❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚❘❺◆ ❚❍➚ ❍➀◆● ❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❳⑩❈ ✣➚◆❍ ❇Ð■ ❍⑨▼ ❖❘▲■❈❩✳ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✺ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚❘❺◆ ❚❍➚ ❍➀◆● ❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❳⑩❈ ✣➚◆❍ ❇Ð■ ❍⑨▼ ❖❘▲■❈❩✳ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ữớ ữợ Pì ❈❍■ ◆❣❤➺ ❆♥ ✲ ✷✵✶✺ ✶ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ✶ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ỗ ữỡ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ỗ ữỡ ởt số tự ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ỗ ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ tr tr ổ ỗ ♣❤÷ì♥❣ ✶✺ ✷ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr ổ ỗ ữỡ ❖r❧✐❝③ ✶✽ ✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr ổ ỗ ữỡ ❖r❧✐❝③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ lM (E)✳ ✳ ✷✽ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷ ▼Ð ✣❺❯ ❚r♦♥❣ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ ❧➔ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❞➣②✳ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ❝ê ữủ t ợ tr tr trữớ ổ ữợ t t ổ ❧➔ ♥❤ú♥❣ ✈➼ ❞ö ❦❤→ ✤✐➸♥ ❤➻♥❤ ❝õ❛ ❣✐↔✐ t➼❝❤ r sỷ ỵ tữ ❖r❧✐❝③✱ ❝→❝ t→❝ ❣✐↔ ❏✳ ▲✐♥❞❡♥str❛✉ss ✈➔ ▲✳ ❚③❛❢r✐r✐ ✤➣ ①➙② ❞ü♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❝→❝ ❞➣② tr ổ ữợ tứ ợ tỹ ✤➦❝ ❜✐➺t ♠➔ ❝❤ó♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ❖r❧✐❝③✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❞➣② ❖r❧✐❝③ ❝ơ♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤→ s➙✉ s➢❝ t❤ỉ♥❣ q✉❛ ❝➜✉ tró❝ ❝õ❛ ❤➔♠ ❖r❧✐❝③ ❜ð✐ ❏✳ ▲✐♥❞❡♥str❛✉ss ✈➔ ▲✳ ❚③❛❢r✐r✐✳ ❚r♦♥❣ ❬✺❪✱ ❞ü❛ tr➯♥ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❏✳ ▲✐♥❞❡♥str❛✉ss ✈➔ ▲✳ ❚③❛❢r✐r✐ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ❖r❧✐❝③ ♥❤➟♥ tr ổ ữợ t ỹ ợ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ t❤✉ ✤÷đ❝ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ①➙② ❞ü♥❣ ❦❤ỉ♥❣ tr tr ổ ỗ ✤à❛ ♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❤➔♠ ❖r❧✐❝③✱ ✈➻ ✈➟② ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐✿ ❱➲ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ tr tr ổ ỗ ữỡ ①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ❖r❧✐❝③✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✤➣ ❜✐➳t ✈➲ ❦❤æ♥❣ ỗ ữỡ ỹ ổ tr tr ổ ỗ ữỡ ✤à♥❤ ❜ð✐ ❝→❝ ❤➔♠ ❖r❧✐❝③ ✈➔ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❈→❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ữủ tr tr ữỡ ổ ỗ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr ổ ỗ ữỡ ữỡ ữỡ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ ❞ị♥❣ ✈➲ s❛✉✱ ✤➦❝ ❜✐➺t ❧➔ ♥❤ú♥❣ ❦➳t q✉↔ ❝➠♥ ❜↔♥ ✈➲ ổ ỗ ữỡ ởt số ợ ổ tr tr ổ ỗ ✤à❛ ♣❤÷ì♥❣✳ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ổ ỗ ữỡ r ❈❤÷ì♥❣ ✷✿ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ữỡ ỹ trú tổổ ỗ ữỡ ❝❤♦ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ỗ ữỡ r ỹ tr ỵ tữ tỹ tr trữớ ủ tr ổ ữợ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✤➣ ✤➲ ❝➟♣ ð tr➯♥✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t✱ ①➨t ♠ët sè ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠ỵ✐ ①➙② ❞ü♥❣ ✈ỵ✐ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ tr tr ổ ỗ ữỡ ✤➣ ❝â✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ữợ sỹ ữợ ❑✐➲✉ P❤÷ì♥❣ ❈❤✐✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ t❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔② t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝✱ ❇❛♥ ❧➣♥❤ ✤↕♦ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ q✉➼ ❚❤➛② ❈ỉ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ❦❤♦❛ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝✲❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ❣✐ó♣ ✤ï tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤✱ ỗ t ❝❛♦ ❤å❝ ❦❤â❛ ✷✶ ❚♦→♥✲●✐↔✐ t➼❝❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❣✐ó♣ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ♥❤✐➺♠ ✈ö tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ▼➦❝ ❞ị ✤➣ ❝â r➜t ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ✈➻ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ữủ ỳ qỵ t ổ ỳ õ ỵ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✺ ✹ ❚r➛♥ ❚❤à ❍➡♥❣ ✺ ❈❍×❒◆● ✶ ❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❱⑨ ❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ Pì ữỡ tr ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ ❞ị♥❣ ✈➲ s❛✉✱ ✤➦❝ ❜✐➺t ❧➔ ♥❤ú♥❣ ❦➳t q✉↔ ❝➠♥ ❜↔♥ ✈➲ ❦❤æ♥❣ ỗ ữỡ ởt số ợ ổ tr tr ổ ỗ ♣❤÷ì♥❣✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ▼ư❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ❣✐↔✐ t➼❝❤ ❝ê ✤✐➸♥ ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝➛♥ ❞ò♥❣ ✈➲ s❛✉✳ ❙❛✉ ✤➙②✱ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✈➲ ❤➔♠ ỗ t q s õ t t t tr♦♥❣ ❬✶❪ ✈➔ ❬✸❪✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ❤➔♠ t❤ü❝ f : (a, b) → R✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ỗ f x + (1 )y ợ ♠å✐ x, y ∈ (a, b) ✈➔ λ λf (x) + (1 − λ)f (y) ✭✶✳✶✮ 1✳ ✶✳✶✳✷ ◆❤➟♥ t tữỡ ữỡ ợ s f (t) − f (s) t−s f (u) − f (t) u−t ✭✶✳✷✮ ✈ỵ✐ ♠å✐ a < s < t < u < b✳ ✶✳✶✳✸ ▼➺♥❤ ✤➲✳ ❈❤♦ f : (a, b) R ỗ c (a, b)✳ ❑❤✐ ✤â✱ − f (c) ❤➔♠ p : (a, b) \ {c} → R ①→❝ ✤à♥❤ ❜ð✐ p(x) = f (x)x − ❧➔ ❦❤ỉ♥❣ c ❣✐↔♠✳ ✻ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✈ỵ✐ ♠å✐ c ∈ (a, b) ❤➔♠ p ổ t f ỗ q ●✐↔ sû f ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ (a, b)✳ õ f ỗ f ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ (a, b)✳ ✶✳✶✳✺ ❍➺ q✉↔✳ ◆➳✉ f f ❝â ✤↕♦ ❤➔♠ ❝➜♣ ✷ tr➯♥ (x) > ✈ỵ✐ ♠å✐ x ∈ (a, b) t❤➻ f ỗ : (a, b) R (a, b) ✈➔ ✶✳✶✳✻ ❱➼ ❞ö✳ ❚ø ❤➺ q✉↔ tr➯♥ t❛ t f (x) = ex ỗ tr R y = xp ỗ tr (0, ) ✈ỵ✐ p 1✳ ✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✼❪✮ ❍➔♠ M : [0, +∞) → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❖r❧✐❝③ ♥➳✉ ✶✮ M ❧➔ ❤➔♠ ❦❤ỉ♥❣ ❣✐↔♠✱ ❧✐➯♥ tư❝❀ ✷✮ M (0) = ✈➔ lim M (t) = ∞❀ t→∞ M ỗ r M s tỗ t t > s❛♦ ❝❤♦ M (t) = 0✳ ✶✳✶✳✾ ❱➼ ❞ö✳ ❈→❝ ❤➔♠ M (t) = tp; M (t) = tet ❧➔ ❤➔♠ ❖r❧✐❝③✳ ✶✳✶✳✶✵ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥ tr÷í♥❣ K✳ ❍➔♠ : E → R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✮ x 0✱ ✈ỵ✐ ♠å✐ x ∈ E ✈➔ x = ⇔ x = 0❀ ✷✮ λx = |λ| x ✱ ✈ỵ✐ ♠å✐ λ ∈ K ✈➔ ✈ỵ✐ ♠å✐ x ∈ E ❀ ✸✮ x + y x + y , ✈ỵ✐ ♠å✐ x, y ∈ E ✳ ❑❤✐ ✤â (E, ) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ d(x, y) = x−y , ∀x, y ∈ E ✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ E ✤➛② ✤õ ✈ỵ✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥✳ ❱ỵ✐ tỉ♣ỉ s✐♥❤ ❜ð✐ ✼ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ õ t ổ ữợ tr E ❧➔ ❧✐➯♥ tư❝✳ ❈❤♦ E, F ❧➔ ❝→❝ ❦❤ỉ♥❣ ỵ L(E, F ) t ❤đ♣ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tø E ✈➔♦ F ✳ ❚❛ ✤➣ ❜✐➳t L(E, F ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈ỵ✐ ❝❤✉➞♥ f = sup f (x) , ∀f ∈ L(E, F ) x =1 ◆➳✉ F ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤➻ L(E, F ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣➦❝ ❜✐➺t✱ L(E, K) := E ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ t❤ù ♥❤➜t ❝õ❛ E ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈→❝ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ q✉❡♥ t❤✉ë❝ s❛✉ ✤÷đ❝ q✉❛♥ t➙♠ ♥❤✐➲✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝õ❛ ❝❤ó♥❣ tỉ✐✳ ✶✳✶✳✶✶ ❱➼ ❞ư✳ ●✐↔ sû K ❧➔ tr÷í♥❣ số tỹ số ự ỵ l = x = (xn ) ⊂ K : (xn ) ❧➔ ❞➣② ❜à ❝❤➦♥ ; C = x = (xn ) ⊂ K : (xn ) ❧➔ ❞➣② ❤ë✐ tö ; C0 = x = (xn ) ⊂ K : lim xn = ; n→∞ ✈➔ ∞ |xn |p < ∞ , p lp = x = (xn ) ⊂ K : n=1 ❱ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ❝→❝ ❞➣② ✈➔ ♥❤➙♥ ♠ët sè ✈ỵ✐ ♠ët ❞➣② t❤ỉ♥❣ t❤÷í♥❣ t❛ ❝â l∞ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈➔ C ✱ C0 ✈➔ lp ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ l∞ ✳ ❍ì♥ ♥ú❛ lp ⊂ C0 ⊂ C ⊂ l∞ ❚❛ ✤➣ ❜✐➳t l∞ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ x = sup |xn |, ∀x ∈ l∞ n ✭✶✳✸✮ ✽ ✣➦❝ ❜✐➺t C0 , C ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ l∞ ✱ ✈➻ t❤➳ ❝❤ó♥❣ ❝ơ♥❣ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ tr➯♥✳ ❚✉② ♥❤✐➯♥ lp ❦❤ỉ♥❣ ✤â♥❣ tr l ố ợ lp ữớ t t ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ ∞ x p |xn |p = 1/p , ∀x ∈ lp ✭✶✳✹✮ n=1 ❑❤✐ ✤â✱ lp ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✶✳✷✳ ổ ỗ ữỡ tr ♥✐➺♠✱ ✈➼ ❞ư ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ỗ ữỡ t q ữủ tê♥❣ ❤ñ♣ ✈➔ tr➼❝❤ r❛ tø ❬✸❪✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝ị♥❣ ✈ỵ✐ ♠ët tỉ♣ỉ tr➯♥ ✤â s❛♦ ❝❤♦ ❝→❝ ♣❤➨♣ t♦→♥ ổ ữợ tử ụ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ❧➔✿ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ♥➳✉ tr➯♥ ✤â ✤➣ ởt tổổ tữỡ t ợ trú số tr➯♥ ❳ s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ tr➯♥ X ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣✮✳ ✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû ❆ ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✳ ❛✮ A ữủ ỗ ợ x, y ∈ A ✈➔ ✈ỵ✐ ♠å✐ t ∈ [0; 1]✱ t❛ ❝â t.x + (1 − t).y ∈ A❀ ❝➙♥ ♥➳✉ αA ⊂ A ✈ỵ✐ ♠å✐ α ∈ K ✈➔ |α| < 1❀ ❝✮ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❜à ợ ộ V tỗ t↕✐ sè ❜✮ ❚➟♣ ❝♦♥ A ✤÷đ❝ ❣å✐ ❧➔ s > s❛♦ ❝❤♦ A ⊂ tV ✈ỵ✐ ♠å✐ t > s✳ ✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ✤÷đ❝ ỗ ữỡ õ ỡ s U ỗ t ỗ sỷ X ổ ỗ ữỡ ❑❤✐ ✤â ∈ X ❝â ❝ì sð ❧➙♥ ❝➟♥ U t❤♦↔ ♠➣♥✿ ✷✸ ✈➔ ∞ v = qα (y) = inf ρα : M n=1 ❑❤✐ ✤â ∞ n=1 ●✐↔ sû t, s ∈ R s❛♦ ❝❤♦ s ♥➯♥ t❛ ❝â ∞ ≤ ✈➔ n=1 ∞ u ✈➔ t n=1 ≤1 pα (yn ) qα (y) ≤ v ✳ ❑❤✐ ✤â✱ ❞♦ M ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ ∞ ≤ M pα (xn ) qα (x) M pα (yn ) qα (y) ≤ n=1 ∞ pα (xn ) t M M n=1 pα (xn ) s M ✈➔ ∞ pα (xn ) qα (x) M pα (xn ) ρα ≤ n=1 ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â pα (xn ) + pα (yn ) s pα (xn ) t pα (yn ) = + t+s s+t s s+t t ứ ỗ s r M p (xn + yn ) s+t pα (xn ) + pα (yn ) s+t s pα (xn ) t M + M s+t s s+t s t ≤ + = s+t s+t ≤M ❉♦ ✤â ∞ s+t∈ ρα : pα (xn + yn ) ρα M n=1 ≤1 pα (yn ) t ❱➻ ✈➟② ∞ qα (x + y) = inf ρα : M n=1 ❱➻ ✭✷✳✸✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ s pα (xn + yn ) ρα qα (x) ✈➔ t ≤1 ≤ s + t qα (y) ♥➯♥ s✉② r❛ ✭✷✳✸✮ ✷✹ qα (x + y) ≤ qα (x) + qα (y) ❙✉② r❛ qα ✈ỵ✐ ❝ỉ♥❣ t❤ù❝ ✤➣ ữủ tr ởt ỷ ợ ♠å✐ α ∈ I ✱ ❤❛② Q = {qα }α∈I ❧➔ ❤å ♥û❛ ❝❤✉➞♥ tr➯♥ lM (E)✳ ❱➻ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ lM (E) ❝â ♠ët ❤å {qα }α∈I ❝→❝ ỷ s r tr õ tỗ t tổổ t tữỡ t ợ trú số ♠å✐ qα ❧✐➯♥ tư❝ ✈➔ ✈ỵ✐ tỉ♣ỉ ♥➔② lM (E) ổ ỗ ữỡ ố ũ sỷ E ❧➔ ❍❛✉s❞♦r❢❢ ✈➔ qα (x) = ✈ỵ✐ ♠å✐ α ∈ I ✳ ❚❛ ❝➛♥ ❝❤➾ r❛ x = 0✳ ❚❛ ❝â ∞ qα (x) = inf ρα > : M n=1 pα (xn ) ρα = ❉♦ t➼♥❤ ❦❤æ♥❣ ❣✐↔♠ ❝õ❛ M ❞➵ ❞➔♥❣ s✉② r❛ ∞ M n=1 pα (xn ) ρα ✭✷✳✹✮ ợ > sỷ tỗ t n s❛♦ ❝❤♦ pα (xn ) = 0✳ ❑❤✐ ✤â✱ ❦❤✐ ρα → ∞✳ ❑➳t ❤đ♣ ✈ỵ✐ lim M (t) = ∞ t❛ s✉② r❛ pα (xn ) →0 ρα t→∞ lim M ρα →0 pα (xn ) = ∞, ρα ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✹✮✳ ❙✉② r❛ pα (xn ) = ✈ỵ✐ ♠å✐ ♥✳ ❚ø E ❍❛✉s❞♦r❢❢ ♥➯♥ xn = ✈ỵ✐ ♠å✐ n ❤❛② x = ❚❛ ❝➛♥ ❜ê ✤➲ s❛✉ ❝❤♦ ❝❤ù♥❣ ♠✐♥❤ ✷✳✶✳✺ ❇ê ✤➲✳ ◆➳✉ ❞➣② (xk ) ⊂ lM (E)✱ tr♦♥❣ ✤â xk = (xk1 , , xkn, ), k = 1, 2, ❤ë✐ tư tỵ✐ tr♦♥❣ lM (E) t❤➻ k→∞ lim xkn = tr♦♥❣ E ✈ỵ✐ ♠å✐ n = 1, 2, ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ ổ ú õ tỗ t n0 s (xkn0 ) ❦❤ỉ♥❣ ❤ë✐ tư tỵ✐ tr♦♥❣ E ✳ tỗ t (kj ) I k s❛♦ ❝❤♦ pα (xnj0 ) r > ❚❛ ❝â ✷✺ ∞ kj qα (x ) = inf{ρα > : n=1 k pα (xnj ) ≤ 1} M ρα ❙✉② r❛ ∞ k k n=1 pα (xnj ) pα (xnj0 ) r M ≥ M ≥ M qα (xkj ) qα (xkj ) qα (xkj ) ✭✷✳✺✮ ✈ỵ✐ ♠å✐ kj ✳ ▼➦t ❦❤→❝✱ ❝❤♦ kj → ∞ ✈➔ tø ❣✐↔ t❤✐➳t xk → ∈ lM (E)) t❛ r ♥❤➟♥ ✤÷đ❝ M → ∞ ▼➙✉ t❤✉➝♥ ợ ữủ q (xkj ) ự ỵ E ổ ❋r❡❝❤❡t t❤➻ lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r❡❝❤❡t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r❡❝❤❡t ♥➯♥ tæ♣æ ❝õ❛ E ✤÷đ❝ s✐♥❤ ❜ð✐ ♠ët ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ♥û❛ ❝❤✉➞♥ P = (pk )✳ ❍ì♥ ♥ú❛✱ tỉ♣ỉ ♥➔② s✐♥❤ ❜ð✐ ♠➯tr✐❝ ∞ d(a, b) = k=1 pk (a − b) 2k + pk (a − b) ✈ỵ✐ ♠å✐ a, b E õ tứ ỵ t õ lM (E) ổ ỗ ♣❤÷ì♥❣ s✐♥❤ ❜ð✐ ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ♥û❛ ❝❤✉➞♥ Q = (qk ) ①→❝ ✤à♥❤ ❜ð✐ ∞ qk (x) = inf ρk > : M n=1 pk (xn ) ρk ✭✷✳✻✮ ✈ỵ✐ ♠å✐ x = (xn ) ∈ lM (E)✳ ❉♦ ✤â✱ lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❦❤↔ ♠➯tr✐❝ ✈ỵ✐ ∞ D(x, y) = k=1 qk (x − y) 2k + qk (x − y) ✈ỵ✐ ♠å✐ x, y ∈ lM (E)✳ ❉➵ t❤➜② D(x, y) ❜➜t ❜✐➳♥✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✤➛② ✤õ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ (lM (E), D)✳ ●✐↔ sû (xm ) ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ lM (E)✳ ❚❛ ❝➛♥ ❝❤➾ r❛ (xm ) ❤ë✐ tư tỵ✐ x ∈ lM (E)✳ ✷✻ ◆❤í ◆❤➟♥ ①➨t ✶✳✷✳✶✾ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ xk ❤ë✐ tư tỵ✐ x ∈ lM (E) t❤❡♦ ❝→❝ ♥û❛ ❝❤✉➞♥ qn ✳ ❚❤➟t ✈➟②✱ ✈➻ (xk ) ❧➔ ❞➣② ❈❛✉❝❤② ♥➯♥ ∞ m l qk (x − x ) = inf ρk > : M n=1 l pk (xm n − xn ) ≤1 →0 ρk ✭✷✳✼✮ ❦❤✐ m, l → ∞ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✺ t❤➻ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â l pk (xm n − xn ) → ❦❤✐ m, l → ∞ ✈➔ ✈ỵ✐ ♠å✐ n = 1, 2, ✳ ❉♦ ✤â✱ (xm n ) ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ E ✈ỵ✐ ♠é✐ n = 1, 2, ✳ ❱➻ E ✤➛② ✤õ ♥➯♥ lim xm n := xn ∈ E ✳ ✣➦t m→∞ x = (x1 , , xn , )✳ ❇➙② ❣✐í✱ t❛ ❝è ✤à♥❤ k = 1, 2, ✳ ❑❤✐ õ ợ > tứ tỗ t m0 s❛♦ ❝❤♦ ∞ m l pk (x − x ) = inf{ρk > : M n=1 ✈ỵ✐ ♠å✐ m, l l pk (xm n − xn ) ≤ 1} < ε ρk m0 ✳ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝è ✤à♥❤ m ✭✷✳✽✮ m0 ❝❤♦ l → ∞ t❛ ♥❤➟♥ ✤÷đ❝ ∞ inf{ρk > : n=1 pk (xm n − xn ) M ≤ 1} < ε ρk ❚❛ ♥❤➟♥ ✤÷đ❝ pk (xm − x) < ε ✈ỵ✐ ♠å✐ m ✭✷✳✾✮ m0 ✱ ∀k = 1, ✳ ❚ù❝ ❧➔ xm ❤ë✐ tư tỵ✐ x✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤➾ r❛ x ∈ lM (E)✳ ❚ø ✭✷✳✾✮ t❛ ❝â ∞ n=1 pk (xm n − xn ) ≤ < ∞, M ρk ✈ỵ✐ ♠å✐ k = 1, 2, ✳ ❙✉② r❛ xm0 − x ∈ lM (E)✳ ❉♦ lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈➔ xm0 ∈ lM (E) ♥➯♥ x = xm0 − (xm0 − x) ∈ lM (E) ❤❛②(lM (E), D) ✤➛② ✤õ✳ ❚❛ ♥❤➟♥ ✤÷đ❝ lM (E) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❋r❡❝❤❡t✳ ❑➳t q✉↔ s❛✉ ✤➙② ❝❤♦ ♠ët ✈➼ ❞ư ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ lM (E) ỵ M (t) = tq (q 1) t❤➻ lM (E) = lq (E)✳ ❈❤ù♥❣ ự t ữợ ữợ t❛ ❝❤➾ r❛ ❤❛✐ t➟♣ ❤ñ♣ lM (E) ✈➔ lq (E) ữợ t r ♥û❛ ❝❤✉➞♥ ①→❝ ✤à♥❤ tr➯♥ ❝❤ó♥❣ trị♥❣ ♥❤❛✉✳ ▲➜② x ❜➜t ❦ý t❤✉ë❝ lM (E)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ α ∈ I t❛ ❝â ∞ M n=1 pα (xn ) ♥➔♦ ✤â✳ ❱➻ M (t) = tq ♥➯♥ t❛ ✤÷đ❝ ∞ n=1 pα (xn ) ρ q : M n=1 ∞ pα (xn ) ρ pα (xn )q = inf ρ > : ρq n=1 ∞ pα (xn )q = inf ρ > : 1/q ρ n=1 = inf ρ > : cα (x) ρ} = cα (x) ✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ lM (E)✳ ▼ư❝ ♥➔② ❞➔♥❤ ❝❤♦ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ lM (E)✳ ❱ỵ✐ ♠é✐ ❤➔♠ ❖r❧✐❝③ M ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ỗ ữỡ E ỷ P = {pα : α ∈ I}✱ t❛ ✤➦t ∞ hM (E) = x = (xn ) ⊂ E : M n=1 pα (xn ) ❚❛ t ữủ t q s ỵ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ t❤➻ hM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ lM (E)✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➛✉ t✐➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ hM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ lM (E)✳ ❉♦ hM (E) ⊂ lM (E) ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ hM (E) ✤â♥❣ ❦➼♥ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t ổ ổ ữợ sỷ x, y ∈ hM (E) ✈➔ λ ∈ K✳ ❑❤✐ ✤â✱ ♥➳✉ λ = t❤➻ λx = ∈ hM (E)✳ ✷✾ ◆➳✉ λ = t❤➻ ✈ỵ✐ ♠é✐ α ∈ I ✱ tø ∞ M n=1 ρ t❛ ❝â |λ| ✈ỵ✐ ♠å✐ ρ > t❛ ❧➜② ρ = ∞ n=1 pα (λxn ) M ρ pα (xn ) 0✳ ❉♦ ✤â✱ ✈ỵ✐ ♠é✐ α ∈ I ∞ n=1 pα (xn + yn ) M ρ ∞ n=1 pα (2xn ) M + ρ ∞ M n=1 pα (2yn ) 0✳ ❱➻ ✈➟② x + y ∈ hM (E) ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ hM (E) ✤â♥❣ tr♦♥❣ lM (E) ●✐↔ sû (xk ) ❧➔ ❞➣② tr♦♥❣ hM (E) ✈➔ xk ❤ë✐ tư tỵ✐ x0 tr♦♥❣ lM (E)✳ ❑❤✐ ✤â✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ x0 ∈ hM (E)✳ ❚❤➟t ✈➟②✱ tø xk ❤ë✐ tư tỵ✐ x0 tr♦♥❣ lM (E) s✉② r❛ ợ > tỗ t k0 s ∞ k qα (x − x ) = inf{ρ > : n=1 pα (xkn − x0n ) ρ M ≤ 1} < ρ ✸✵ ✱ ✈ỵ✐ ♠å✐ k k0 ✈➔ ♠å✐ α ∈ I ✳ ❙✉② r❛ ∞ pα (xkn0 − x0n ) M ρ n=1 ≤ ∀α ∈ I ▼➦t ❦❤→❝✱ ✈➻ xk0 ∈ hM (E) ♥➯♥ ∞ M pα (xkn0 ) n=1 ρ s ❝❤♦ M (t0 ) = 0✳ ❚ø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ M ✈➔ lim M (t) = ∞ s✉② r❛ tỗ t T0 tr ợ t s M (T0 ) = 0✳ t→∞ ●✐↔ sû x = (xn ) ∈ l∞ (E)✳ ❱ỵ✐ ♠é✐ α ∈ I ✱ t❛ ✤➦t kα = sup pα (xn ) < ∞ n ✸✶ ▲➜② ρ = 2kα t❛ t❤✉ ✤÷đ❝ T0 pα (xn ) T0 pα (xn ) = ρ 2kα T0 ❚ø t➼♥❤ ❝❤➜t ❦❤æ♥❣ ❣✐↔♠ ❝õ❛ M (t) t❛ ❝â pα (xn ) ρ M M T0 =0 ✈ỵ✐ ♠å✐ n✳ ❚❛ ♥❤➟♥ ✤÷đ❝ ∞ pα (xn ) = ρ M n=1 ❍❛② x = (xn ) ∈ lM (E)✳ ❱➻ ✈➟② l (E) = lM (E) õ ỗ ♥❤➜t s➩ ❧➔ ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤ ❣✐ú❛ l∞ (E) ✈➔ lM (E)✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ lM (E) ✤➥♥❣ ❝➜✉ ✈ỵ✐ l∞ (E) t❛ ❝á♥ ♣❤↔✐ ❝❤➾ r❛ ❝→❝ ♥û❛ ú s s ữủ ợ ①➨t l∞ (E) ✈ỵ✐ ♥û❛ ❝❤✉➞♥ bα (x) = sup pα (xn ) n ❚ø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ❝â✱ ✈ỵ✐ ρ = ∞ M n=1 2k t❤➻ T0 pα (xn ) =0 : M n=1 ◆❤÷ ✈➟②✱ bα (x) ✈ỵ✐ ♠å✐ x ∈ lM (E)✳ pα (xn ) ρ T0 qα (x) 1} 2kα 2bα (x) = T0 T0 ✭✷✳✶✶✮ ✸✷ ◆➳✉ x ∈ lM (E) ✈➔ qα (x) = 0✳ ❑❤✐ ✤â ∞ pα (xn ) qα (x) M n=1 ✈ỵ✐ ♠å✐ x ∈ lM (E) ✈➔ x = 0✳ ◆➯♥ t❛ ❝â M pα (xn ) qα (x) ✈ỵ✐ ♠å✐ n✳ ●å✐ T1 ❧➔ sè ❧ỵ♥ ♥❤➜t s❛♦ ❝❤♦ M (T1 ) = T1 tỗ t t tử ❝õ❛ M ✱ lim M (t) = ∞ ✈➔ M (0) = 0✮✳ ❉♦ t➼♥❤ ❦❤æ♥❣ ❣✐↔♠ ❝õ❛ M ♥➯♥ t→∞ pα (xn ) qα (x) T1 ✈ỵ✐ ♠å✐ n✳ ❚❛ t❤✉ ✤÷đ❝ bα (x) = sup pα (xn ) ✭✷✳✶✷✮ T1 qα (x) n ✈ỵ✐ ♠å✐ qα (x) = 0✳ ◆➳✉ qα (x) = t❤➻ tø ∞ qα (x) = inf ρ > : M n=1 s✉② r❛ ∞ M n=1 pα (xn ) ρ pα (xn ) ρ 1} = ✈ỵ✐ ♠å✐ ρ > 0✳ ❑❤✐ ✤â✱ ♥➳✉ pα (xn ) = t tứ lim M (t) = s r tỗ t→∞ pα (xn ) > ▼➙✉ t❤✉➝♥ ✈ỵ✐ t↕✐ ρ > s❛♦ ❝❤♦ M ρ ∞ M n=1 pα (xn ) ρ ❱➻ ✈➟② pα (xn ) = ✈ỵ✐ ♠å✐ n✱ tù❝ ❧➔ bα (x) = sup pα (xn ) = n ❱➻ ✈➟②✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✷✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ lM (E) ứ ỵ t õ lM (E) ✤➥♥❣ ❝➜✉ ✈ỵ✐ l∞ (E)✳ ✸✸ ✷✮ ❱➻ C0 (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ l∞ (E) ✈➔ hM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ lM (E)✱ ❦❤✐ M s✉② ❜✐➳♥ lM (E) ✤➥♥❣ ❝➜✉ ✈ỵ✐ l∞ (E) ♥➯♥ ✤➸ ❝❤ù♥❣ ♠✐♥❤ hM (E) ✤➥♥❣ ❝➜✉ ✈ỵ✐ C0 (E) t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ hM (E) = C0 (E) ❦❤✐ M s✉② ❜✐➳♥✳ ●✐↔ sû x = (xn ) ∈ hM (E)✳ ❑❤✐ ✤â ∞ M n=1 pα (xn ) x / C0 (E) t tỗ t ∈ I s❛♦ ❝❤♦ pα (xn ) n → ∞✳ r tỗ t (xnk ) s pα (xnk ) nk ✳ ▲➜② ρ s❛♦ ❝❤♦ < ρ r 2T0 ∞ M n=1 ❚✉② ♥❤✐➯♥✱ ❞♦ pα (xnk ) ρ 2T0 pα (xn ) < ∞ ρ pα (xnk ) ρ ✈ỵ✐ ♠å✐ nk ✳ ❉♦ ✤â lim M ∞ n=1 M r ρ 2T0 ✈ỵ✐ ♠å✐ nk ♥➯♥ s✉② r❛ M ❝❤✉é✐ r > ✈ỵ✐ ♠å✐ ❦❤✐ ✤â pα (xnk ) ρ ❙✉② r❛ ❦❤✐ nk →∞ pα (xn ) ✳ ❱➻ ρ M (2T0 ) > pα (xxk ) ρ = 0✳ ▼➙✉ t❤✉➝♥ ✈ỵ✐ sü ❤ë✐ tư ❝õ❛ ✈➟② hM (E) ⊂ C0 (E) ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû x = (xn ) ∈ C0 (E)✳ ❚❛ ❝❤➾ r❛ x ∈ hM (E)✳ ❚❤➟t ✈➟②✱ (xn ) ∈ Co (E) s✉② r❛ lim xn = ⇒ lim pα (xn ) = ✈ỵ✐ ♠é✐ α ∈ I ✳ n→∞ n õ > tũ ỵ ợ ộ I tỗ t n0 s p (xn ) < ρT0 pα (xn ) pα (xn ) ✈ỵ✐ ♠å✐ n n0 ✳ ❍❛② < T0 ✈ỵ✐ ♠å✐ n n0 ✳ ❱➻ ✈➟② M ( ) ρ ρ M (T0 ) = ✈ỵ✐ ♠å✐ n n0 ❚❛ t❤✉ ✤÷đ❝ ∞ n=1 pα (xn ) M = ρ n0 −1 M k=1 pα (xn ) < ∞ ρ ✸✹ ❱➻ ✈➟② x = (xn ) ∈ hM (E)✳ ❉♦ ✤â C0 (E) ⊂ hM (E)✳ ❚ø ✤â t❛ ❝â C0 (E) = hM (E) ❙❛✉ ✤➙② ❧➔ ♠ët ❧ỵ♣ ❤➔♠ ❖r❧✐❝③ q✉❛♥ trå♥❣✳ ✷✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✼❪✮ ❍➔♠ ❖r❧✐❝③ M ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ M (qt) < ∞ ✈ỵ✐ q > ♥➔♦ ✤â✳ t→0 M (t) ∆q t↕✐ ♥➳✉ lim ❘ã r➔♥❣ M (t) = ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆q t↕✐ 0✳ ✷✳✷✳✹ ❇ê ✤➲✳ ✭❬✼❪✮ ❍➔♠ ❖r❧✐❝③ M t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆q t↕✐ ✈ỵ✐ ♠å✐ q>0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆2 t↕✐ 0✳ ỵ s ữ r hM (E) = lM (E) t❤æ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ∆q ❝õ❛ ❤➔♠ r ỵ sỷ M r ổ s E ổ ỗ ♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❤å ♥û❛ ❝❤✉➞♥ P = {pα : α ∈ I} ✳ ❑❤✐ ✤â✱ ♥➳✉ M t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆2 t↕✐ t❤➻ lM (E) = hM (E)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆2 t↕✐ 0✳ ❑❤✐ ✤â✱ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✹ t❛ ❝â M t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∆q ✈ỵ✐ ♠é✐ q > 0✳ ▲➜② x ∈ lM (E)✳ ❑❤✐ ✤â✱ ợ ộ I tỗ t > s❛♦ ❝❤♦ ∞ M n=1 ❚❛ t❤✉ ✤÷đ❝ lim M n→∞ pα (xn ) ρ0α pα (xn ) ρ0α pα (xn ) ρα0 < ∞ = 0✳ ❉♦ M ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✈➔ ❧✐➯♥ tư❝ t↕✐ pα (xn ) < ✈ỵ✐ ♠å✐ n→∞ ρ0α ρ0 n n0 ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ ρ > →♣ ❞ư♥❣ ✤✐➲✉ q ợ q = t õ tỗ t↕✐ K > s❛♦ ❝❤♦ ♥➯♥ lim = õ tỗ t n0 s t M < KM (t) ρ ✸✺ ✈ỵ✐ ♠å✐ < t 1✳ ◆❤÷ ✈➟② pα (xn ) ρ0α pα (xn ) M =M ρ ρ ρ0α ✈ỵ✐ ♠å✐ n ∞ M n=1 KM pα (xn ) ρ0α n0 ✳ ❚❛ t❤✉ ✤÷đ❝ pα (xn ) ρ n0 M = n=1 n0 ≤ M n=1 ◆❤÷ ✈➟② + pa (xn ) ρ +K ∞ M n=1 ∞ pα (xn ) ρ pα (xn ) ρ M n=n0 +1 ∞ M n=n0 +1 pα (xn ) ρ0α < ∞ pα (xn ) 0✱ tù❝ ❧➔ x ∈ hM (E)✳ ❉♦ ✤â lM (E) ⊂ hM (E)✳ ❱➻ ✈➟② lM (E) = hM (E) ✸✻ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ỗ ữỡ ỹ trú t t ỵ trú ỗ ữỡ ỵ ổ tr tr ổ ỗ ữỡ ❤➔♠ ❖r❧✐❝③ ✈➔ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ú t q ỵ ỵ ữ r ởt số t t ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ tr tr ổ ỗ ữỡ r t ỵ ỵ ỵ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❚r➛♥ ❱➠♥ ❹♥ ✈➔ ❑✐➲✉ P❤÷ì♥❣ ❈❤✐ ✭✷✵✶✹✮✱ →♥ ♣❤→t tr✐➸♥ ❣✐→♦ ✈✐➯♥ ❚❍P❚✳ ❬✷❪ ❍➔ ❍✉② ❇↔♥❣ ✭✷✵✵✸✮✱ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ✣ë ✤♦ ✈➔ t ỹ ỵ tt ổ r ✣↕✐ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✈➔ ▲➯ ▼➟✉ ❍↔✐ ✭✷✵✵✷✮✱ ❈ì sð ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❚➟♣ ■ ✈➔ ❚➟♣ ■■✱ ỵ tt ❤➔♠ ✈➔ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ổ ỗ ữỡ õ tốt rữớ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❱➲ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ❖r❧✐❝③✱ ▲✉➟♥ ✈➠♥ ❬✺❪ ❚r÷ì♥❣ ❚❤à ❚❤✉ ❍✐➲♥ ✭✷✵✶✹✮✱ ❚❤↕❝ s➽ ❚♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❚♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s ■✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✳ ❬✼❪ ❏✳ ▲✐♥❞❡♥str❛✉ss ❛♥❞ ▲✳ ❚③❛❢r✐r✐ ✭✶✾✼✼✮✱ ❈❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s✳ ■✳ ❙❡q✉❡♥❝❡ s♣❛❝❡s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲◆❡✇ ❨♦r❦✳ ❬✽❪ ❘✳ ▼❡✐s❡ ❛♥❞ ❉✳ ❱♦❣t ✭✶✾✾✼✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ❬✻❪ ●✳ ❑☎♦t❤❡✭✶✾✻✾✮✱ ❈❧❛❞❡r♦♥ Pr❡ss✱ ❖①❢♦r❞✳ ❬✾❪ ❆✳ P✐❡ts❝❤ ✭✶✾✼✷✮✱ ◆✉❝❧❡❛r ▲♦❝❛❧❧② ❈♦♥✈❡① ❙♣❛❝❡s✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✳ ... ♣❤÷ì♥❣✳ ❚❤➟t ✈➟②✱ ♥➳✉ ♥❣÷đ❝ ❧↕✐ t❤➻ ♥â ❧➔ ❦❤ỉ♥❣ õ tỗ t tr R s❛♦ ❝❤♦ tỉ♣ỉ s✐♥❤ r❛ ❜ð✐ ❝❤✉➞♥ trị? ??❣ ✈ỵ✐ tỉ♣ỉ s✐♥❤ r❛ ❜ð✐ {pn }✳ ❳➨t B(0, 1) = {x R : x < 1} õ tỗ t V = {x ∈ R∞ : pi (x) = |xi... ự t ữợ ữợ t❛ ❝❤➾ r❛ ❤❛✐ t➟♣ ❤ñ♣ lM (E) ✈➔ lq (E) ữợ t r ♥û❛ ❝❤✉➞♥ ①→❝ ✤à♥❤ tr➯♥ ❝❤ó♥❣ trị? ??❣ ♥❤❛✉✳ ▲➜② x ❜➜t ❦ý t❤✉ë❝ lM (E)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ α ∈ I t❛ ❝â ∞ M n=1 pα (xn )