➜➵■ ❍ä❈ ❚❍➳■ ◆●❯❨↔◆ tr➢ ê♥❣ ➤➵✐ ❤ä❝ s➢ ♣❤➵♠ P❤➵♠ ❚✉✃ ♥ ✈✐Öt ❤✐Ö✉ ❝❤Ø♥❤ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ t❤❡♦ ♣❤➢ ➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị ▲✉❐♥ ✈➝♥ t❤➵❝ sü ❝❤✉②➟♥ ♥❣➭♥❤ t♦➳♥ ❣✐➯✐ tÝ❝❤ ❚❤➳✐ ◆❣✉②➟♥ ✲ ◆➝♠ ✷✵✶✶ ✶ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ➜➵■ ❍ä❈ ❚❍➳■ ◆●❯❨↔◆ tr➢ ê♥❣ ➤➵✐ ❤ä❝ s➢ ♣❤➵♠ P❤➵♠ ❚✉✃ ♥ ✈✐Öt ❤✐Ö✉ ❝❤Ø♥❤ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ t❤❡♦ ♣❤➢ ➡♥❣ ♣❤➳♣ ➤✐Ó♠ ❣➬♥ ❦Ò ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ ❚Ý❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶ ▲✉❐♥ ✈➝♥ t❤➵❝ sü ❝❤✉②➟♥ ♥❣➭♥❤ t♦➳♥ ❣✐➯✐ tÝ❝❤ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ●❙✳❚❙❑❍ ❧➟ ❞ị♥❣ ♠➢✉ ❚❤➳✐ ◆❣✉②➟♥ ✲ ◆➝♠ ✷✵✶✶ ✷ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ▼ô❝ ❧ô❝ ▲ê✐ ♥ã✐ ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✹ ❈❤➢➡♥❣ ✶✿ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✻ ✶✳✶ ❑❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✶ ❈❤✉➮♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✷ ❚Ý❝❤ ✈➠ ❤➢í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ ❑❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✹ ▼ét sè ✈Ý ❞ơ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✺ ❚Ý♥❤ trù❝ ❣✐❛♦ ✈➭ ❤×♥❤ ❝❤✐Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✻ ❍Ö trù❝ ❝❤✉➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✼ P❤✐Õ♠ ❤➭♠ t✉②Õ♥ tÝ♥❤ ✈➭ s♦♥❣ t✉②Õ♥ tÝ♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✽ ❚♦➳♥ tö ➤è✐ ①ø♥❣ ❤♦➭♥ t♦➭♥ ❧✐➟♥ tô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❣✐➯✐ tÝ❝❤ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❈❤➢➡♥❣ ✷✿ ❇➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶ ❇➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ✈➭ sù tå♥ t➵✐ ♥❣❤✐Ö♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❈➳❝ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✶ ❇➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ◆❛s❤ tr♦♥❣ trß ❝❤➡✐ ❦❤➠♥❣ ❤ỵ♣ t➳❝ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✷ ❇➭✐ t♦➳♥ tè✐ ➢✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✳✸ ❇➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✹ ❇➭✐ t♦➳♥ ❜ï ♣❤✐ t✉②Õ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✺ ❇➭✐ t♦➳♥ ➤✐Ó♠ ❜✃t ➤é♥❣ ❑❛❦✉t❛♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷✳✻ ❇➭✐ t♦➳♥ ➤✐Ó♠ ②➟♥ ♥❣ù❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❈❤➢➡♥❣ ✸✿ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ➤✐Ĩ♠ ❣➬♥ ❦Ị ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✶ P❤➢➡♥❣ ♣❤➳♣ ❜➭✐ t♦➳♥ ♣❤ô ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❍✐Ư✉ ❝❤Ø♥❤ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸ ▼ét sè ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸✳✶ ❚è✐ ➢✉ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸✳✷ ❇❛♦ ❤➭♠ t❤ø❝ ➤➡♥ ➤✐Ö✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸✳✸ ❱✃♥ ➤Ò ❝➞♥ ❜➺♥❣ ◆❛s❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ▲ê✐ ♥ã✐ ➤➬✉ ❇➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ tõ ❧➞✉ ❜ë✐ ❝➳❝ ❝➠♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝ñ❛ ❑② ❋❛♥✱ ❇r♦✇❞❡r✱ ❖❡tt❧✐ ✈➭ ♠ét sè t➳❝ ❣✐➯ ❦❤➳❝✳ ●➬♥ ➤➞② ❜➭✐ t♦➳♥ ♥➭② ❝➭♥❣ ➤➢ỵ❝ q✉❛♥ t➞♠ ứ ề t ị tí ị ợ ì ữ ứ ụ rộ r ủ ề ❜➺♥❣✳ ❚r➟♥ t❤ù❝ tÕ✱ ❝ã t❤Ó ♥ã✐ ♠ä✐ sù ✈❐t✱ ❤✐Ư♥ t➢ỵ♥❣ tr♦♥❣ ❝✉é❝ sè♥❣ tù ♥❤✐➟♥✱ ①➲ ❤é✐ ➤Ị✉ ❤➢í♥❣ ➤Õ♥ sù ❝➞♥ ❜➺♥❣✳ ➜➷❝ ❜✐Ưt tr♦♥❣ t❤ê✐ ➤➵✐ t❤➠♥❣ t✐♥ ❤✐Ư♥ ♥❛②✱ ♠ä✐ ❤♦➵t ➤é♥❣ ➤Ị✉ ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥❤✐Ị✉ ➤è✐ t➳❝ ✈➭ ❧ỵ✐ Ý❝❤ ❝đ❛ ❝➳❝ ➤è✐ t➳❝ ➤Ị✉ ♣❤ơ t❤✉é❝ ♥❤❛✉✱ ♥❤✐Ị✉ ❦❤✐ ♠➞✉ t❤✉➱♥✱ ➤è✐ ❦❤➳♥❣ ♥❤❛✉✳ ▼ét ❣✐➯✐ ♣❤➳♣ tèt ❝❤♦ ➤è✐ t➳❝ ♥➭② ❧➵✐ ❝ã t❤Ó ❦❤➠♥❣ tèt ❝❤♦ ➤è✐ t➳❝ ❦❤➳❝✳ ❉♦ ➤ã ➤Ó ❣✐➯✐ q✉②Õt ♠➞✉ t❤✉➱♥✱ ♠ét ❣✐➯✐ ♣❤➳♣ ❝➞♥ ❜➺♥❣ t❤➢ê♥❣ ❞Ơ ➤➢ỵ❝ ♠ä✐ ➤è✐ t➳❝ ❝❤✃♣ ♥❤❐♥✳ ❱Ị ♠➷t t♦➳♥ ❤ä❝✱ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ➤➢ỵ❝ ♣❤➳t ❜✐Ĩ✉ ❦❤➳ ➤➡♥ ❣✐➯♥ ❞➢í✐ ❞➵♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❑② ❋❛♥✳ ❚✉② ♥❤✐➟♥ ♥❤✐Ò✉ ❜➭✐ t♦➳♥ r✃t q✉❛♥ trä♥❣ ♥❤➢ ❜➭✐ t♦➳♥ tè✐ ➢✉✱ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❑❛❦✉t❛♥✐✱ ❜➭✐ t♦➳♥ ♠✐♥✐♠❛①✱ ✈➭ ♥❤✐Ị✉ ♠➠ ❤×♥❤ ❝➞♥ ❜➺♥❣✱ t➝♥❣ tr➢ë♥❣ ❦✐♥❤ tÕ✱ ❣✐❛♦ t❤➠♥❣ ✈❐♥ t➯✐ ✈✳✈✳✳✳ ➤Ị✉ ❝ã t❤Ĩ ♠➠ t➯ ❞➢í✐ ❞➵♥❣ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✳ ▼ét ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ tr♦♥❣ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ❧➭ ✈✃♥ ➤Ị ❤✐Ư✉ ❝❤Ø♥❤✳ ❍✐Ư✉ ❝❤Ø♥❤ ❧➭ ♠ét ❦ü t❤✉❐t ❝➡ ❜➯♥ ➤Ó ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ỉ♥ ➤Þ♥❤✱ t❤❡♦ ♥❣❤Ü❛ ❧➭ ❝➳❝ s❛✐ sè ♥❤á ❝đ❛ ❞÷ ❧✐Ư✉✱ ❝ã t❤Ĩ ❞➱♥ ➤Õ♥ ❝➳❝ s❛✐ ❧Ư❝❤ ❧í♥ ✈Ị ❧ê✐ ❣✐➯✐✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❦ü t❤✉❐t ❤✐Ư✉ ❝❤Ø♥❤ ❧➭ t❤❛② t❤Õ ❜➭✐ t♦➳♥ ❦❤➠♥❣ ỉ♥ ➤Þ♥❤✱ ❦❤ã ❣✐➯✐ q✉②Õt✱ ❜➺♥❣ ❝➳❝ ❜➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ❞Ơ ❣✐➯✐ q✉②Õt ❤➡♥✳ ❈ã ♠ét sè ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤✱ tr♦♥❣ ➤ã ❤✐Ö✉ ❝❤Ø♥❤ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị ➤➢ỵ❝ sư ❞ơ♥❣ r✃t ♥❤✐Ị✉ tr♦♥❣ ❝➳❝ ❧Ü♥❤ ✈ù❝ ❦❤➳❝ ♥❤❛✉✳ ●➬♥ ➤➞② ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ♥➭② ➤➢ỵ❝ ♠ë ré♥❣ ❝❤♦ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ♥❤✃t ✈Ị ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▲✉❐♥ ✈➝♥ ♥❤✃♥ ♠➵♥❤ ✈➭♦ ♠è✐ ❧✐➟♥ q✉❛♥ ❣✐÷❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ✈➭ ❝➳❝ ❜➭✐ t♦➳♥ ➤➲ ♥➟✉ ë tr➟♥✳ ❚✐Õ♣ ➤Õ♥ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ t❤❡♦ ♥❣✉②➟♥ ❧ý ❜➭✐ t♦➳♥ ♣❤ô✳ ❈✉è✐ ❝ï♥❣✱ ❜➯♥ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ✈✃♥ ➤Ị ❤✐Ư✉ ❝❤Ø♥❤ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị ➤➢ỵ❝ ➤Ị ①✉✃t ❜ë✐ ▼♦✉❞❛❢✐ ❝❤♦ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✳ ◆❣✉②➟♥ ❧ý ❜➭✐ t♦➳♥ ♣❤ơ sÏ ➤➢ỵ❝ sư ❞ơ♥❣ ➤Ĩ ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ ➤➲ ➤➢ỵ❝ ❤✐Ư✉ ❝❤Ø♥❤ tr♦♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị✳ ❇➯♥ ❧✉❐♥ ✈➝♥ ợ trì tr ể trì ế tứ ổ trợ ề ❍✐❧❜❡rt✱ ❝➳❝ ❦✐Õ♥ t❤ø❝ ✈Ị ❣✐➯✐ tÝ❝❤ ❧å✐ sÏ ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ s❛✉✳ ✹ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❈❤➢➡♥❣ ✷ ❞➭♥❤ ❣✐í✐ t❤✐Ư✉ ✈Ị ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✱ ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ♥❣❤✐Ư♠ ✈➭ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✳ ❈❤➢➡♥❣ ✸✱ tr➢í❝ ❤Õt ❣✐í✐ t❤✐Ư✉ ♥❣✉②➟♥ ❧ý ❜➭✐ t♦➳♥ ♣❤ơ ❝❤♦ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ➤➡♥ ➤✐Ö✉ ♠➵♥❤✳ ❙❛✉ ➤ã✱ ë ❝✉è✐ ❝❤➢➡♥❣✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ✈Ị ❤✐Ư✉ ❝❤Ø♥❤ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ề ù t ố ỗ ❧ù❝ s♦♥❣ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭② ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ ❤➵♥ ❝❤Õ✳ ❘✃t ♠♦♥❣ ♥❤❐♥ ➤➢ỵ❝ sù ➤ã♥❣ ❣ã♣ ý ❦✐Õ♥ ❝ñ❛ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦✱ ❝➳❝ ♥❤➭ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ❜➵♥ ➤ä❝ q✉❛♥ t➞♠ ➤Õ♥ ✈✃♥ ➤Ò ♥➭②✳ ❚➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ▲➲♥❤ ➤➵♦ ✈✐Ö♥ ❚♦➳♥ ❤ä❝✳ ❑❤♦❛ s❛✉ ➤➵✐ ❤ä❝✱ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ tr➢ê♥❣ ➜❍❙P✲ ➜❍❚◆ ➤➲ t❐♥ t×♥❤ ❣✐➯♥❣ ❞➵②✱ ❣✐ó♣ ➤ì✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ tr♦♥❣ q✉➳ tr×♥❤ t➳❝ ❣✐➯ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤✐Ö♥ ❧✉❐♥ ✈➝♥ ✳ ❚➳❝ ❣✐➯ ①✐♥ ợ tỏ ò ết s s➽❝ ♥❤✃t tí✐ ♥❣➢ê✐ t❤➬②✿ ●❙✳❚❙❑❍ ▲➟ ❉ị♥❣ ▼➢✉ ✭❱✐Ư♥ ❚♦➳♥ ❤ä❝ ❱✐Ưt ◆❛♠✮ ➤➲ ❤➢í♥❣ ❞➱♥✱ ❣✐ó♣ ➤ì ✈➭ t❐♥ t×♥❤ ❝❤Ø ❜➯♦ ❝❤♦ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✶ ❍ä❝ ✈✐➟♥ P❤➵♠ ❚✉✃♥ ❱✐Ưt ✺ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❈❤➢ ➡♥❣ ✶ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ t❛ sÏ ①Ðt X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ù❝✳ ❙❛✉ ➤➞②✱ t❛ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥✳ ❈➳❝ ➤Þ♥❤ ❧ý ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝ã t❤Ĩ t❤❛♠ ❦❤➯♦ tr♦♥❣ ❬✸❪✿ ✶✳✶ ❑❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✶✳✶✳✶ ❈❤✉➮♥ ❝ñ❛ ♠ét ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤✳ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤ X✱ ❦ý ❤✐Ư✉ ❧➭ ♠ét ❤➭♠ ①➳❝ ➤Þ♥❤ tr➟♥ X ✱ trị ữ ó tí ❝❤✃t✿ ✐✮ x ≥ 0, ∀x ∈ X; x = ⇔ x = ✐✐✮ αx =| α | · x , ∀α ∈ R, ∀x ∈ X ✐✐✐✮ ∀x, y ∈ X, x + y ≤ x + y ✭❇✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✮ t♦➭♥ ❦❤➠♥❣ ❣✐❛♥ ✶✳✶✳✷ ❚Ý❝❤ ✈➠ ❤➢í♥❣✳ ❚r♦♥❣ ❦❤➠♥❣ ❣✐❛♥ Rk ✱ tÝ❝❤ ✈➠ ❤➢í♥❣ ❝đ❛ ❤❛✐ ✈Ð❝ t➡ x = (x1 , ✳✳✳, xk )✱ y = (y1 , ✳✳✳, yk ) ợ ị xy = x1 y1 + x2 y2 + + xk yk ✈➭ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ (x, y) = (y, x)✳ ✐✐✮ (x + y, z) = (x, z) + (y, z)✳ ✐✐✐✮ (αx, y) = α(x, y)✳ ✐✈✮ (x, x) ≥ ✈➭ (x, x) = ⇔ x = 0✳ ✈✮ (x, x) = x ✳ ✐✮ ✶✳✶✳✸ ❑❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ét ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ ♠➭ tr♦♥❣ ➤ã ❝ã t❤Ĩ ị ợ ột ế (x, y) tỏ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ tÝ❝❤ ✈➠ ❤➢í♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ t✐Ò♥ ❍✐❧❜❡rt ✭❤❛② ❦❤➠♥❣ ❣✐❛♥ ❯♥✐t❛✮✳ ❚õ ❜❛ tÝ♥❤ ❝❤✃t ➤➬✉ tr♦♥❣ ✶✳✶✳✷ t❛ ❝ã✿ (x + y, x + y) + (x − y, x − y) = 2(x, x) + 2(y, y) ❑Õt ❤ỵ♣ ✈í✐ tÝ♥❤ ❝❤✃t ✈✮ ❝ñ❛ ✶✳✶✳✷ t❛ s✉② r❛ ❝❤✉➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ t✐Ị♥ ❍✐❧❜❡rt ♣❤➯✐ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥✿ x+y + x−y = 2( x + y ) ✻ Số hóa Trung tâm Học liệu – Đại học Thái Ngun http://www.lrc-tnu.edu.vn ➜➻♥❣ t❤ø❝ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ❜×♥❤ ❤➭♥❤✳ ❱í✐ ❤➭♠ X x (x, x) ①➳❝ ➤Þ♥❤ ♠ét ❝❤✉➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ X ✱ ❦❤✐ ➤ã ❂ trë t❤➭♥❤ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❚❤❐t ✈❐②✱ ✈í✐ ♠ä✐ sè t❤ù❝ α t❛ ❝ã✿ ≤ (x − αy, x − αy) = (x, x) − 2α(x, y) + α2 (y, y), ❤❛② | (x, y) |≤ x y ❚õ ➤ã✿ (x + y)(x + y) = (x, x) + 2(x, y) + (y, y) ≤ x +2 x y + y =( x + y )2 ❱❐② x+y ≤ x + y ◆❣❤Ü❛ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❚õ ➤ã s✉② r❛✿ x >0 ♥Õ✉ x = x =0 ♥Õ✉ x = 0, ✈➭ αx =| α | · ❉♦ ➤ã x ➤ó♥❣ ❧➭ ♠ét ❝❤✉➮♥✳ ◆❤➢ ✈❐② x X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝✳ ❑❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ ✈í✐ tÝ❝❤ ✈➠ ❤➢í♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✶✳✹ ▼ét sè ✈Ý ❞ơ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✮ ❑❤➠♥❣ ❣✐❛♥ Rnp ✈í✐ x = (x1 , ✳✳✳, xn ) ✈➭ ❝❤✉➮♥ p n | xi |p x = , i=1 ≤ p ≤ +∞ n ❑❤✐ p = t❛ t❤➢ê♥❣ ❦ý ❤✐Ö✉ E ✈➭ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❊✉❝❧✐❞ n ❝❤✐Ị✉✳ ✈í✐ ♣ ❧➭ sè t❤ù❝ ❜✃t ❦ú✱ ✼ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ✷✮ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❞➲② sè lp ✈í✐ x = (x1 , ✳✳✳, xn ✳✳✳) ✈➭ ❝❤✉➮♥ p n | xi |p x = < +∞, i=1 ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❦❤✐ p = 2✳ ✶✳✶✳✺ ❚Ý♥❤ trù❝ ❣✐❛♦ ✈➭ ❤×♥❤ ❝❤✐Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❚r♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❛ ♥ã✐ ❤❛✐ ✈Ð❝ t➡ ❦ý ❤✐Ư✉ x, y trù❝ ❣✐❛♦ ✈í✐ ♥❤❛✉ ✈➭ x ⊥ y ✱ ♥Õ✉ (x, y) = 0✳ ❚❛ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t ➤➡♥ ❣✐➯♥ s❛✉✿ x ⊥ y t❤× y ⊥ x✳ ❚❛ ❝ã x ⊥ x ⇔ x = 0✳ ❱Ð❝ t➡ trù❝ ❣✐❛♦ ✈í✐ ♠ä✐ ✈Ð❝ t➡ x✳ ✐✐✮ ◆Õ✉ x ⊥ y1 , y2 , ✳✳✳, yn t❤× x ⊥ α1 y1 + α2 y2 + ✳✳✳ + αn yn ✐✐✐✮ ◆Õ✉ x ⊥ yn , yn −→ y(n −→ ∞) t❤× x ⊥ y ✳ ⊥ ✐✈✮ ◆Õ✉ t❐♣ M trï ♠❐t tr♦♥❣ X t❤× M ❣å♠ ♠ét ♣❤➬♥ tư ❞✉② ♥❤✃t ❧➭ 0✱ ♥❣❤Ü❛ ❧➭✿ x ⊥ M ⇒ x = 0✳ ✈✮ ◆Õ✉ x ⊥ y t❤× x + y = x + y ✳ ✭➜Þ♥❤ ❧ý P②t❤❛❣♦r❡✮ ✐✮ ◆Õ✉ ❚✐Õ♣ t❤❡♦ t❛ ❝ã ❦❤➳✐ ♥✐Ö♠ ì ế ột ị ý M ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ➤ã♥❣ ❝ñ❛ ♠ét ❦❤➠♥❣ x ♥➭♦ ❝đ❛ X ❝ị♥❣ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♠ét ⊥ ❝➳❝❤ ❞✉② ♥❤✃t ❞➢í✐ ❞➵♥❣ x = y + z ✈í✐ y ∈ M, z ∈ M tr♦♥❣ ➤ã y ❧➭ ♣❤➬♥ tư ❝đ❛ M ❣➬♥ x ♥❤✃t✱ tø❝ ❧➭ x − y ≤ x − u , ∀u ∈ M ✳ ❣✐❛♥ ❍✐❧❜❡rt X✳ ❈❤♦ ❇✃t ❦ú ♣❤➬♥ tư ✶✳✶✳✻ ❍Ư trù❝ ❝❤✉➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ {en } ❝➳❝ ♣❤➬♥ tư ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt X ❣ä✐ ❧➭ ❤Ö trù❝ ❝❤✉➮♥ ♥Õ✉ (ei , ej ) = δij tr♦♥❣ ➤ã δij = ✈í✐ i = j ✈➭ δij = ✈í✐ i = j ✳ ◆❤➢ ✈❐② ♠ét ❤Ö trù❝ ❝❤✉➮♥ ❧➭ ♠ét ❤Ö trù❝ ❣✐❛♦ ✈➭ ❝❤✉➮♥ ❤ã❛ ei = 1, ∀i✳ ❑❤✐ {en } ❧➭ ♠ét ❤Ư trù❝ ❝❤✉➮♥ t❤× ✈í✐ ♠ä✐ x ∈ X ✱ sè ζi = (x, ei ) ➤➢ỵ❝ ❣ä✐ ▼ét ❤Ư ∞ ❧➭ ❤Ư sè ❋♦✉r✐❡r ❝đ❛ x ➤è✐ ei ỗ i ei ợ ọ ỗ ❋♦✉r✐❡r i=1 ❝đ❛ x t❤❡♦ ❤Ư {en }✳ ❚❛ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ∞ ✐✮ ζ ≤ x ✳ ✭❇✃t ➤➻♥❣ t❤ø❝ ❇❡s❡❧✮ i=1 ∞ ∞ ✐✐✮ ỗ i ei ộ tụ (x i ei ) ⊥ en ✳ i=1 i=1 ▼ét ❤Ö trù❝ ❝❤✉➮♥ {en } ❣ä✐ ❧➭ ➤➬② ➤ñ ❦❤✐ ❝❤Ø ❝ã ✈Ð❝ t➡ ♠í✐ trù❝ ❣✐❛♦ ✈í✐ t✃t ❝➯ ❝➳❝ ♣❤➬♥ tư ❝đ❛ ❤Ư✿ x ⊥ en (n = 1, 2, .) ⇒ x = 0✳ ✽ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn {en } ❧➭ ♠ét ❤Ö trù❝ ❝❤✉➮♥✱ ζn = (x, en ) ❤Ư sè ❋♦✉r✐❡r ❝đ❛ x ➤è✐ ✈í✐ en ✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ t➢➡♥❣ ➤➢➡♥❣✿ ✶✮ {en } ❧➭ ❤Ư trù❝ ❝❤✉➮♥ ➤➬② ➤đ✳ ➜Þ♥❤ ❧ý ✶✳✶✳✷❬✸❪ ❈❤♦ ❧➭ ❝➳❝ ∞ ✷✮ ζi ei , ∀x ∈ X ✳ x= i=1 ✸✮ x ∞ = ζi2 , ∀x ∈ X ✭➤✐Ị✉ ❦✐Ư♥ ➤ã♥❣✮✳ i=1 ∞ ζi ηi , ∀x, y ∈ X ✭ηi ❧➭ ❤Ư sè ❋♦✉r✐❡r ❝đ❛ y ➤è✐ ✈í✐ ei ✮✳ i=1 ✺✮ ❍Ư {en } t✉②Õ♥ tÝ♥❤ trï ♠❐t tr♦♥❣ X ✱ ♥❣❤Ü❛ ❧➭ ❤ä ❝➳❝ tỉ ❤ỵ♣ t✉②Õ♥ ✹✮ (x, y) = tÝ♥❤ ❝đ❛ ❝➳❝ en ✭❜❛♦ t✉②Õ♥ tÝ♥❤ ❝đ❛ ❤Ư ➜Þ♥❤ ❧ý ✶✳✶✳✸❬✸❪ {en } ✮ ❧➭ trï ♠❐t tr♦♥❣ X ✳ ✭❘✐❡s③✲ ❋✐s❝❤❡r✮ ❈❤♦ ➤ñ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt {en } ❧➭ ♠ét ❤Ö trù❝ ❝❤✉➮♥ ➤➬② X ✳ ◆Õ✉ ♠ét ❞➲② sè {ξ} t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ n ξi2 < ∞ i=1 t❤× sÏ ❝ã ♠ét ✈Ð❝ t➡ ❞✉② ♥❤✃t x∈X ♥❤❐♥ ❝➳❝ {ξi } ❧➭♠ ❤Ö sè ❋♦✉r✐❡r ✈➭ ∞ n x= ξi ei , x ξi2 = i=1 i=1 ✶✳✶✳✼ P❤✐Õ♠ ❤➭♠ t✉②Õ♥ tÝ♥❤ ✈➭ s♦♥❣ t✉②Õ♥ tÝ♥❤ ✳ ❚❛ ❝ã ❞➵♥❣ tỉ♥❣ q✉➳t ❝đ❛ ♣❤✐Õ♠ ❤➭♠ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♥❤➢ s❛✉✿ ị ý rt s ỗ é t➡ a ❝è ➤Þ♥❤ t❤✉é❝ ♠ét ❦❤➠♥❣ X ✱ ❤Ư t❤ø❝ f (x) = (a, x), ①➳❝ ➤Þ♥❤ ♠ét ♣❤✐Õ♠ ❤➭♠ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ f (x) tr➟♥ ❦❤➠♥❣ ❣✐❛♥ X ✱ ✈í✐ f = a ◆❣➢ỵ❝ ❧➵✐✱ ❜✃t ❦ú ♣❤✐Õ♠ ❤➭♠ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ f (x) ♥➭♦ tr➟♥ ♠ét ❦❤➠♥❣ X ❝ị♥❣ ➤Ị✉ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♠ét ❝➳❝❤ ❞✉② ♥❤✃t ❞➢í✐ f (x) = (a, x) tr♦♥❣ ➤ã a ❧➭ ♠ét ✈Ð❝ t➡ ❝ñ❛ X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ f = a ❣✐❛♥ ❍✐❧❜❡rt ❞➵♥❣ ❚õ ➜Þ♥❤ ❧ý ✶✳✶✳✹ t❛ s✉② r❛ ❤Ư q✉➯ s❛✉✿ ✾ Số hóa Trung tâm Học liệu – Đại hc Thỏi Nguyờn http://www.lrc-tnu.edu.vn ệ q ỗ t tử t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ A tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ X ①➳❝ ➤Þ♥❤ f (x, y) = (Ax, y) ♠ét ♣❤✐Õ♠ ❤➭♠ s♦♥❣ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ f (x, y) ♥❣❤✐Ư♠ ➤ó♥❣ f = A ✳ ◆❣➢ỵ❝ ❧➵✐ ❜✃t ❦ú ♣❤✐Õ♠ ❤➭♠ s♦♥❣ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ f (x, y) ♥➭♦ tr➟♥ X ❝ị♥❣ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♠ét ❝➳❝❤ ❞✉② ♥❤✃t ❞➢í✐ ❞➵♥❣ f (x, y) = (Ax, y)✳ ❚r♦♥❣ ➤ã A ❧➭ ♠ét t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ tr➟♥ X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ f = A ❍✐❧❜❡rt ✶✳✶✳✽ ❚♦➳♥ tư ➤è✐ ①ø♥❣ ❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝✳ ❈❤♦ ♠ét t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ ❝ã A tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt X✳ ❚❛ (Ax, y) ❧➭ ♠ét ♣❤✐Õ♠ ❤➭♠ s♦♥❣ t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝✱ ❝❤♦ ♥➟♥ ❝ã ♠ét ∗ t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ ❞✉② ♥❤✃t A ➤Ó ❝❤♦✿ (Ax, y) = (x, A∗ y) A∗ ❣ä✐ t tử ợ ủ A ì A ❝❤✉➮♥ ❤➭♠ (Ax, y)✱ ♠➭ ❝❤✉➮♥ ❝ñ❛ ♣❤✐Õ♠ ❤➭♠ ♥➭② ❧➵✐ ❜➺♥❣ a ♥➟♥✿ ❚♦➳♥ tư ❝đ❛ ♣❤✐Õ♠ A∗ = A ❚❛ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ∗ ∗ ✐✮ (A ) = A ∗ ∗ ✐✐✮ (A + B) = A + ∗ ∗ ∗ ✐✐✐✮ (AB) = B A B ∗ , (αA)∗ = αA∗ ▼ét t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ A ❣ä✐ ❧➭ ➤è✐ ①ø♥❣ ♥Õ✉ t❛ ❝ã ✈í✐ ♠ä✐ x, y ✿ (Ax, y) = (x, Ay) A = A∗ ❝❤♦ ♥➟♥ A ❝ị♥❣ ❣ä✐ ❧➭ t♦➳♥ tư tù ❧✐➟♥ ❤ỵ♣✳ ❚❛ ♥ã✐ ♠ét sè λ ❧➭ trÞ r✐➟♥❣ ❝đ❛ t♦➳♥ tư A✱ ♥Õ✉ ♣❤➢➡♥❣ tr×♥❤ A = λx ❝ã ♥❣❤✐Ư♠ x ❦❤➠♥❣ t➬♠ t❤➢ê♥❣✳ ❑❤✐ ✃② ♥❣❤✐Ö♠ x ♥➭② ❣ä✐ ❧➭ ♠ét ✈Ð❝ t➡ r✐➟♥❣ ❝đ❛ A✱ ø♥❣ ✈í✐ trÞ r✐➟♥❣ λ✳ ✐✈✮ ❚❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ✈Ð❝ t➡ r✐➟♥❣ ❝đ❛ t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tơ❝ A ø♥❣ ✈í✐ ❝ï♥❣ ♠ét trÞ r✐➟♥❣ λ ❧➭♠ t❤➭♥❤ ✭❝ï♥❣ ✈í✐ ♣❤➬♥ tư ✵✮ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ➤ã♥❣ ❝đ❛ X ❜✃t ❜✐Õ♥ ➤è✐ ✈í✐ A✳ ❑❤➠♥❣ ❣✐❛♥ ❝♦♥ ♥➭② ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ r✐➟♥❣ ø♥❣ ✈í✐ trÞ r✐➟♥❣ λ✳ ✈✮ ◆Õ✉ A ❧➭ ♠ét t♦➳♥ tư ➤è✐ ①ø♥❣ t❤× ❝➳❝ ✈Ð❝ t➡ r✐➟♥❣ ❝đ❛ A ø♥❣ ✈í✐ ❑❤✐ ➤ã ❤❛✐ trÞ r✐➟♥❣ ❦❤➳❝ ♥❤❛✉ ❜❛♦ ❣✐ê ❝ị♥❣ trù❝ ❣✐❛♦ ✈í✐ ♥❤❛✉✳ A ❧➭ ♠ét t♦➳♥ tư ➤è✐ ①ø♥❣ t❤× ♣❤➬♥ ❜ï trù❝ ❣✐❛♦ ❝ñ❛ ♠ä✐ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❜✃t ❜✐Õ♥ ➤è✐ ✈í✐ A ❝ị♥❣ ❜✃t ❜✐Õ♥ ➤è✐ ✈í✐ A✳ ✈✐✮ ◆Õ✉ ❚✐Õ♣ t❤❡♦ ❧➭ ❦❤➳✐ ♥✐Ư♠ t♦➳♥ tư ❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝✳ ✶✵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❚❛ ❝ã ❤Ö q✉➯ trù❝ t✐Õ♣ s❛✉ ➤➞②✿ ❍Ư q✉➯ ✸✳✶✳✶❬✼❪ ➜✐Ĩ♠ x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝ù❝ trÞ✿ miny∈K {εF (x∗ , y) + H(x∗ , y)} tr♦♥❣ ➤ã ε > ❜✃t ❦ú ❝❤♦ tr➢í❝✳ ❉Ơ t❤✃② r➺♥❣ ❦❤✐ t❛ ❝❤ä♥✿ H(x, y) = G(y) − G(x) − G (x), y − x tr♦♥❣ ➤ã G : K −→ R ❧➭ ❤➭♠ ❧å✐ ♠➵♥❤✱ ❦❤➯ ✈✐ tr➟♥ K✱ t❤× H t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ♥➟✉ tr♦♥❣ ♠Ư♥❤ ➤Ị tr➟♥✳ ❱× ✈❐② t❛ ①Ðt t❤✉❐t t♦➳♥ s❛✉✿ ❚❤✉❐t t♦➳♥ ✸✳✶✳✶❬✼❪ ✐✮ ▲✃② x0 ∈ K, ε > 0, k = ✐✐✮ ●✐➯✐ ❜➭✐ t♦➳♥✿ miny∈K {εF (xk , y) + G (xk ), y + G(y)}, k t×♠ ♥❣❤✐Ư♠ tè✐ ➢✉ ❞✉② ♥❤✃t y ✳ k k ◆Õ✉ y = x t❤× t❤✉❐t t♦➳♥ ❞õ♥❣ ✈➭ ❦Õt ❧✉❐♥ ❝➞♥ ❜➺♥❣✳ ❚r➳✐ ❧➵✐✱ t❛ ❝❤✉②Ó♥ s❛♥❣ ❜➢í❝ ✐✐✐✮✳ k k+1 ✐✐✐✮ ●➳♥ ♥❣❤✐Ư♠ y ✈õ❛ t×♠ ❧➭ x ✳ ▲✃② k xk ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ = k + ✈➭ q✉❛② ❧➵✐ ❜➢í❝ ✐✐✮✳ ➜Þ♥❤ ❧ý s❛✉ ➤➞② ❝❤♦ t❛ ❦❤➻♥❣ ➤Þ♥❤ ✈Ị sù ❤é✐ tơ ❝đ❛ ❞➲② ①✃♣ ①Ø ①➞② ❞ù♥❣ ❜ë✐ ❚❤✉❐t t♦➳♥ ✸✳✶✳✶ ❦❤✐ ➜Þ♥❤ ❧ý ✸✳✶✳✶❬✼❪ F ✈➭ G t❤á❛ ♠➲♥ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ♥❤✃t ➤Þ♥❤✳ ●✐➯ sư ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② ➤➢ỵ❝ t❤á❛ ♠➲♥✿ F (x, ) ❧➭ ❤➭♠ ❧å✐ ❝❤➷t✱ ❦❤➯ ✈✐ ✈í✐ ♠ä✐ x ∈ K ✳ ✐✐✮ F ❝ã tÝ♥❤ ❝❤✃t ❜➳♥ ❧✐➟♥ tô❝ tr➟♥✳ ✐✐✐✮ F ➤➡♥ ➤✐Ư✉ ♠➵♥❤ tr➟♥ K ✈í✐ ❤Ư sè τ ✳ ✐✈✮ G ❧å✐ ♠➵♥❤ tr➟♥ K ✈í✐ ❤Ư sè η ✳ ✈✮ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tù❛ ▲✐♣s❝❤✐t③✱ tø❝ ❧➭ L1 , L2 > t❤á❛ ♠➲♥✿ ✐✮ tå♥ t➵✐ ❝➳❝ ❤➺♥❣ sè F (x, y) + F (y, z) ❑❤✐ ≥ F (x, z) − L1 x − y − L2 y − z , ∀x, y, z ∈ K η k ➤ã✱ ♥Õ✉ ε < ✈➭ L1 < τ t❤× ❞➲② ①✃♣ ①Ø {x } ①➞② ❞ù♥❣ 2L2 ❜ë✐ t❤✉❐t t♦➳♥ tr➟♥ ❤é✐ tơ ♠➵♥❤ tí✐ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP )✳ ✸✹ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✐✮✱ ✐✐✮✱ ✐✐✐✮ ✈➭ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶ t❛ ∗ s✉② r❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) ❝ã ♥❣❤✐Ö♠ ❞✉② ♥❤✃t✱ ❦ý ❤✐Ö✉ ❧➭ x ✳ ❳Ðt ❤➭♠ sè✿ Λ(x) = G(x∗ ) − G(x) − G (x), x∗ − x η ∗ ❉♦ G ❧å✐ ♠➵♥❤ ✈í✐ ❤Ư sè η ♥➟♥ Λ(x) ≥ x − x ≥ 0, ∀x ∈ K ✳ ❳Ðt ❤✐Ö✉✿ Λ(xk ) − Λ(xk+1 ) = [G(x∗ ) − G(xk ) − G (xk ), x∗ − xk ] −[G(x∗ ) − G(xk+1 ) − G (xk+1 ), x∗ − xk+1 ] = G(xk+1 ) − G(xk ) − G (xk ), x∗ − xk + G (xk+1 ), x∗ − xk+1 = G(xk+1 ) − G(xk ) − G (xk ), xk+1 − xk + G (xk+1 ) − G (xk ), x∗ − xk+1 ❉♦ G ❧å✐ ♠➵♥❤ ✈í✐ ❤Ư sè η ♥➟♥✿ G(xk+1 ) − G(xk ) − G (xk ), xk+1 − xk ≥ ❉♦ xk+1 ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ η k+1 x − xk (AEP ) ♥➟♥ t❛ ❝ã✿ εF2 (xk , xk+1 ) + G (xk+1 ) − G (xk ), y − xk+1 ≥ 0, ∀y ∈ K ❈❤ä♥ y = x∗ t❛ s✉② r❛✿ G (xk+1 ) − G (xk ), x∗ − xk+1 ≥ ε F2 (xk , xk+1 ), xk+1 − x∗ ❉♦ F (xk , ) ❧å✐ ♥➟♥ t❛ ❝ã✿ F2 (xk , xk+1 ), xk+1 − x∗ ≥ F (xk , xk+1 ) − F (xk , x∗ ) ▼➷t ❦❤➳❝✱ t❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ➤Þ♥❤ ❧ý ✈➭ tÝ♥❤ ❝❤✃t ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) t❛ ❝ã✿ F (xk , xk+1 ) − F (xk , x∗ ) = [F (x∗ , xk ) + F (xk , xk+1 ) − F (x∗ , xk+1 )] −[F (xk , x∗ ) + F (x∗ , xk )] + F (x∗ , xk+1 ) ≥ −L1 x∗ − xk − L2 xk − xk+1 = (τ − L1 ) x∗ − xk 2 + τ x∗ − xk − L2 xk − xk+1 ❱❐② t❛ ❝ã✿ Λ(xk ) − Λ(xk+1 ) ≥ η k+1 x − xk 2 ✸✺ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn +ε(τ − L1 ) x∗ − xk ◆❤➢ − εL2 xk − xk+1 η = ( − εL2 ) xk+1 − xk + ε(τ − L1 ) x∗ − xk k ✈❐②✱ ❞➲② Λ(x ) ❧➭ ❞➲② ❣✐➯♠✱ ❜Þ ❝❤➷♥ ❞➢í✐ ✭❜ë✐ ✵✮✳ ≥ ❉♦ ➤ã ❤é✐ tô✱ s✉② r❛✿ lim (Λ(xk ) − Λ(xk+1 )) = k→+∞ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ lim x∗ − xk k→0 = {xk } ❤é✐ tơ ♠➵♥❤ tí✐ x∗ ✳ x ❇➞② ❣✐ê t❛ ①Ðt tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt✱ G(x) = ✱ ❦❤✐ k ❤Ư sè ✶ ✈➭ sù ❤é✐ tơ ❝đ❛ ❞➲② {x } ❧➭ t✉②Õ♥ tÝ♥❤✳ ❇✐Ó✉ t❤ø❝ tr➟♥ ❝❤ø♥❣ tỏ ị ý L1 < tì ❞➲② x k+1 ➤ã G ❱í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➤Þ♥❤ ❧ý tr➟♥✱ ❦❤✐ ❧å✐ ♠➵♥❤ ✈í✐ 0 ❧➭ t❤❛♠ sè ❝❤♦ tr➢í❝✱ ❞➲② {x } ①➳❝ ➤Þ♥❤ k+1 ❜ë✐ q✉② t➽❝✿ x ❧➭ ♥❣❤✐Ư♠ ❝ñ❛ ❜➭✐ t♦➳♥ (AEP )✿ ✐✮ εF (xk+1 , x) + H (xk+1 ) − H (xk ), x − xk+1 ≥ 0, ∀x ∈ K ❑❤✐ ➤ã {xk } ❤é✐ tơ ②Õ✉ tí✐ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP )✳ ❚❛ ❝➬♥ tí✐ ❤❛✐ ❜ỉ ➤Ị s❛✉ ➤➞②✿ {xn } ❧➭ ❞➲② ❜Þ ❝❤➷♥ tr♦♥❣ X ✈➭ ❝➳❝ t❐♣ ➤✐Ĩ♠ ∗ n ∗ tơ ②Õ✉ ❝đ❛ ♥ã ❝❤Ø ❝ã ➤ó♥❣ ♠ét ➤✐Ĩ♠ x ✳ ❑❤✐ ➤ã ❞➲② {x } ❤é✐ tơ ②Õ✉ tí✐ x ✳ ❇ỉ ➤Ị ✸✳✷✳✶❬✽❪ ●✐➯ sư ❞➲② w ❜✃t ❦ú t❤✉é❝ X ✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✱ ❣✐➯ sö w, x −→ w, x ✳ ❑❤✐ ➤ã✱ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❣✐í✐ ❤➵♥✱ tå♥ t➵✐ ε > n ✈➭ ❞➲② ❝♦♥ {x k }✱ ✈í✐ k ➤đ ❧í♥ s❛♦ ❝❤♦✿ ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt n | w, xnk − w, x |> ε, ∀k ≥ ▼➷t ❦❤➳❝✱ ❞♦ ❞➲② {xnk } (5) ị t trí r ợ ột ❞➲② ❝♦♥ ∗ ❤é✐ tơ ②Õ✉✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ ❣✐í✐ ❤➵♥ ♥➭② ♣❤➯✐ ❧➭ x ✱ ❞♦ ➤ã t❛ ❝ã✿ lim k→+∞ {xnk } w, xnk = w, x (6) ❚❛ t❤✃② ✭✺✮ ✈➭ ✭✻✮ ❧➭ ♠➞✉ t❤✉➱♥✱ tø❝ ❧➭✿ lim k→+∞ w, xn = w, x ✸✽ Số hóa Trung tâm Học liệu – Đại học Thái Ngun http://www.lrc-tnu.edu.vn ➜✐Ị✉ ♥➭② ➤ó♥❣ ✈í✐ ❜✃t ❦ú w∈X ♥➟♥ t❛ ❝ã ➤✐Ị✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư f : K −→ R ∪ {+∞} ❧➭ ❤➭♠ ❧å✐✱ ♥ö❛ ❧✐➟♥ tơ❝ k ❞➢í✐✳ ❑❤✐ ➤ã ♥Õ✉ ❞➲② {x } ❤é✐ tơ ②Õ✉ tí✐ x t❤× t❛ ❝ã✿ ❇ỉ ➤Ị ✸✳✷✳✷❬✽❪ f (x) ≤ lim inf f (xn ) n→+∞ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ f ⇔ t❐♣ ♠ø❝ ❞➢í✐ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ Lf (α) := {x | f (x) ≤ α} ❧➭ t❐♣ ➤ã♥❣✳ Lf (α) ❧➭ t❐♣ ❱× ✈❐② f ❧➭ ❤➭♠ ♥ư❛ ❧➟♥ tơ❝ ❞➢í✐ ②Õ✉✳ ◆➟♥ ♥Õ✉ x x t❤× ❉♦ f ❧➭ t❐♣ ❧å✐ ♥➟♥ Lf (α) ❝ò♥❣ ❧➭ t❐♣ ❧å✐✱ s✉② r❛ ➤ã♥❣ ②Õ✉✳ k f (x) ≤ lim inf f (xn ) n→+∞ ❈❤ø♥❣ ♠✐♥❤✳ ✭ ➜Þ♥❤ ❧ý ✸✳✷✳✶✮ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✐✮✱ ✐✐✮ ✈➭ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ t❛ s✉② r❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) ❝ã ♥❣❤✐Ö♠ ❞✉② ♥❤✃t✱ ❦ý ❤✐Ö✉ ❧➭ x∗ ✳ ❳Ðt ❤➭♠ sè Λ : K −→ R ①➳❝ ➤Þ♥❤ ❜ë✐✿ Λ(x) = H(x∗ ) − H(x) − H (x), x∗ − x ❚õ ❣✐➯ t❤✐Õt H ❧å✐ ♠➵♥❤ ✈í✐ ❤Ư sè Λ(x) ≥ α✱ t❛ s✉② r❛✿ α ∗ x −x 2 ≥ (7) ❚❛ ❝ã✿ Λ(xk ) − Λ(xk+1 ) = H(xk+1 ) − H(xk ) − H (xk ), x∗ − xk + H (xk+1 ), x∗ − xk+1 = H(xk+1 ) − H(xk ) − H (xk ), xk+1 − xk + H (xk+1 ) − H (xk ), x∗ − xk+1 ❉♦ H ❧å✐ ♠➵♥❤ ✈í✐ ❤Ư sè α ♥➟♥ ✿ H(xk+1 ) − H(xk ) − H (xk ), xk+1 − xk ≥ ❚❤❡♦ ❝➳❝❤ ①➳❝ ➤Þ♥❤ xk+1 (8) α k+1 x − xk (9) t❛ ❝ã✿ H (xk+1 ) − H (xk ), x∗ − xk+1 ≥ −εF (xk+1 , x∗ ) ✸✾ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn (10) ▼➷t ❦❤➳❝✱ ❞♦ ❤➭♠ F ➤➡♥ ➤✐Ư✉ ✈➭ x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) ♥➟♥✿ F (xk+1 , x∗ ) = F (x∗ , xk+1 ) − [F (x∗ , xk+1 ) + F (xk+1 , x∗ )] ≥ ❑Õt ❤ỵ♣ (8), (9), (10), (11) t❛ ➤➢ỵ❝✿ Λ(xk ) − Λ(xk+1 ) ≥ ❉➲② (11) α k+1 x − xk Λ(xk ) ❧➭ ❞➲② ❣✐➯♠ ✈➭ ❜Þ ❝❤➷♥ ❞➢í✐ ✭❜ë✐ ✵✮ ♥➟♥ ❣✐í✐ ❤➵♥✿ l(x∗ ) = lim Λ(xk ), k→+∞ tå♥ t➵✐ ❤÷✉ ❤➵♥✳ ❚❛ ❝ã✿ lim k→+∞ ▼➷t ❦❤➳❝✱ tõ xk+1 − xk = (7)✿ xk − x∗ ❱× ✈❐②✱ s✉② r❛ ❞➲② ≤ Λ(xk ) α {xk } ❜Þ ❝❤➷♥✳ x ❧➭ ♠ét ➤✐Ĩ♠ tơ ②Õ✉ ❜✃t ❦ú ❝đ❛ ❞➲② {xk }✱ tø❝ ❧➭ tå♥ t➵✐ ❞➲② ❝♦♥ {xki } k+1 ❤é✐ tô ②Õ✉ ➤Õ♥ x✳ ❉♦ x ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ♣❤ơ (AEP ) ✈➭ ❞♦ tÝ♥❤ ➤➡♥ ➤✐Ư✉ ❝đ❛ F t❛ ❝ã✿ ●ä✐ εF (x, xk+1 ) ≤ −εF (xk+1 , x) ≤ H (xk+1 ) − H (xk ), x − xk+1 ▲➵✐ ❞♦ tÝ♥❤ ❧✐➟♥ tơ❝ ▲✐♣s❝❤✐t③ ❝đ❛ ❤➭♠ H ♥➟♥✿ H (xk+1 ) − H (xk ), x − xk+1 ≤ x − xk+1 · H (xk+1 ) − H (xk ) ≤ β x − xk+1 · xk+1 − xk ❑Õt ❤ỵ♣ ❤❛✐ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛✿ εF (x, xk+1 ) ≤ β x − xk+1 · xk+1 − xk ➜✐Ị✉ ♥➭② ➤ó♥❣ ✈í✐ ♠ä✐ k ♥➟♥ ➤ó♥❣ ✈í✐ ❞➲② ❝♦♥ ki ✱ tø❝ ❧➭✿ εF (x, xk+1 ) ≤ β x − xki +1 · xki +1 − xki ❱Õ ♣❤➯✐ t✐Õ♥ ❞➬♥ tí✐ ✵ ❦❤✐ ki −→ +∞✱ ➳♣ ❞ơ♥❣ ❜ỉ ➤Ị t❤ø ❤❛✐ t❛ t❤✉ ➤➢ỵ❝✿ F (x, x) ≤ lim inf F (x, xki +1 ) ≤ 0, ∀x ∈ K ki →+∞ ✹✵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ➜➷t xt = tx + (1 − t)x ✈í✐ < t ≤ t❛ ➤➢ỵ❝✿ = F (xt , xt ) ≤ tF (xt , x) + (1 − t)F (xt , x) ≤ tF (xt , x) F (xt , x) ≥ 0, ∀t > 0✳ ❈❤♦ t ↓ t❤× xt → x✱ tr➟♥ ❝ñ❛ F t❛ ❝ã✿ F (x, x) ≥ 0, ∀x ∈ K ❉♦ ✈❐② ❤❛② x ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP )✳ ❉♦ ❞♦ tÝ♥❤ ❜➳♥ ❧✐➟♥ tơ❝ F ➤➡♥ ➤✐Ư✉ ❝❤➷t ♥➟♥ ∗ ❧➭ ❞✉② ♥❤✃t ❤❛② x = x ✳ ❱❐② ❞➲② ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP ) {xk } t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❜ỉ ➤Ị t❤ø ♥❤✃t ✈➭ ❞♦ ➤ã ♥ã ❤é✐ tô ②Õ✉ ∗ ➤Õ♥ x ✳ ▼Ư♥❤ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ♠Ư♥❤ ➤Ị tr➟♥ ♥Õ✉ t❤❛② ➤✐Ị✉ ❦✐Ư♥ ➤➡♥ ➤✐Ư✉ ❝đ❛ F ❜ë✐ ➤✐Ị✉ ❦✐Ư♥ ➤➡♥ ➤✐Ư✉ ♠➵♥❤✱ t❛ sÏ ❝ã ➤➢ỵ❝ sù ❤é✐ tơ ♠➵♥❤✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ♠➵♥❤ ✈í✐ ❤Ư sè F ➤➡♥ ➤✐Ö✉ τ ✱ ❦❤✐ ➤ã t❛ ❝ã✿ F (xk+1 , x∗ ) = F (x∗ , xk+1 ) − [F (x∗ , xk+1 ) + F (xk+1 , x∗ )] ≥ τ xk+1 − x∗ ▲❐♣ ❧✉❐♥ t➢➡♥❣ tù ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ➤➢ỵ❝✿ Λ(xk ) − Λ(xk+1 ) ≥ ❚õ ➤ã ✈➭ sù ❤é✐ tơ ❝đ❛ ❞➲② α k+1 x − xk + ετ xk+1 − x∗ Λ(xk ) t❛ s✉② r❛✿ lim k→+∞ x ❈❤ä♥ H(x) = 2 xk − x∗ = t❤× H (x) = x ✈➭ H ❤é✐ tơ t✉②Õ♥ tÝ♥❤ ❝đ❛ ❞➲② ➜Þ♥❤ ❧ý ✸✳✷✳✷❬✽❪ ❧å✐ ♠➵♥❤ ✈í✐ ❤Ư sè ✶✳ ❑❤✐ ➤ã sù {xk } ợ t ể tr ị ý s sư ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② ➤➢ỵ❝ t❤á❛ ♠➲♥✿ F (x, ) ❧å✐ ❝❤➷t✱ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ∀x ∈ K ✳ ✐✐✮ F ❝ã tÝ♥❤ ❝❤✃t ❜➳♥ ❧✐➟♥ tô❝ tr➟♥ ✈➭ ➤➡♥ ➤✐Ư✉ ♠➵♥❤ ✈í✐ ❤Ư sè τ ✳ k k+1 ❳➞② ❞ù♥❣ ❞➲② {x } ❜ë✐ q✉② t➽❝✿ x ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ (AEP (k))✿ ✐✮ εF (xk+1 , x) + xk+1 − xk , x − xk+1 ≥ 0, ∀x ∈ K ❑❤✐ ➤ã ❞➲② {xk } t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝✿ xk+1 − x∗ ≤ r xk − x∗ , ∀k ≥ ✹✶ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❱í✐✿ 0 0✱ t❛ ❝ã✿ 2Λ(x) ≤ a2 | w |2 ❦❤✐ F (x, y) + w, y − x ≥ 0, ∀y ∈ K, | w |≤ τ ➜Þ♥❤ ❧ý ✸✳✷✳✸❬✽❪ ●✐➯ sư {xk } ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ♣❤ơ (AEP )✳ ◆Õ✉ tết ủ ị ý ợ tỏ ♠➲♥ ✈➭ F ❧➭ k ➤å♥❣ ❧✐➟♥ tơ❝ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ t➵✐ ➤è✐ ✈í✐ Λ✱ t❤× {x } ❤é✐ tơ ♠➵♥❤ ∗ tí✐ x ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP )✳ ❍➡♥ t❤Õ ♥÷❛✱ tå♥ t➵✐ ♠ét βa √ ❝❤Ø sè k0 s❛♦ ❝❤♦ ∀k ≥ k0 ✱ ✈í✐ ε > β − α✳ ❚❛ ❝ã✿ α β | xk − x¯ | α | xk+1 − x¯ |≤ θ tr♦♥❣ ➤ã θ= ❈❤ø♥❣ ♠✐♥❤✳ ●ä✐ βa β a2 + αε2 w = H (xk+1 ) − H (xk )✱ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✸✳✷✳✶ t❛ ➤➲ ❝ã✿ β | xk − xk+1 | · | x − xk+1 | ✹✷ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ≥ H (xk+1 ) − H (xk ), x − xk+1 ≥ εF (x, xk+1 ) ✈➭ ❞♦ tÝ♥❤ ❧✐➟♥ tơ❝ ▲✐♣s❝❤✐t③ ❝đ❛ H t❛ ❝ã✿ lim | wk |= k→+∞ ε−1 | wk |< τ, ∀k ≥ k0 ✳ ❚õ ➤å♥❣ ❧✐➟♥ tơ❝ ▲✐♣s❝❤✐t③ ❝đ❛ F t➵✐ t❛ s✉② r❛✿ ❈❤ä♥ k0 t❤á❛ ♠➲♥ ❜➭✐ t♦➳♥ (AEP ) ✈➭ tÝ♥❤ 2Λ(xk+1 ) ≤ a2 | ε−1 wk | ▼➷t ❦❤➳❝✱ ✈í✐ Λ(xk ) − Λ(xk+1 ) ≥ ✈➭ ❞♦ tÝ♥❤ ▲✐♣s❝❤✐t③ ❝ñ❛ H k+1 Λ(x α k+1 |x − xk |2 ♥➟♥✿ β a2 )≤ 2 Λ(xk ) β a + αγ ❉♦ ➤ã✿ lim Λ(xk ) = k→+∞ ✈➭ lim | xk − x∗ |= k→+∞ ✈× θ < ✈➭ | xk − x∗ |2 ≤ | xk − x∗ |2 ≤ Λ(xk ) ❚❛ ❝ã t❤Ó s✉② r❛✿ α 2 β Λ(xk ) ≤ θ2 Λ(xk+1 ) ≤ θ2 | xk+1 − x |2 ị ý ợ ứ ♠✐♥❤✳ ◆❤➢ ✈❐②✱ ë t❤✉❐t t♦➳♥ ❞♦ ▼♦✉❞❛❢✐ ➤➢❛ r❛ tr t ỗ t (AEP (k))✿ εF (xk+1 , x) + xk+1 − xk , x − xk+1 ≥ 0, ∀x ∈ K ◆Õ✉ ➤➷t✿ F k (x, y) = εF (x, y) + x − xk , y − x , t❤× xk+1 trë t❤➭♥❤ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥✿ F k (xk+1 , x) ≥ 0, ∀x ∈ K ❚❤ù❝ ❝❤✃t ➤➞② ❝ò♥❣ ❧➭ ♠ét ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✳ ❚✉② ♥❤✐➟♥ ♠Ư♥❤ ➤Ị s❛✉ ➤➞② k ❝❤Ø r❛ r➺♥❣ ❤➭♠ F sÏ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t tèt ❤➡♥ F ✿ ✹✸ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ▼Ư♥❤ ➤Ị ✸✳✷✳✶❬✽❪ ◆Õ✉ F ➤➡♥ ➤✐Ư✉ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tù❛ ▲✐♣s❝❤✐t③ ✈í✐ ❤Ư sè ❞➢➡♥❣ L1 , L2 t❤×✿ k ✐✮ F ➤➡♥ ➤✐Ư✉ ♠➵♥❤ ✈í✐ ❤Ö sè ✶✳ k ✐✐✮ ∀t > 0✱ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tù❛ ▲✐♣s❝❤✐t③ ✈í✐ ❤Ư sè✿ L1 (k) = εL1 + ❈❤ø♥❣ ♠✐♥❤✳ ❑❤❛✐ tr✐Ó♥ Fk 4t ✈➭ t❤❡♦ F L2 (k) = εL2 + t t❛ ❝ã✿ F k (x, y) + F k (y, x) = εF (x, y) + x − xk , y − x + εF (y, x) + y − xk , x − y = ε(F (x, y) + F (y, x)) + x − y, y − x = ε(F (x, y) + F (y, x)) − x − y ≤ − x − y ❇✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ã ➤➢ỵ❝ ❞♦ tÝ♥❤ ➤➡♥ ➤✐Ư✉ ❝đ❛ F✳ ❱❐② ✐✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ➜➷t Gk (x, y) := x − xk , y − x ✱ t❛ ❝ã✿ Gk (x, y) + Gk (y, z) + Gk (x, z) = x − xk , y − x + y − xk , z − y − x − xk , z − x = x − xk , y − z − x − xk , z − x + y − xk , z − y − x − xk , z − x = x − xk , y − z + y − xk , z − y = x − y, y − z ≥− x−y · y−z ❚õ ❜✃t ➤➻♥❣ t❤ø❝✿ x−y · y−z ≤ ❚❛ s✉② r❛✱ Gk x−y 4t + t y − z t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ tù❛ ▲✐♣s❝❤✐t③✿ Gk (x, y) + Gk (y, z) ≥ Gk (x, z) − x−y 4t F k (x, y) = F (x, y) + Gk (x, y) ♠➭ F ✈í✐ ❤Ư sè L1 , L2 ♥➟♥ t❛ ❝ã ✐✐✮✳ ❉♦ − t y − z t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tù❛ ▲✐♣s❝❤✐t③ ✸✳✸ ▼ét sè ø♥❣ ❞ô♥❣✳ ❚r♦♥❣ ♣❤➬♥ ♥➭②✱ t❛ ♥➟✉ ♠ét sè ø♥❣ ❞ô♥❣ ♠➭ ♥é✐ ❞✉♥❣ ❝❤đ ②Õ✉ ➤➢ỵ❝ t❤❛♠ ❦❤➯♦ tr♦♥❣ ❬✽❪✳ ✹✹ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ✸✳✸✳✶ ❚è✐ ➢✉ ❧å✐✳ ❈❤♦ f ❧➭ ♠ét ❤➭♠ ❧å✐✱ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✳ ◆Õ✉ t❛ ❧✃② F (x, y) = f (y) − f (x), h(x) = ✈➭ | x |2 , X = K ✱ ❜➭✐ t♦➳♥ (AEP ) q✉② ✈Ò✿ f (x) ≥ f (xk+1 ) + ε−1 (xk − xk+1 ), x − xk+1 , ∀x ∈ X, tø❝ ❧➭✿ ε−1 (xk − xk+1 ) ∈ ∂f (xk+1 ), ✈í✐ ∂f ❧➭ ❝➳❝ ✈✐ ♣❤➞♥ ❧å✐ ❝ñ❛ f✳ ❚õ ➤ã s✉② r❛✿ xk+1 = arg f (x) + | x − xk |2 2ε ✸✳✸✳✷ ❇❛♦ ❤➭♠ t❤ø❝ ➤➡♥ ➤✐Ö✉✳ ➜➬✉ t✐➟♥ ❜➺♥❣ ❝➳❝❤ ❧✃②✿ F (x, y) = sup ξ, y − x , ∀y, x ∈ K ξ∈T (x) T ⇒ X ❧➭ ♠ét t♦➳♥ tư ➤➡♥ ➤✐Ư✉✱ ✈✃♥ ➤Ị ❝đ❛ ✈✐Ư❝ t×♠ ❦✐Õ♠ ❦❤➠♥❣ ➤✐Ĩ♠ ❝đ❛ T ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ✈✐Ư❝ t×♠ ♥❣❤✐Ư♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ (EP )✳ ❉♦ F ❧➭ ➤➡♥ ➤✐Ư✉✱ t❤á❛ ♠➲♥✿ ♠➭ ∀(x, y) ∈ K × K : F (x, y) + F (y, x) ≤ F ợ ị (, x) X × K ✿ F (y, x) ≤ −ζ, y − x , ∀y ∈ K, ♥❣❤Ü❛ ❧➭ ≤ F (x, y) + −ζ, y − x , ∀x ∈ K ❈➬♥ ❝❤ó ý r➺♥❣ F ❧➭ ❤➭♠ ➤➡♥ ➤✐Ư✉✱ ❧å✐ t❤❡♦ ➤è✐ sè t❤ø ❤❛✐ ✈➭ ❝ã tÝ♥❤ ❝❤✃t ❜➳♥ ❧✐➟♥ tô❝ tr➟♥ t❤❡♦ ➤è✐ sè t❤ø ♥❤✃t✳ ❇➞② ❣✐ê t❛ ①Ðt ε K = X ✱ ❜➭✐ t♦➳♥ (AEP ) trë t❤➭♥❤✿ sup ξ, x − xk+1 + xk+1 − xk , x − xk+1 ≥ ξ∈T (xk+1 ) ✹✺ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❉♦ tÝ♥❤ ➤➡♥ ➤✐Ö✉ ❝đ❛ − sup F t❛ ❝ã t❤Ĩ ✈✐Õt✿ ξ, xk+1 − x + ε−1 xk+1 − xk , x − xk+1 ≥ ξ∈T (x) ❚➢➡♥❣ ➤➢➡♥❣ ✈í✐✿ ε−1 (xk − xk+1 ) − ξ, xk+1 − x ≥ 0, ∀ξ ∈ T (x) ❚õ T ❧➭ ➤➡♥ ➤✐Ö✉ ❝ù❝ ➤➵✐✱ t❛ ❝ã ➤➢ỵ❝✿ k (x − xk+1 ) ∈ T (xk+1 ) ε ❚õ ➤ã s✉② r❛✿ xk+1 = (I + εT )−1 (xk ) ✸✳✸✳✸ ❱✃♥ ➤Ò ❝➞♥ ❜➺♥❣ ◆❛s❤✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✱ ❜➭✐ t♦➳♥ (AEP ) ❝ã ❞➵♥❣ s❛✉✿ (fi (xik+1 ) − fi (xk+1 ) + (xk+1 )i − (xk )i , xi − (xk+1 )i ) ≥ ε i∈I ◆Õ✉ ❝❤♦ ♠ét sè i∈I ♠➭ t❛ ❝❤ä♥ x∈K t❤á❛ ♠➲♥ xi = xik+1 ✱ ❦❤✐ ➤ã✿ ε(fi (xik+1 , xi ) − fi (xk+1 ) + (xk+1 )i − (xk )i , xi − (xk+1 )i ) ≥ ∀i ∈ I, xi ∈ Ki ✱ ❈ã t❤Ó ✈✐Õt ❧➵✐ t❤❡♦ ✈✐ ♣❤➞♥ r✐➟♥❣ ♥❤➢ s❛✉✿ ((xk )i − (xk+1 )i ) ∈ ∂i fi (xk+1 ), ∀i ∈ I ε ➜ã ❧➭✿ (xk+1 )i = arg minxi ∈Ki {f (xik+1 , xi ) + | (xk )i − xi |2 }, ∀i ∈ I 2ε ✹✻ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❑Õt ▲✉❐♥ ◆❤➢ ➤➲ ♥ã✐ ë tr➟♥✱ ♠ô❝ ➤Ý❝❤ ❝❤Ý♥❤ ❝đ❛ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ♥❤✃t ✈Ò ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❇➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ❧➭ ❜➭✐ t tổ qt ì ó tể ợ ứ t✐Õ♣ ❝❐♥ t❤❡♦ ♥❤÷♥❣ ❝➳❝❤ ❦❤➳❝ ♥❤❛✉ t❤➠♥❣ q✉❛ ❝➳❝ ❜➭✐ t♦➳♥ q✉❡♥ t❤✉é❝ ♥❤➢✿ ❜➭✐ t♦➳♥ tè✐ ➢✉✱ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ◆❛s❤✱ ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✱ ❜➭✐ t♦➳♥ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❑❛❦✉t❛♥✐✳✳✳ ◆é✐ ❞✉♥❣ ❝đ❛ ❧✉❐♥ ✈➝♥ ❝❤đ ②Õ✉ tr×♥❤ ❜➭② ✈Ị ✈✐Ư❝ ❤✐Ư✉ ❝❤Ø♥❤ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ➤✐Ĩ♠ ❣➬♥ ❦Ị✳ ◆❣♦➭✐ r❛✱ ❜➯♥ ❧✉❐♥ ✈➝♥ ❝ß♥ ❣✐í✐ t❤✐Ư✉ ♥❣✉②➟♥ ❧ý ❜➭✐ t♦➳♥ ♣❤ơ✱ ➤➢ỵ❝ ❞ï♥❣ ➤Ĩ ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ♣❤ơ ♥➯② s✐♥❤ tr♦♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ t❤❡♦ ❦ü t❤✉❐t ➤✐Ĩ♠ ❣➬♥ ❦Ị✳ ❚r♦♥❣ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ➤Þ♥❤ ❧ý ➤➲ ➤➢ỵ❝ ♣❤➳t ❜✐Ĩ✉ ✈➭ ❝❤ø♥❣ ♠✐♥❤ s➡ ❧➢ỵ❝ tr♦♥❣ ❬✽❪✱ ➤å♥❣ t❤ê✐ ❝❤Ø r❛ ❝❤♦ t❛ sù ♥í✐ ❧á♥❣ ➤✐Ị✉ ❦✐Ư♥ r➭♥❣ ❜✉é❝ ❣✐÷❛ ❝➳❝ ❤Ư sè tù❛ ▲✐♣s❝❤✐t③ ✈➭ ❤Ö sè ➤➡♥ ➤✐Ö✉ ♠➵♥❤✳ ❈➞✉ ❤á✐ ➤➷t r❛ ❧➭ ❧✐Ư✉ ❝ß♥ ❝ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ♥➭♦ ❦❤➳❝ ❝ị♥❣ ❝❤♦ t❛ tÝ♥❤ ❝❤✃t t➢➡♥❣ tù❄ ❉♦ ❦✐Õ♥ t❤ø❝ ❝ñ❛ ❜➯♥ t❤➞♥ ✈➭ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ❝ß♥ ♥❤✐Ị✉ ❤➵♥ ❝❤Õ ♥➟♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ❝ß♥ ♥❤✐Ị✉ t❤✐Õ✉ sãt✳ ❚➳❝ ❣✐➯ ♠♦♥❣ ♠✉è♥ ❝ã t❤Ó ♥❣❤✐➟♥ ❝ø✉ s➞✉ ❤➡♥ ✈Ị ❤➢í♥❣ ➤✐ ♥➭②✳ ❘✃t ♠♦♥❣ ♥❤❐♥ ➤➢ỵ❝ sù ❝❤Ø ❞➱♥ ❝ñ❛ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ❝ï♥❣ ❜➵♥ ➤ä❝✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥✳ ✹✼ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn t➭✐ ❧✐Ö✉ t ế ệt ỗ P ❑❤➯✐ ✭✶✾✾✾✮✱ ●✐➯✐ tÝ❝❤ ❧å✐✱ ◆❤➭ ①✉✃t ❜➯♥ ❦❤♦❛ ❤ä❝ ❦ü t❤✉❐t✳ ❬✷❪ ▲➟ ❉ò♥❣ ▼➢✉ ✭✶✾✾✾✮✱ ◆❤❐♣ ♠➠♥ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ tè✐ ➢✉✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ❦ü t❤✉❐t✳ ❬✸❪ ❍♦➭♥❣ ❚ô② ✭✷✵✵✸✮✱ ❍➭♠ t❤ù❝ ✈➭ ❣✐➯✐ tÝ❝❤ ❤➭♠✱ ◆❤➭ ①✉✃t ❜➯♥ ➜❍◗● ❍➭ ◆é✐✳ ❬✹❪ ❍♦➭♥❣ ❚ô② ✭✷✵✵✸✮✱ ▲ý t❤✉②Õt tè✐ ➢✉✱ ❱✐Ö♥ ❚♦➳♥ ❤ä❝✳ ❚✐Õ♥❣ ❆♥❤ ❬✺❪ ◆✳❍❛❞❥✐s❛✈✈❛s✱ ❙✳❑♦♠❧♦s✐✱ ❙✳❙❝❤❛✐❜❧❡ ✭✷✵✵✺✮✱ ❍❛♥❞❜♦♦❦ ♦❢ ●❡♥❡r✲ ❛❧✐③❡❞ ❈♦♥✈❡①✐t② ❛♥❞ ●❡♥❡r❛❧✐③❡ ▼♦♥♦t♦♥✐❝✐t②✱ ❙♣r✐♥❣❡r Pr❡ss✳ ❬✻❪ ■❣♦r ❑♦♥♥♦✈ ✭✷✵✵✶✮✱ ❈♦♠❜✐♥❡❞ ❘❡❧❛①❛t✐♦♥ ▼❡t❤♦❞s ❢♦r ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s✱ ❙♣r✐♥❣❡r✳ ❬✼❪ ●✳ ▼❛str♦❡♥✐ ✭✷✵✵✸✮✱ ●❛♣ ❋✉♥❝t✐♦♥ ❢♦r ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠s✱ ❏♦✉r✲ ♥❛❧ ♦❢ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✷✼✱ ✹✶✶✲ ✹✷✻✳ ❬✽❪ ❆✳ ▼♦✉❞❛❢✐ ✭✶✾✾✾✮✱ Pr♦①✐♠❛❧ P♦✐♥t ❆❧❣♦r✐t❤♠ ❊①t❡♥❞❡❞ t♦ ❊q✉✐❧✐❜✲ r✐✉♠ Pr♦❜❧❡♠s✱ ❏♦✉r♥❛❧ ♦❢ ◆❛t✉r❛❧ ●❡♦♠❡tr②✱ ✾✶✲ ✶✵✵✳ ❬✾❪ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r ✭✶✾✼✻✮✱ ▼♦♥♦t♦♥❡ ❖♣❡r❛t♦rs ❛♥❞ Pr♦①✐♠❛❧ P♦✐♥t ❆❧❣♦r✐t❤♠✱ ❙✐❛♠ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✐③❛t✐♦♥✱ ✽✼✼✲✽✾✽✳ ✹✽ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn