H ✮✳ ❍å −1 SW,v Wi , vi i∈I ✤÷đ❝ ❣å✐ ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣ ✤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 ✭✷✳✹✮ ◆❤➟♥ ①➨t ✷✳✶✳✷✳ ❈â ♠ët ✈➜♥ ố ợ ữỡ tr (2.4) õ ❧✉æ♥ ❧✉æ♥ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❧➯♥ T W ✱ ✭p ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ tø H ❧➯♥ K ♥➳✉ p2 = p✮✱ ♥❤÷♥❣ ♥â ❦❤ỉ♥❣ ❧➔ ♣❤➨♣ ❝❤✐➳✉ 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♦♥❣ ▼ư❝ 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✐ ♥❣➝✉ ❝õ❛ ❦❤✉♥❣ ❦➳t ❤đ♣ ❈ỉ♥❣ tự (2.4) tữỡ ữỡ ợ T W T = T π W ✣➛✉ t✐➯♥✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ s❛✉ ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ t♦→♥ tû✳ ✸✹ ❈❤♦ 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 ) ữ T h, T v = 0, ợ ♠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 πW f2 = T W f ỗ t ổ rt H t tỷ ❦❤↔ ♥❣❤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) ∈ ✸✺ ✈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❛ ❝â 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 ✸✻ ❈❤♦ {(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 ≤ T ∗−1 vi2 πWi T ∗ f i∈I ✸✼ ≤ T ∗−1 D T ∗ f ≤ D · T ∗−1 2 T∗ f ❑❤✉♥❣ ❦➳t ❤đ♣ ✤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 ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ T = SW,v tr ỵ 2.2.1 {(Wi, vi)}iI ởt ❦➳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✉↔ ự ỵ 2.2.1 ✈ỵ✐ 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 = i∈I f ∈ H, i∈I tr õ SUW,v ỵ t tỷ ❦❤✉♥❣ ❦➳t ❤đ♣ {(UWi , vi )}i∈I ❇➙② ❣✐í ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ❝ỉ♥❣ t❤ù❝ ❦❤ỉ✐ ♣❤ư❝ ♥❤í ❦❤✉♥❣ ❦➳t ❤ñ♣ ✤è✐ ♥❣➝✉✳ ❚❤❡♦ ❇ê ✤➲ 2.2.1✱ t❛ ❝â −1 −1 −1 πWi SW,v = πWi SW,v πSW,v Wi , ✸✽ ▲➜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 ❝æ♥❣ 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 ✸✾ −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 ❤ñ♣ ❇❡ss❡❧✱ t❛ s✉② r❛ {(Vi , vi )}i∈I ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ✷✳✸ ❚♦→♥ tû ❦❤✉♥❣ ❝õ❛ ❝➦♣ ủ ss r ữợ ú tổ ①➨t ❤❛✐ ❞➣② ❦➳t ❤ñ♣ ❇❡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 | ≤ D1 D2 g f , ✭✷✳✽✮ ✹✵ ❝❤♦ ♥➯♥ 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 ữợ ()K L(H) s {Ti }iI tr t tỷ ỗ t tr♦♥❣ ✤â ✣à♥❤ ❧➼ ✷✳✸✳✶✳ Ti = vi wi KπVi πWi , i ∈ I ◆➳✉ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ①↔② r❛ t❤➻ {(Wi, wi)}i∈I ❧➔ ♠ët ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ⇒ ✭✐✐✮✳ ◆➳✉ SV W ❜à ữợ t tỗ t K L(H) s KSV W = IH ✳ ❚ø ✤â f= vi wi KπVi πWi f, i∈I ✹✶ 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 ①↔② 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 }iI tr t tỷ ỗ t ụ ❝â ♥❣❤➽❛ ❧➔ IH = KSV W ❚ø ✹✷ ✤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 | , ✹✸ ♥➯♥ λ1 f + λ2 SV W f ≥ f − SV W (f ) , ❝❤♦ ♥➯♥ − λ1 f + λ2 ❧➔ ❦❤✉♥❣ ❦➳t ❤ñ♣✳ ❚ø (2.9) t t ữủ SV W f ứ ỵ 2.3.1✱ {(Wi , wi )}i∈I wi2 π 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 ❤ñ♣ 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♥❣ ✣à♥❤ ỵ 2.3.1 t ữủ ự f ✹✹ ❑➳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❡❧✳ ✹✺ ❚➔✐ ❧✐➺✉ 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✑✱ ... 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