❯❇◆❉ ❚➓◆❍ ✣➬◆● ◆❆■ ❚❘×❮◆● ✣❸■ ❍➴❈ ✣➬◆● ◆❆■ ◆●❯❨➍◆ ❍❖⑨◆● ❍■➏P ❚➷P➷ ❨➌❯ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ✣➚◆❍ ❈❍❯❽◆ ❇⑨■ ❚❾P ▲❰◆ ▼➷◆✿ ●■❷■ ❚➑❈❍ ❍⑨▼ ✶ ✣➬◆● ◆❆■ ✲ ✷✵✶✽ ❯❇◆❉ ❚➓◆❍ ✣➬◆● ◆❆■ ❚❘×❮◆● ✣❸■ ❍➴❈ ✣➬◆● ◆❆■ ◆●❯❨➍◆ ❍❖⑨◆● ❍■➏P ❚➷P➷ ❨➌❯ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ✣➚◆❍ ❈❍❯❽◆ ◆❣➔♥❤ ❤å❝✿ ❙÷ ♣❤↕♠ ❚♦→♥ ❇⑨■ ❚❾P ▲❰◆ ▼➷◆✿ ●■❷■ ❚➑❈❍ ❍⑨▼ ✶ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿ ❚❙✳ ❇Ị■ ❚❍➌ ◗❯❹◆ ✣➬◆● ◆❆■ ✲ ✷✵✶✽ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✷ ✸ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ⑩♥❤ ①↕ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✲ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❙✐➯✉ ♣❤➥♥❣ ✲ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❚æ♣æ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✼ ✷✳✶ ◆❤➢❝ ❧↕✐ tæ♣æ ②➳✉ ①→❝ ✤à♥❤ ❜ð✐ ♠ët ❤å ❤➔♠ ✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ tæ♣æ ②➳✉ ✷✳✸ ❚æ♣æ ②➳✉ ✯ σ(X , X) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ σ(X, X ) ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✶✼ ✶✽ ✶ ▲í✐ ♥â✐ ✤➛✉ ●✐↔✐ t➼❝❤ ❧➔ ♠ët ♠ỉ♥ ❤å❝ ❝â tø ❧➙✉ ✈➔ ✤➣ ❣➦t ❤→✐ ✤÷đ❝ ♥❤✐➲✉ t❤➔♥❤ tü✉✱ tr♦♥❣ ✤â ❝â ❣✐↔✐ t➼❝❤ ❤➔♠✳ ●✐↔✐ t➼❝❤ ❤➔♠ ✤➣ ①➙♠ ♥❤➟♣ ✈➔♦ t➜t ❝↔ ❝→❝ ♥❣➔♥❤ ❝õ❛ ❚♦→♥ ❤å❝ ❝â ❧✐➯♥ q✉❛♥ ✈➔ sû ❞ö♥❣ ✤➳♥ ❝ỉ♥❣ ❝ư ❝õ❛ ●✐↔✐ t➼❝❤ ✈➔ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝t♦r✳ ❚ỉ♣ỉ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ◗✉❛ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ t❛ ❝â ❝ì ❤ë✐ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ tæ♣æ✱ ♠ët ♥ë✐ ❞✉♥❣ ❦❤→ q✉❡♥ t❤✉ë❝ ✈➔ ❜❛♦ ❤➔♠ ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ✈➔ tê♥❣ q✉→t ❝õ❛ ❣✐↔✐ t ữỡ ữỡ ỗ ❝❤÷ì♥❣✳ tỉ♣ỉ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝t♦r✳ tỉ♣ỉ ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ỗ tự ❑❤ỉ♥❣ ❣✐❛♥ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❚r♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ tổ ổ ũ t ỡ ố ợ ữợ ❝õ❛ t✐➳♥ s➽ ❇ò✐ ❚❤➳ ◗✉➙♥ ✤➣ ✤å❝✱ ❦✐➸♠ tr❛ tt tổ õ ỵ ổ ũ ❣✐→ trà✳ ❈✉è✐ ❝ò♥❣✱ tỉ✐ ♠✉è♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱❜↕♥ ❜➧ ♥❤ú♥❣ ♥❣÷í✐ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ ✈➲ ♠➦t t✐♥❤ t❤➛♥✳ ❉♦ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠✱ t➔✐ ❧✐➺✉ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât ✈➔ ❤↕♥ qỵ t ổ s õ ỵ ỗ ũ ✷✵✶✽ ◆❣✉②➵♥ ❍♦➔♥❣ ❍✐➺♣ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ❈❤♦ t➟♣ X ❜➜t ❦➻✱ ♠ët ❤å τ ♥❤ú♥❣ t➟♣ ❝♦♥ ❝õ❛ X ❧➔ ♠ët tæ♣æ tr➯♥ X ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ✐✳ ∅, X ∈ τ ✐✐✳ ◆➳✉ Ai ∈ τ, ∀i ∈ I Ai ∈ τ t❤➻ i∈I n ✐✐✐✳ ◆➳✉ Ai ∈ τ, ∀i = 1, n Ai ∈ τ t❤➻ i=1 ❑❤✐ ✤â✱ ❝➦♣ (X, τ ) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ✳ ▼é✐ ♣❤➛♥ tû x∈X ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❙♦ s→♥❤ tæ♣æ ❈❤♦ ❤❛✐ tæ♣æ σ ❤❛② tæ♣æ σ τ ✈➔ σ X ✳ ◆➳✉ τ ⊂ σ t❤➻ t❛ ♥â✐ tæ♣æ τ ✭❤♦➦❝ ♠à♥ ❤ì♥✮ τ ✳ tr➯♥ ♠↕♥❤ ❤ì♥ ②➳✉ ❤ì♥ ✭❤♦➦❝ t❤ỉ ❤ì♥✮ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈ì sð ✈➔ t✐➲♥ ❝ì sð ❈❤♦ τ ❧➔ ♠ët tæ♣æ tr➯♥ ▼ët ❤å ❝♦♥ β ❝õ❛ τ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝ì sð ❝õ❛ τ ♥➳✉ β ✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❤å ❝♦♥ β ❝õ❛ τ ❧➔ ❝ì sð ❝õ❛ τ ♥➳✉ ∀A ∈ τ, ∀x ∈ A, ∃V ∈ β : x ∈ V ⊂ A✳ ▼ët ❤å ❝♦♥ σ ❝õ❛ τ ✤÷đ❝ ❣å✐ ❧➔ t✐➲♥ ❝ì sð ❝õ❛ τ ♥➳✉ ❤å t➜t ❝↔ ❝→❝ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ t❤✉ë❝ σ ❧➔ ♠ët ❝ì sð ❝õ❛ τ ✳ ◆❤÷ ✈➟②✱ ❤å ❝♦♥ σ ❝õ❛ τ ❧➔ t✐➲♥ ❝ì sð ❝õ❛ τ ♥➳✉ ∀A ∈ τ, ∀x ∈ A, ∃W1 , W2 , , Wn ∈ σ : x ∈ W1 ∩ W2 ∩ · · · ∩ Wn ⊂ A✳ ♠å✐ t➟♣ t❤✉ë❝ τ X✳ ✤➲✉ ❜➡♥❣ ❤ñ♣ ❝õ❛ ♠ët ❤å ❝→❝ t➟♣ t❤✉ë❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ▲➙♥ ❝➟♥ x ∈ X ✳ ❚➟♣ ❝♦♥ V ❝õ❛ X ✤÷đ❝ ởt x tỗ t↕✐ t➟♣ ♠ð G s❛♦ ❝❤♦ x ∈ G ⊂ V ✳ ◆➳✉ ❧➙♥ ❝➟♥ V ❝õ❛ x ❧➔ t➟♣ ♠ð t❤➻ V ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ♠ð ❝õ❛ x✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ ✸ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ P❤➛♥ tr♦♥❣ ✲ ❇❛♦ ✤â♥❣ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ ✐✳ P❤➛♥ tr♦♥❣ ❝õ❛ A ❧➔ ❤ñ♣ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ♠ð ❝❤ù❛ tr♦♥❣ A◦ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â✿ ✈➔ A A ❦➼ ❤✐➺✉ ❧➔ t❤➻ A◦ ✳ A◦ ⊂ B ◦ A = A◦ ✳ ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ✤â♥❣ ❝❤ù❛ A ❧➔ t➟♣ A = A✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â✿ ✤â♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A✱ A❀ A ⊂ B ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t ❝❤ù❛ tr♦♥❣ ❧➔ t➟♣ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✐✐✳ ❇❛♦ ✤â♥❣ ❝õ❛ A⊂X ✤â♥❣ ♥❤ä ♥❤➜t ❝❤ù❛ A✱ ❦➼ ❤✐➺✉ A❀ A ⊂ B A✳ t❤➻ A⊂B A ✈➔ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ❈→❝ ✤✐➸♠ tæ♣æ q✉❛♥ trå♥❣ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X✱ t➟♣ ❝♦♥ A ✈➔ ✤✐➸♠ x∈X ✐✳ ✣✐➸♠ x ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ A ♥➳✉ x ❝â ♠ët ❧➙♥ ❝➟♥ V s❛♦ ❝❤♦ V ⊂ A✳ ✐✐✳ ✣✐➸♠ x ❣å✐ ❧➔ ✤✐➸♠ ♥❣♦➔✐ ❝õ❛ A ♥➳✉ x ❝â ♠ët ❧➙♥ ❝➟♥ V s❛♦ ❝❤♦ V ∩ A = ∅✳ x ❣å✐ ❧➔ ✤✐➸♠ V ∩ (X\A) = ∅✳ ✐✐✐✳ ✣✐➸♠ ❜✐➯♥ ❝õ❛ A ♥➳✉ ♠å✐ ❧➙♥ ❝➟♥ V ❝õ❛ x ✤➲✉ ❝â V ∩A=∅ ✈➔ ✶✳✷ ⑩♥❤ ①↕ ❧✐➯♥ tö❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ⑩♥❤ ①↕ ❧✐➯♥ tö❝ X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ →♥❤ ①↕ f : X → Y ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ x ∈ X ♥➳✉ ✈ỵ✐ ♠å✐ V f (x) tr Y tỗ t↕✐ ❧➙♥ ❝➟♥ U −1 ❝õ❛ x tr♦♥❣ X s❛♦ ❝❤♦ f (U ) ⊂ V ✱ ❤❛② f (V ) ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x✳ ⑩♥❤ ①↕ f ❧✐➯♥ tö❝ tr➯♥ X ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ♠å✐ x X ỵ X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ →♥❤ ①↕ f : X → Y ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ✐✳ f ❧✐➯♥ tư❝✳ ✐✐✳ f −1(G) ♠ð tr♦♥❣ X ✈ỵ✐ ♠å✐ t➟♣ G ♠ð tr♦♥❣ Y ✳ ✐✐✐✳ f −1(U ) ✤â♥❣ tr♦♥❣ X ✈ỵ✐ ♠å✐ t➟♣ U ✤â♥❣ tr♦♥❣ Y ✳ ✐✈✳ f (A) ⊂ f (A) ✈ỵ✐ ♠å✐ A ⊂ X ✳ ✹ ✶✳✸ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ f :X→Y ✐✳ ✐✐✳ ❈❤♦ X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r tr➯♥ tr÷í♥❣ K✳ ⑩♥❤ ①↕ ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ♥➳✉ ♥â t❤ä❛ ♠➣♥✿ f (x + y) = f (x) + f (y), ∀x, y ∈ X ✳ f (kx) = kf (x), ∀k ∈ K, ∀x ∈ X ✳ ◆➳✉ Y =X t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ t tỷ t t tỹ ỗ t t ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤✮✳ ◆➳✉ Y =K t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✳ ✶✳✹ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✲ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ❚❛ ♥â✐ · ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r ❧➔ ❝❤✉➞♥ tr➯♥ X X tr➯♥ tr÷í♥❣ K ✈➔ →♥❤ ①↕ · : X → R✳ ♥➳✉ t❤ä❛ ♠➣♥✿ x ≥ 0, ∀x ∈ X ✐✳ x = ⇔ x = θ✳ ✐✐✳ ✐✐✐✳ kx = |k| x , ∀k ∈ K, ∀x ∈ X ✳ ✐✈✳ x + y ≤ x + y , ∀x, y ∈ X ✳ ❑❤✐ ✤â (X, · ) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ✭ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✮✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✷✳ ❈❤♦ ❛✮ ▼ët ❞➣② ❝→❝ ✈❡❝t♦r X {xn } ❧➔ ❦❤æ♥❣ ữủ tử tợ xX ♥➳✉ lim xn − x = n→∞ ∀ε > 0, ∃N > : ∀n ≥ N, xn − x ≤ ε✳ xn → x ❤♦➦❝ lim xn = x✳ ◆❣❤➽❛ ❧➔ ❑➼ ❤✐➺✉ n→∞ ❜✮ ▼ët ❞➣② ❝→❝ ✈❡❝t♦r ◆❣❤➽❛ ❧➔ ❝✮ X {xn } ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ lim m,n→∞ xm − xn = ∀ε > 0, ∃N > : ∀m, n ≥ N, xm − xn ≤ ε✳ ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② ✤➲✉ ❤ë✐ tö✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✤➛② ✤õ ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✸✳ ✐✳ {xn } ❈❤♦ X ữợ ổ inf xn > 0✳ ✺ {xn } ⊂ X ✱ ❦❤✐ ✤â✿ ✐✐✳ {xn } ❜à ❝❤➦♥ tr➯♥ ♥➳✉ ✐✐✐✳ {xn } ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ ❤â❛ ♥➳✉ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✹✳ ❣✐❛♥ ✤è✐ ♥❣➝✉✮ ❝õ❛ tö❝ tr➯♥ X✳ X ❈❤♦ X sup xn < ∞✳ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ✭❦❤ỉ♥❣ ✭❦➼ ❤✐➺✉ X ✮ ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ ❝â ❝❤✉➞♥ ✤è✐ ♥❣➝✉ f f ∈ X ✈➔ x ∈ X ♥❣➝✉ X , X ✳ ❑❤✐ ✤è✐ xn = 1, ∀n✳ X t❛ ❦➼ ❤✐➺✉ = |f (x)| x x∈X,x=θ f, x sup t❤❛② ✈➻ f (x)✳ ❚❛ õ ã, ã t ổ ữợ ỗ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ H = {x ∈ X; f (x) = α} ♣❤÷ì♥❣ tr [f = ] ỗ t ổ õ t➟♣ ❚❛ ♥â✐ H ❧➔ s✐➯✉ ♣❤➥♥❣ ❝â f : X → R✱ ❦❤→❝ →♥❤ ✤÷đ❝ ❣å✐ ❧➔ s✐➯✉ ♣❤➥♥❣✳ ỗ t ỗ AX ữủ t ỗ tx + (1 − t)y ∈ A, ∀x, y ∈ A, ∀t ∈ [0, 1] ●✐↔ sû A ✈➔ B A ⊂ E, B ⊂ E ✳ ❚❛ ♥â✐ r➡♥❣ s✐➯✉ ♣❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤ [f = α] t→❝❤ t❤❡♦ ♥❣❤➽❛ rë♥❣ ♥➳✉ f (x) ≤ α, ∀x ∈ A H H t→❝❤ A ✈➔ B ✈➔ f (x) ≥ α, ∀x B t t tỗ t f (x) ≤ ε − α, ∀x ∈ A ✈➔ ε>0 s❛♦ ❝❤♦ f (x) ≥ ε + α, ∀x ∈ B ỵ ỵ tự ♥❤➜t ❈❤♦ A ⊂ E ✈➔ B ⊂ E ❧➔ t ỗ rộ rớ sỷ A õ tỗ t ởt s ✤â♥❣ t→❝❤ A ✈➔ B t❤❡♦ ♥❣❤➽❛ rë♥❣✳ ✐✐✳ ❉↕♥❣ ❤➻♥❤ ❤å❝ t❤ù ❤❛✐ ❈❤♦ A ⊂ E ✈➔ B E t ỗ rộ rớ ♥❤❛✉✳ ●✐↔ sû A ✤â♥❣ ✈➔ B ❝♦♠♣❛❝t✳ ❑❤✐ ✤â tỗ t ởt s õ t A B t❤❡♦ ♥❣❤➽❛ ❝❤➦t✳ ✻ ❈❤÷ì♥❣ ✷ ❚ỉ♣ỉ ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✷✳✶ ◆❤➢❝ ❧↕✐ tæ♣æ ②➳✉ ①→❝ ✤à♥❤ ❜ð✐ ♠ët ❤å ❤➔♠ f : X → Y ✳ ◆➳✉ tr➯♥ Y ❝â ♠ët tæ♣æ τY t❤➻ ❞♦ f −1 ❜↔♦ t♦➔♥ ❝→❝ −1 ♣❤➨♣ t♦→♥ t➟♣ ❤ñ♣✱ ♥➯♥ f (τY ) = {f −1 (G)|G ∈ τY } ❧➔ ♠ët tæ♣æ tr➯♥ X ✳ −1 ◆➳✉ tr➯♥ X ❝â s➤♥ tỉ♣ỉ τX t❤➻ →♥❤ ①↕ f ❧✐➯♥ tư❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (τY ) ⊂ τX ✱ −1 ♥❣❤➽❛ ❧➔ ♥❣❤à❝❤ ↔♥❤ ❝õ❛ tæ♣æ tr➯♥ Y (f (τY )) ②➳✉ ❤ì♥ tỉ♣ỉ tr➯♥ X(τX )✳ ❉♦ ✤â t❛ t❤➜② r➡♥❣✱ ♥➳✉ tr➯♥ Y ✤➣ ❝â ♠ët tæ♣æ ♠➔ tr➯♥ X ❝❤÷❛ ❝â tỉ♣ỉ t❤➻ ❝â −1 t❤➸ ❜✐➳♥ X t❤➔♥❤ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ❜➡♥❣ ❝→❝❤ ❣→♥ ❝❤♦ ♥â tæ♣æ f (τY )✿ ✤â ❧➔ tæ♣æ ②➳✉ ♥❤➜t ❜↔♦ ✤↔♠ sü ❧✐➯♥ tö❝ ❝õ❛ →♥❤ ①↕ f ✳ ❚ê♥❣ q✉→t ❤ì♥✱ ❝❤♦ ♠ët ❤å →♥❤ ①↕ {fα } ✈ỵ✐ fα : X → Yα ✱ tr♦♥❣ ✤â ♠é✐ Yα ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ✱ ✈ỵ✐ tỉ♣ỉ τα ✳ ❑❤✐ ➜②✱ tæ♣æ tr➯♥ X s✐♥❤ ❜ð✐ ❤å Φ = fα−1 (τα ) ❈❤♦ →♥❤ ①↕ α s➩ ❧➔ tæ♣æ ②➳✉ ♥❤➜t ✤↔♠ ❜↔♦ sü ❧✐➯♥ tö❝ ❝õ❛ t➜t ❝↔ ❝→❝ →♥❤ ①↕ ②➳✉ ①→❝ ✤à♥❤ tr➯♥ X ❜ð✐ ❤å fα ✱ t❛ ❣å✐ ♥â ❧➔ tæ♣æ fα ✳ ✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ tæ♣æ ②➳✉ σ(X, X ) X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ f ∈ X ✳ ❑➼ ❤✐➺✉ ϕf : X → R ❧➔ →♥❤ ①↕ ①→❝ ✤à♥❤ ❜ð✐ ϕf (x) = f, x ✳ ❑❤✐ f ❝❤↕② ❦❤➢♣ X t❛ ✤÷đ❝ ♠ët ❤å (ϕf )f ∈X ❝→❝ →♥❤ ①↕ tø X ✈➔♦ R✳ ❚❛ ❜ä q✉❛ tæ♣æ s✐♥❤ ❜ð✐ · ✈➔ ✤à♥❤ ♥❣❤➽❛ ởt tổổ ợ ữ s sỷ tæ♣æ ②➳✉ σ(X, X ) tr➯♥ (ϕf )f ∈X trð t❤➔♥❤ ❧✐➯♥ tư❝ ✭tỉ♣ỉ σ(X, X ) ❧➔ tỉ♣ỉ ②➳✉ ♥❤➜t tr➯♥ X ❝→❝ →♥❤ ①↕ ❚æ♣æ f ∈X X ❧➔ tæ♣æ ②➳✉ ♥❤➜t tr➯♥ s✐♥❤ ❜ð✐ ❤å →♥❤ ①↕ X ❜✐➳♥ t➜t ❝↔ (ϕf )f ∈X ✮✳ ✤↔♠ ❜↔♦ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ✤➲✉ ❧✐➯♥ tö❝✳ ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❚æ♣æ ②➳✉ σ(X, X ) ❧➔ t→❝❤ ✭❍❛✉s❞♦r❢❢✮✳ x1 , x2 ∈ X, x1 = x2 ✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝â ❤❛✐ t➟♣ ♠ð O1 , O2 ✤è✐ ✈ỵ✐ tæ♣æ ②➳✉ σ(X, X ) s❛♦ ❝❤♦ x1 ∈ O1 , x2 ∈ O2 ✈➔ O1 ∩ O2 = ∅✳ {x1 }, {x2 } t ỗ ré♥❣✱ rí✐ ♥❤❛✉ ✈➔ {x1 } ✤â♥❣✱ {x2 } ❝♦♠♣❛❝t ự t ỵ tự tỗ t s õ t t {x1 }, {x2 } r tỗ t↕✐ f ∈X ✈➔ k∈R s❛♦ ❝❤♦✿ f, x1 < k < f, x2 ✣➦t O1 = {x ∈ X, f, x < k} = ϕ−1 f ((−∞; k)) O2 = {x ∈ X, f, x > k} = ϕ−1 f ((k; +∞)) ❑❤✐ ✤â✱ O1 , O2 ❧➔ ♠ð ✤è✐ ✈ỵ✐ ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ❈❤♦ x σ(X, X ) t❤ä❛ ♠➣♥ x1 ∈ O1 , x2 ∈ O2 ∈ X, ε > 0 ✈➔ O1 ∩ O2 = ∅✳ ✈➔ t➟♣ {fi} ⊂ X ✈ỵ✐ i ∈ I ❤ú✉ ❤↕♥✳ ❳➨t t➟♣ V = {x ∈ X; | fi , x − x0 | < ε, ∀i ∈ I} ❑❤✐ ✤â✱ V ❧➔ ♠ët ❝ì sð ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x0 ✤è✐ ✈ỵ✐ tỉ♣ỉ σ(X, X )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤➣ ❜✐➳t r➡♥❣ ❤å x0 ♥➳✉ x0 ∈ V ⊂ U ✳ ❝➟♥ ❝õ❛ Vx0 ♥➔♦ ✤â ♥❤ú♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ U ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ ❜➜t ❦ý x0 ❝õ❛ x0 ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❧➙♥ ổ tỗ t V Vx0 s | fi , x0 − x0 | = < ε, ∀i ∈ I ♥➯♥ x0 ∈ V ✳ ▲➜② O ❧➔ ♠ët t➟♣ ♠ð ❝❤ù❛ x0 ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ tổổ (X, X ) s r tỗ t ỳ ❤↕♥ t✉②➳♥ t➼♥❤ f1 , f2 , , fn ✈➔ ❝→❝ t➟♣ ❝♦♥ ♠ð ❝õ❛ R ❧➔ O1 , O2 , , On s❛♦ ❚❛ t❤➜② r➡♥❣ ②➳✉ ❜➜t ❦➻✱ ❝→❝ →♥❤ ①↕ n ❝❤♦ x0 ∈ fj−1 (Oj ) j=1 ⊂ O✳ ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ j = 1, n t❤➻ fj , x0 ∈ Oj ✈➔ ❞♦ ♠é✐ Oj ❧➔ εj s❛♦ ❝❤♦ ( fj , x0 − εj , fj , x0 + εj ) ⊂ Oj ✳ ε = εj > 0✱ t❤➻ ✈ỵ✐ ♠å✐ j = 1, n, ( fj , x0 − ε, fj , x0 + ) Oj tỗ t 1jn r❛ ✈ỵ✐ x∈V ❜➜t ❦➻ t❤➻ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ V t❛ ✤÷đ❝ fj , x ∈ ( fj , x0 − ε, fj , x0 + ε) ⊂ Oj , ∀j = 1, n n ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ x∈ j=1 fj−1 (Oj ) ⊂ O ❤❛② V ⊂O ▼➺♥❤ ✤➲ ✷✳✷✳✹✳ ❈❤♦ ❞➣② x tr♦♥❣ X ✱ ❦❤✐ ✤â x ϕi (xn ) → ϕi (x), ∀i ∈ I ✳ n n →x ✭t❤❡♦ tæ♣æ τ ✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❈❤ù♥❣ ♠✐♥❤✳ • • ◆➳✉ xn → x t❤➻ ✣↔♦ ❧↕✐✱ ❣✐↔ sû ϕi (xn ) → ϕi (x) ✈ỵ✐ ♠å✐ i ∈ I ✈➻ ♠é✐ ϕi ❧✐➯♥ tö❝✳ U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ U = ϕ−1 (Vi ) i∈J J ⊂ I ❧➔ ❤ú✉ ❤↕♥✳ ❱ỵ✐ ♠é✐ i J tỗ t số Ni s ϕi (xn ) ∈ Vi n ≥ Ni ✳ ●✐↔ sû N = max Ni ✱ t❛ ❝â xn ∈ U ✈ỵ✐ n ≥ N ✳ ❍❛② xn → x✳ ✈ỵ✐ i∈J ✽ ✈ỵ✐ ❑➼ ❤✐➺✉✿ ❈❤♦ ❞➣② ❜ð✐ xn (xn ) ❝õ❛ X✱ t❛ ❦➼ ❤✐➺✉ sü ❤ë✐ tö ❝õ❛ xn ✈➲ x ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉ σ(X, X ) x✳ xn x ②➳✉ ✤è✐ ✈ỵ✐ σ(X, X )✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥❤➜♥ ♠↕♥❤✱ t❛ ♥â✐ rã✿ xn → x ♠↕♥❤✱ ✤➸ ❝❤➾ r➡♥❣ xn −x → 0✳ ✣➸ tr→♥❤ ♥❤➛♠ ❧➝♥✱ t❛ t❤÷í♥❣ ❝❤➼♥❤ ①→❝ ❤â❛✿ ▼➺♥❤ ✤➲ ✷✳✷✳✺✳ ❈❤♦ (x ) ❧➔ ❞➣② tr♦♥❣ X ✳ ❚❛ ❝â n ✐✳ xn x ✤è✐ ✈ỵ✐ σ(X, X ) ⇔ [ f, xn → f, x , ∀f ∈ X ]✳ ✐✐✳ ◆➳✉ xn → x ♠↕♥❤✱ t❤➻ xn x ②➳✉ ✤è✐ ✈ỵ✐ σ(X, X )✳ ✐✐✐✳ ◆➳✉ xn x ②➳✉ ✤è✐ ✈ỵ✐ σ(X, X ) t❤➻ xn ❜à ❝❤➦♥ ✈➔ xn ≤ lim inf xn ✳ ✐✈✳ ◆➳✉ xn x ②➳✉ ✤è✐ ✈ỵ✐ σ(X, X ) ✈➔ fn → f ♠↕♥❤ tr♦♥❣ X ✭t✳❧ fn − f → 0✮✱ t❤➻ fn, xn → f, x ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✐✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✹ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ tæ♣æ ②➳✉ ♥➯♥ f ❧➔ ❧✐➯♥ tö❝✱ ✈➻ ✈➟② t❛ s✉② r❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ | f, xn − f, x | = | f, xn − x | ≤ f · xn − x ✳ ✤â✱ ♥➳✉ xn → x t❤➻ f, xn → f, x ✱ t❤❡♦ ✐✳ t❛ ❝â ✤✐➲✉ ✐✐✳ ❚❛ ❝â✿ ❉♦ ✐✐✐✳ ❚❤❡♦ ỵ ts ợ ộ n → +∞ f ∈E✱ t➟♣ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ( f, xn )n ❧➔ ❜à ❝❤➦♥ ♥➯♥ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ | f, xn | ≤ f · xn t❛ ✤÷đ❝ | f, x | ≤ f lim inf xn ❞♦ ✤â x ≤ sup | f, x | ≤ lim inf xn f ≤1 ✐✈✳ ❚❛ ❝â✿ | fn , xn − f, x | ≤ | fn − f, xn | + | f, xn − x | ≤ fn − f · xn + | f, xn − x | ❚❤❡♦ ✐✐✐✱ ❚❤❡♦ ✐✱ xn ❜à ❝❤➦♥ ♥➯♥ fn − f · xn → 0✳ | f, xn − x | = | f, xn − f, x | → 0✳ ❉♦ ✤â✱ t❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✷✳✻✳ ❚æ♣æ ②➳✉ σ(X; X ) ✈➔ tỉ♣ỉ t❤÷í♥❣ trò♥❣ ♥❤❛✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ✾ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ t❛ ✤➣ ❜✐➳t r➡♥❣ tæ♣æ ②➳✉ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ tỉ♣ỉ ♠↕♥❤✱ ❞♦ ✤â t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❦❤✐ X ❤ú✉ ❤↕♥ ❝❤✐➲✉ t❤➻ tæ♣æ ②➳✉ ❝❤ù❛ tæ♣æ ♠↕♥❤✳ ❱➻ X X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ♥➯♥ ❝â ❝ì sð {e1 , e2 , , en } ợ xX t tỗ t n ❞✉② ♥❤➜t ❜ë sè (x1 , x2 , , xn ) ❞✉② ♥❤➜t s❛♦ ❝❤♦ x= x i ei ✳ i=1 x ❚❛ ✤à♥❤ ♥❣❤➽❛ X✳ ∞ = max |xi |✱ 1≤i≤n ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ · ∞ ❧➔ ❝❤✉➞♥ tr➯♥ ❚❤➟t ✈➟②✿ x ✐✳ ∞ kx ✐✐✳ = max |xi | ≥ 0✱ x ∞ 1≤i≤n ∞ = max |kxi | = k max |xi | = k x 1≤i≤n x= xi ei x+y ✈➔ y= ♠↕♥❤ tr➯♥ ❦➻ t❤➻ O X n yi ei ✱ ❦❤✐ ✤â x+y = i=1 ∞ ❚❛ ✤➣ ❜✐➳t✱ ❦❤✐ ∈R n i=1 ❙✉② r❛ ∞, k 1≤i≤n n ✐✐✐✳ ●✐↔ sû = ⇔ x = θ✳ X (xi + yi )ei i=1 = max |xi + yi | ≤ max |xi | + max |yi | = x 1≤i≤n 1≤i≤n 1≤i≤n ∞ + y ∞ ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ t➜t ❝↔ ❝→❝ ❝❤✉➞♥ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✈➔ tæ♣æ O ❧➔ t➟♣ ♠ð ♠↕♥❤ ❜➜t · ∞ ✳ ◆❣❤➽❛ ❧➔ ✈ỵ✐ ♠å✐ x ∈ O, ∃εx > s❛♦ ❝❤♦ B∞ (x, εx ) ⊂ O✳ O= B∞ (x, εx )✳ ❧➔ tỉ♣ỉ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❜➜t ❦➻ ❝❤✉➞♥ ♥➔♦✳ ◆➳✉ ❧➔ ♠ð tr♦♥❣ ❉♦ ✤â✱ t❛ ❝â t❤➸ ✈✐➳t x∈O ◆➯♥ ♥➳✉ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❜➜t ❦➻ ❤➻♥❤ ❝➛✉ ♠ð ❧➔ ♠ð ②➳✉✱ t❛ s➩ ❝â ✤÷đ❝ t➟♣ O ♠ð ♠↕♥❤ ❧➔ ♠ð ②➳✉ ✈➻ ♥â ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ t➟♣ ♠ð ②➳✉✳ ợ xX t >0 tũ ỵ t õ B∞ (x, ε) = {y ∈ X : y − x ∞ < ε} = {y ∈ X : ∀i = 1, n, |yi − xi | < ε} n ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ x i ei ∈ X x= ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ f1 , f2 , , fn ✤÷đ❝ i=1 ①→❝ ✤à♥❤ ❜ð✐ fi , x = xi ✱ t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✸ t❛ ❝â t❤➸ ✈✐➳t✿ B∞ (x, ε) = {y ∈ X : | fi , y − x | < ε, ∀i = 1, n} ✈➔ B∞ (x, ε) ❧➔ ♠ð ②➳✉✳ ❉♦ ✤â O ❧➔ ♠ð ②➳✉✳ ●✐í t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ X ✈æ ❤↕♥ ❝❤✐➲✉✱ tæ♣æ ②➳✉ ✈➔ tæ♣æ ♠↕♥❤ X ✱ S = {x ∈ X : x = 1}✳ ❑❤✐ ✤â S ❧➔ ✤â♥❣ ②➳✉✱ ♥❤÷♥❣ ✵ t❤✉ë❝ ✈➔♦ ❜❛♦ ✤â♥❣ ②➳✉ ❝õ❛ S ✱ t❤➟t ✈➟②✱ ✈ỵ✐ O ❧➔ ❧➙♥ ❝➟♥ ②➳✉ ❝õ❛ t tỗ t > ✈➔ f1 , f2 , , fn ∈ X s❛♦ ❝❤♦ ❦❤ỉ♥❣ trò♥❣ ♥❤❛✉✳ ❱ỵ✐ S ❧➔ ♠➦t ❝➛✉ ✤ì♥ ✈à tr♦♥❣ W = {x ∈ X : | fi , x | < ε} ⊂ O ⑩♥❤ ①↕ ✈➔ φ : X → Rn ①→❝ ✤à♥❤ ❜ð✐ φ(x) = ( f1 , x , f2 , x , , fn , x ) n ker φ = {x ∈ X : fi , x = 0, i = 1, n} = ker fi ✳ i=1 ✶✵ ❧➔ t✉②➳♥ t➼♥❤ X t❤➻ dim ker φ + dim Imφ = dim X = ∞✳ dim ker φ = ∞ ♥➯♥ ker φ = {0}✳ ❚ù❝ tỗ t x = ỵ sè ❝❤✐➲✉ ❝õ❛ dim Imφ ≤ n✱ s✉② r❛ s❛♦ ❝❤♦ fi , x = 0, ∀i = 1, n ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ λ ∈ R t❤➻ | fi , λx | = < ε, ∀i = 1, n ♥➯♥ λx ∈ W ⊂ O, ∀λ ∈ R✳ = t O ữủ ợ S ✳ ❱➟② ✈ỵ✐ ❜➜t ❦➻ ❧➙♥ ❝➟♥ ②➳✉ ❝õ❛ ✵ ❣✐❛♦ ✈ỵ✐ x S t❤➻ ✵ ♥➡♠ tr♦♥❣ ❜❛♦ ✤â♥❣ ②➳✉ ❝õ❛ S ✱ tù❝ ❧➔ ❜❛♦ ✤â♥❣ ②➳✉ ✈➔ ♠↕♥❤ ❝õ❛ S ❧➔ ❦❤→❝ ▼➔ ♥❤❛✉✳ ◆❤➟♥ ①➨t ✷✳✷✳✼✳ ❚➟♣ ♠ð ✭t➟♣ ✤â♥❣✮ ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉ σ(X; X ) ❧✉ỉ♥ ❧✉ỉ♥ ♠ð ✭✤â♥❣✮ ✤è✐ ✈ỵ✐ tỉ♣ỉ ♠↕♥❤✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ ❜➜t ❦ý✱ tæ♣æ ②➳✉ tỹ sỹ tổ ỡ tổổ tự tỗ t t➟♣ ♠ð ✭✤â♥❣✮ t❤❡♦ tæ♣æ ♠↕♥❤ ♠➔ ❦❤æ♥❣ ♠ð ✭✤â♥❣✮ t❤❡♦ tỉ♣ỉ ②➳✉✳ ❱➼ ❞ư ✷✳✷✳✽✳ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉✱ ❦❤✐ ✤â ♠➦t ❝➛✉ ✤ì♥ ✈à S = {x ∈ X : x = 1} ❦❤æ♥❣ ❧➔ t➟♣ ✤â♥❣ ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉ σ(X, X )✳ ❚❤➟t σ (X,X ) ✈➟②✱ t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ❜❛♦ ✤â♥❣ ❝õ❛ S t❤❡♦ tæ♣æ σ(X, X ) ❧➔ S ❜➡♥❣ BX = {x ∈ X : x ≤ 1} ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ X ✳ σ (X,X ) ✱ ❣✐↔ sû x0 ∈ BX ✈➔ V ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 t❤❡♦ ✣➸ ❝❤ù♥❣ ♠✐♥❤ BX ⊂ S tæ♣æ σ(X, X )✱ t❛ ❝❤ù♥❣ ♠✐♥❤ V ∩ S = ∅✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t V = {x ∈ X; | fi , x − x0 | < ε, ∀i ∈ I} ✈ỵ✐ ε > 0, fi ∈ X ✈➔ I ❤ú✉ ❤↕♥✳ ❈è ✤à♥❤ y0 ∈ X, y0 = : f, y0 = 0, ∀i ∈ I ✳ k sỷ y0 ữ ổ tỗ t õ →♥❤ ①↕ ϕ : X → R ①→❝ ✤à♥❤ ❜ð✐ ϕ(x) = ( fi , x )i=1,k ❧➔ ♠ët ✤ì♥ →♥❤ ✭✈➻ ker ϕ = {0}✮ ❤✐➸♥ ♥❤✐➯♥ ϕ ❧➔ t♦➔♥ →♥❤ ♥➯♥ ϕ ❧➔ s♦♥❣ →♥❤ ❤❛② ϕ ❧➔ ✤➥♥❣ ❝➜✉ tø X ✈➔♦ ϕ(X)✳ ❉♦ ✤â dim X k ổ ỵ X ổ ổ y0 ữ tr tỗ t ✤â✱ ❤➔♠ sè g(t) = x0 + ty0 ❧✐➯♥ tö❝ tr➯♥ [0; +∞) ✈ỵ✐ g(0) < ✈ỵ✐ lim g(t) = + tỗ t t0 > s x0 + t0 y0 = 1✱ ❞♦ ✤â x0 + t0 y0 ∈ V ∩ S ✳ ❈❤♦ t→+∞ ❙✉② r❛ S ⊂ BX ⊂ S σ (X,X ) ✳ ▼➔ t❛ t❤➜② {x ∈ X : | f, x | ≤ 1} BX = ❧➔ f ∈X , f ≤1 ❣✐❛♦ ❝õ❛ ❝→❝ t➟♣ ✤â♥❣ ②➳✉ ♥➯♥ σ (X,X S ❱➟② S ▼➔ ) BX ✤â♥❣ t❤❡♦ ❧➔ t➟♣ ✤â♥❣ ❜➨ ♥❤➜t ❝❤ù❛ S ❈❤♦ X σ(X, X )✱ σ(X, X )✳ t❤❡♦ ❦❤ỉ♥❣ ❧➔ t➟♣ ✤â♥❣ ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉ ❱➼ ❞ư ✷✳✷✳✾✳ σ(X, X )✳ ❞♦ ✤â BX = S σ (X,X ) ✳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✱ ❦❤✐ ✤â ❤➻♥❤ ❝➛✉ U = {x ∈ X : x < 1} ❦❤æ♥❣ ❧➔ t➟♣ ♠ð t❤❡♦ tỉ♣ỉ ②➳✉ ❚❤➟t ✈➙②✱ t❛ ❣✐↔ sû ♥❣÷đ❝ σ(X, X )✳ ❧↕✐✱ ♥➳✉ U ❧➔ t➟♣ ♠ð t❤❡♦ tæ♣æ ②➳✉ t❤➻ ♣❤➛♥ ❜ò U C = {x ∈ X : x ≥ 1} ❧➔ ✤â♥❣ ②➳✉✳ C ❑❤✐ ✤â S = BX ∩ U ❝ô♥❣ ❧➔ ✤â♥❣ ②➳✉ ✭♠➙✉ ❱➟② U ❦❤æ♥❣ ❧➔ ♠ð t❤❡♦ tæ♣æ ②➳✉ σ(X, X )✳ ✶✶ t❤✉➝♥ ✈ỵ✐ ✈➼ ❞ư ✷✳✷✳✽✮✳ ◆❤➟♥ ①➨t ✷✳✷✳✶✵✳ ▼å✐ t➟♣ ✤â♥❣ ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉ σ(X, X ) ❧➔ ✤â♥❣ ✤è✐ ✈ỵ✐ tỉ♣ỉ ♠↕♥❤✳ ❚r♦♥❣ ❝→❝ ✈➼ ❞ư ð tr➯♥✱ t❛ ✤➣ ❝❤➾ r❛ ♣❤➛♥ ✤↔♦ s❛✐ tr♦♥❣ trữớ ủ ổ ố ợ t ỗ t trũ õ ỵ s ỵ sỷ C E t ỗ õ C õ ố ✈ỵ✐ σ(X, X ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ✤â♥❣ ♠↕♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû C ✤â♥❣ ♠↕♥❤✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥â ✤â♥❣ ②➳✉ ✈➻ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ✤➣ ✤÷đ❝ x0 ∈ C ✱ t❤❡♦ ❞↕♥❣ ❤➻♥❤ ❤å❝ ❝õ❛ ✤à♥❤ t→❝❤ {x0 } ✈➔ C t❤❡♦ ♥❣❤➽❛ ❝❤➦t✱ tù❝ ự sỷ s õ ỵ tỗ t f X , R : f, x0 < α < f, y , ∀y ∈ C V = {x ∈ X : f, x < α}✱ ❦❤✐ ✤â x0 ∈ V ❦❤æ♥❣ ❣✐❛♦ C ✳ ❉♦ ✤â✱ C ❧➔ ✤â♥❣ ②➳✉✳ ✣➦t ◆❤➟♥ ①➨t ✷✳✷✳✶✷✳ X V ❧➔ ♠ët ❧➙♥ ❝➟♥ ②➳✉ ❝õ❛ x0 ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✱ tæ♣æ ②➳✉ ❧➔ ❦❤ỉ♥❣ t❤➸ ♠❡tr✐❝ ❤♦→ ✤÷đ❝✱ tù❝ ❧➔ ❦❤ỉ♥❣ ❝â ♠ët ♠❡tr✐❝ ♥➔♦ tr♦♥❣ ♥❤✐➯♥✱ ♥➳✉ ❤❛② X ❧➔ t→❝❤ ✤÷đ❝ t❤➻ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ♠ët ❝❤✉➞♥ s✐♥❤ tr➯♥ ❝→❝ t➟♣ ❤đ♣ ❜à ❝❤➦♥ ❝õ❛ X✱ tỉ♣ỉ ②➳✉ σ(X, X )✳❚✉② tr♦♥❣ X ♠➔ ❝↔♠ s✐♥❤ r❛ tæ♣æ ②➳✉ σ(X, X )✳ ✷✳✸ ❚æ♣æ ②➳✉ ✯ σ(X , X) ●✐↔ sû ✤è✐ ♥❣➝✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ f = sup x∈X sup f ∈X | f, x | x ✈➔ X X ❧➔ ✤è✐ ♥❣➝✉ ❝õ❛ ❧➔ ✤è✐ ♥❣➝✉ ❝õ❛ ✤÷đ❝ tr❛♥❣ ❜à ❝❤✉➞♥ ξ = ✤÷đ❝ tr❛♥❣ ❜à ❝❤✉➞♥ | ξ, f | ✳ f ❱➲ ♠➦t ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ tr÷í♥❣ ❤đ♣ X X X X ❝â sè ❝❤✐➲✉ ❤ú✉ ❤↕♥ t❤➻ ❝❤➦t ❝❤➩ ❤ì♥✿ X X ∼ =X X ✈➔ X ❦❤æ♥❣ rã r➔♥❣ ❧➢♠✱ trø X X ✳ ✳ ❚✉② ♥❤✐➯♥✱ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❝♦♥ ❝õ❛ ✈➔ ❱➻ x ❝õ❛ X ✱ ♥❣♦➔✐ ❜↔♥ ❝❤➜t ❧➔ ♠ët ✈❡❝tì t❤ỉ♥❣ t❤÷í♥❣✱ ♥â ❝á♥ ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ①→❝ ✤à♥❤ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✧❧ỵ♥ ❤ì♥✧ ❧➔ X ✱ t❤❡♦ ❝æ♥❣ t❤ù❝ x ∈ X ⇒ x ∈ X : x(f ) = f (x), ∀f ∈ X ◆➳✉ X = X t❤➻ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ◆❤÷ ✈➟②✱ t❛ ✤➣ ❝â ❤❛✐ tæ♣æ tr➯♥ X ❧➔ ✈➟② ♠é✐ ♣❤➛♥ tû ✐✳ ❚ỉ♣ỉ ♠↕♥❤ ❧✐➯♥ ❦➳t ✈ỵ✐ ❝❤✉➞♥ ❝õ❛ ✐✐✳ ❚æ♣æ ②➳✉ X x ∈ X✱ ✳ σ(X , X )✳ ❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ tỉ♣ỉ t❤ù ❜❛ tr➯♥ ♠é✐ X X ϕx : X → R ①→❝ ✤à♥❤ ❜ð✐ ϕx (f ) = f, x ①↕ (ϕx )x∈X tø X ✈➔♦ R✳ ①➨t →♥❤ ①↕ t❛ ✤÷đ❝ ❤å ❝→❝ →♥❤ σ(X , X)✳ ❱ỵ✐ ❑❤✐ x ❝❤↕② ❦❤➢♣ ✱ tæ♣æ ②➳✉✯ ✈➔ t❛ ❦➼ ❤✐➺✉ ❧➔ ✶✷ ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ❝❤♦ ♠å✐ →♥❤ ①↕ ❱➻ X⊂X ❚æ♣æ ②➳✉ ✯✭ ❦➼ ❤✐➺✉ ❧➔ (ϕx )x∈X ♥➯♥ tæ♣æ ❝â ➼t t➟♣ ♠ð ✭✤â♥❣✮ ❤ì♥ σ(X , X)✮ ❧➔ tỉ♣ỉ t❤æ ♥❤➜t tr➯♥ E ❧➔♠ trð t❤➔♥❤ →♥❤ ①↕ ❧✐➯♥ tư❝✳ σ(X , X) ❧➔ t❤ỉ ❤ì♥ tỉ♣ỉ σ(X , X )✱ tù❝ ❧➔ tæ♣æ σ(X , X) tæ♣æ σ(X , X ) ⇒ σ(X , X) ❝â ➼t t➟♣ ♠ð ✭✤â♥❣✮ ❤ì♥ tỉ♣ỉ ♠↕♥❤✳ ◆❤➟♥ ①➨t ✷✳✸✳✷✳ ◆❣✉②➯♥ ♥❤➙♥ ♣❤↔✐ ♥❣❤✐➯♥ ❝ù✉ tæ♣æ ②➳✉✯ ❧➔ ✈➻✿ ✏ ♠ët tæ♣æ t❤ỉ ❤ì♥ ✭❝❤ù❛ ➼t t➟♣ ♠ð ❤ì♥✮ s➩ ❝â ♥❤✐➲✉ t➟♣ ❝♦♠♣❛❝t ❤ì♥✑✳ ❈❤➥♥❣ ❤↕♥ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣✿ BX tr♦♥❣ X ❦❤æ♥❣ ❧➔ t➟♣ ❝♦♠♣❛❝t t❤❡♦ tæ♣æ ♠↕♥❤ ✭trø tr÷í♥❣ ❤đ♣ ♥❤÷♥❣ ❧➔ t➟♣ ❝♦♠♣❛❝t t❤❡♦ tỉ♣ỉ ②➳✉✯ dim X < ∞✮✱ σ(X , X)✳ ▼➺♥❤ ✤➲ ✷✳✸✳✸✳ ❚æ♣æ ②➳✉ ✯ σ(X , X) ❧➔ tæ♣æ t→❝❤ ✭❍❛✉s❞♦r❢❢✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f1 ; f2 ∈ X , f1 = f2 ✳ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❝♦✐ r➡♥❣ ∃x ∈ X : f1 , x = f2 , x f1 , x < f2 , x ✱ ❝❤å♥ α s❛♦ ❝❤♦ ❑❤✐ ✤â✱ ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ f1 , x < α < f2 , x ✣➦t O1 = {f ∈ X : f, x < α} = ϕ−1 x (−∞; α) O2 = {f ∈ X : f, x > α} = ϕ−1 x (α; +∞) ❑❤✐ ✤â✱ O1 , O2 ❧➔ ❝→❝ t➟♣ ♠ð tr♦♥❣ σ(E , E) : f1 ∈ O1 , f2 ∈ O2 ▼➺♥❤ ✤➲ ✷✳✸✳✹✳ ❚❛ s➩ ✤÷đ❝ ❝ì sð ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ f σ(X , X) ❦❤✐ ①➨t t➜t ❝↔ ❝→❝ ✈ỵ✐ I ❤ú✉ ❤↕♥ ✈➔ ε > 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❑➼ ❤✐➺✉ t➟♣ ❤ñ♣ ❝â ❞↕♥❣ ✈➔ O1 ∩ O2 = ∅✳ ∈ X ✤è✐ ✈ỵ✐ tỉ♣ỉ ②➳✉✯ V = {f ∈ X : | f − f0 , xi | < ε, ∀i ∈ I} ●✐è♥❣ ♠➺♥❤ ✤➲ ✷✳✷✳✸ ∗ (fn ) tr♦♥❣ X ✱ t❛ ❦➼ ❤✐➺✉ fn → f ❧➔ sü ❤ë✐ tö ❝õ❛ fn ✈➲ f ∗ tæ♣æ ②➳✉✯ σ(X , X)✳ ✣➸ tr→♥❤ ♥❤➛♠ ❧➝♥ t❛ t❤÷í♥❣ ❝❤➼♥❤ ①→❝ ❤â❛✿ fn f σ(X , X)❀ fn f ✤è✐ ✈ỵ✐ σ(X , X ) fn f trữợ ố ✈ỵ✐ ✤è✐ ✈ỵ✐ ▼➺♥❤ ✤➲ ✷✳✸✳✺✳ ●✐↔ sû f ❧➔ ❞➣② tr♦♥❣ X ✳ ❚❛ ❝â✿ n ✐✳ fn ∗ f ✤è✐ ✈ỵ✐ σ(X , X) ⇔ fn, x →∗ f, x ∀x ∈ X ✳ ✐✐✳ ◆➳✉ fn →∗ f ♠↕♥❤✱ t❤➻ fn f ✤è✐ ✈ỵ✐ σ(X , X )✳ ◆➳✉ fn f ✤è✐ ✈ỵ✐ σ(X , X ) t❤➻ fn ∗ f ✤è✐ ✈ỵ✐ σ(X , X)✳ ✐✐✐✳ ◆➳✉ fn ∗ f ✤è✐ ✈ỵ✐ σ(X , X) t❤➻ f ❜à ❝❤➦♥ ✈➔ f ≤ lim inf fn ✳ ✐✈✳ ◆➳✉ fn ∗ f ✤è✐ ✈ỵ✐ σ(X , X) ✈➔ xn → x ♠↕♥❤ tr♦♥❣ X t❤➻ fn, xn → ✶✸ f, x ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐è♥❣ ♠➺♥❤ ✤➲ ✷✳✷✳✺ ◆❤➟♥ ①➨t ✷✳✸✳✻✳ σ(X , X )✮ ✈➔ xn ◆❤➟♥ ①➨t ✷✳✸✳✼✳ ∗ f ✤è✐ ✈ỵ✐ σ(X , X) ụ ữ t fn f ố ợ x ✤è✐ ✈ỵ✐ σ(X, X ) t❛ ❦❤ỉ♥❣ t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ fn , xn → f, x ✳ ◆➳✉ fn ❑❤✐ X trò♥❣ ♥❤❛✉✳ ❚❤➟t ✈➟②✱ ♣❤➨♣ ♥❤ó♥❣ ❝❤➼♥❤ t➢❝ dim X ✮✱ ❞♦ ✤â σ(X , X ) ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ❜❛ tæ♣æ ✭♠↕♥❤✱ J :X→X σ(X , X)✮ ✭✈➻ dim X = ✈➔ ❧➔ t♦➔♥ →♥❤ σ(X , X ) = σ(X , X)✳ ❇ê ✤➲ ✷✳✸✳✽✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r ✈➔ ϕ, ϕ , ϕ , , ϕ ❧➔ ❝→❝ →♥❤ ①↕ t✉②➳♥ n t➼♥❤ tr➯♥ X ✳ ❑❤✐ ✤â✱ ϕ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ϕ1, ϕ2, , ϕn ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ n ker ϕi ⊂ ker ϕ i=1 ❈❤ù♥❣ ♠✐♥❤✳ n (⇒) ϕ ❜✐➸✉ t❤à t✉②➳♥ t➼♥❤ ✤÷đ❝ q✉❛ ϕi , i = 1, n ❦❤✐ ✤â t❛ ❝â t❤➸ ✈✐➳t ϕ = i=1 n ✈ỵ✐ αi ∈ R✳ αi ϕi ker ϕi ⊂ ker ϕ✳ ❚ø ✤➙② t❛ s✉② r❛ ✤÷đ❝ i=1 (⇐) ◆❣÷đ❝ ❧↕✐✱ ①➨t →♥❤ ①↕ F : X → Rn+1 ①→❝ ✤à♥❤ ❜ð✐ F (u) = (ϕ(u), ϕ1 (u), ϕ2 (u), , ϕn (u)) n ❚❛ t❤➜② r➡♥❣ (1, 0, 0, , 0) ❦❤æ♥❣ t❤✉ë❝ ✈➔♦ ImF ✱ ker ϕi ⊂ ker ϕ ❞♦ ♥➯♥ ♥➳✉ i=1 ♥❤÷ ϕi , u = 0, i = 1, n n+1 R tỗ ImF ✱ tù❝ ❧➔ t❤➻ ϕ, u = 0✳ ❱➻ ✈➟②✱ ImF ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝❤➦t ❝õ❛ t↕✐ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❦❤→❝ ❦❤æ♥❣ tr➯♥ Rn+1 ✱ tr✐➺t t✐➯✉ tr➯♥ n n+1 ∃(α, α1 , , αn ) ∈ R \{θ} : (x, x1 , , xn ) ∈ ImF ⇒ αx + αi xi = i=1 n αi ϕi , u = 0, ∀u ∈ X ✳ α ϕ, u + ❚ø ✤â t❛ ❝â i=1 n ⇔ αϕ + n α i ϕi , u = 0, ∀u ∈ X ✳ ❙✉② r❛ i=1 αϕ + αi ϕi = i=1 ϕ1 , ϕ2 , , ϕn ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ s✉② r❛ α = 0✱ tø ✤â t❛ ❝â t❤➸ ❝❤✐❛ ✷ ✈➳ ❝❤♦ α✳ ❙✉② r❛ ϕ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ϕ1 , ϕ2 , , ϕn ✳ ❚r÷í♥❣ ❤đ♣ ϕ1 , ϕ2 , , ϕn ❧➔ ♣❤ư t❤✉ë❝ t✉②➳♥ t➼♥❤ t❛ ❝â t❤➸ rót r❛ ✤÷đ❝ ❜ë ϕ1 , ϕ2 , , ϕp (p < n) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ϕp+1 , , ϕn ❜✐➸✉ t❤à t✉②➳♥ t➼♥❤ ✤÷đ❝ q✉❛ ϕ1 , ϕ2 , , ϕp ✳ ❚❤❡♦ ❝❤✐➲✉ (⇒) t❛ ✤÷đ❝ ❚r÷í♥❣ ❤đ♣ p n ker ϕi ⊂ ker ϕ ker ϕi = i=1 ❙✉② r❛ ϕ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ i=1 ϕ1 , ϕ2 , , ϕn ✳ ✶✹ ▼➺♥❤ ✤➲ ✷✳✸✳✾✳ ●✐↔ sû ϕ : X ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ✤è✐ ✈ỵ✐ tỉ♣ỉ σ(X , X) õ tỗ t x X s ❝❤♦ ϕ(f ) = f, x , ∀f ∈ X ✳ → R ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ θ ϕ ❧✐➯♥ tö❝ ợ ố ợ (X , X) tỗ t V = {f ∈ X : |ϕ(f ) < 1|} ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ σ(X , X)✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✸✳✹✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ W = {f ∈ X : | f, xi | < ε, ∀i = 1, n} ⊂ V, xi ∈ X, ε > xi ∈ ker f, ∀i = 1, n t❤➻ ∀λ ∈ R, ∀1 ≤ i ≤ n, λf, xi = < ε ♥➯♥ λf ∈ W ⊂ V ✈ỵ✐ ♠å✐ λ ∈ R✳ ❉♦ ✤â |λ|.|ϕ(f )| < 1, ∀λ ∈ R✳ ❑❤✐ ❝❤♦ λ → ∞ t❛ s✉② r❛ ✤÷đ❝ f ∈ ker ϕ✳ ⑩♣ ❞ư♥❣ ❜ê ✤➲ ✷✳✸✳✽✱ s✉② r❛ ϕ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ x1 , x2 , , xn ✱ ❤❛② ◆➳✉ n ϕ(f ) = n λi f, xi = i=1 f, λi xi , ∀f ∈ X i=1 ❍➺ q✉↔ ✷✳✸✳✶✵✳ ●✐↔ sû H ❧➔ s✐➯✉ ♣❤➥♥❣ ❝õ❛ X ✤â♥❣ ✤è✐ ✈ỵ✐ tỉ♣ỉ σ(X , X)✳ ❑❤✐ ✤â tỗ t x X, R s H ❝â ❞↕♥❣ H = {f ∈ X : f, x = α}✳ ❈❤ù♥❣ ♠✐♥❤✳ H ❝â ❞↕♥❣ H = {f ∈ X : ϕ(f ) = α} ✈ỵ✐ ϕ ❧➔ ♣❤✐➳♠ X ✈➔♦ R, ϕ ≡ 0✳ ●✐↔ sû f0 ∈ H ✈➔ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ f0 ✤è✐ ✈ỵ✐ ❝❤♦ V ⊂ CH ✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✸✳✹✱ ❣✐↔ sû r➡♥❣✿ ❚➟♣ ❤ñ♣ ❤➔♠ t✉②➳♥ t➼♥❤ tø tæ♣æ σ(X , X) s❛♦ V = {f ∈ X : | f − f0 , xi | < ε, ∀i = 1, n} f0 ∈ H ✱ t❛ ❝â ϕ(f0 ) = α✳ ❉♦ ✤â ϕ(f0 ) > α (f0 ) < rữớ ủ (f0 ) < α ●✐↔ sû ✈ỵ✐ f ∈ V ✱ t❛ ❝â ϕ(f ) > α✱ t❤➻ →♥❤ ①↕ φ : t −→ ϕ(tf + (1 − t)f0 ) ❧➔ ❧✐➯♥ tư❝ tr➯♥ [0; 1]✳ ❍ì♥ ♥ú❛ t❛ ♥❤➟♥ t❤➜② φ(0) = ϕ(f0 ) < α ✈➔ φ(1) = ϕ(f ) > ỵ tr tr tỗ t g [f0 , f ] s g ∈ H ✳ ▼➦t ❦❤→❝✱ t❛ ♥❤➟♥ t❤➜② r➡♥❣ V t ỗ [f0 , f ] V ✱ ❞♦ ✤â V ∩ H = ∅ ✭✈æ ỵ V CH õ , f < α, ∀f ∈ V ✳ ❇➙② ❣✐í✱ t❛ ①➨t V − f0 = {f − f0 : f ∈ V } = {f − f0 : f ∈ X , | f − f0 , xi | < ε, ∀i = 1, n} ⇒ V − f0 = {f ∈ X : | f, xi | < ε, ∀i = 1, n} ❧➔ ❧➙♥ ❝➟♥ ②➳✉ ❝õ❛ ✵✳ ◆❣♦➔✐ r❛ ❚ø ∀f ∈ V − f0 , ϕ(f ) = ϕ(f + f0 ) − ϕ(f0 ) < α − ϕ(f0 ) (∗) f ∈ V − f0 t❤➻ −f ∈ V − f0 ✱ (∗) ⇒ |ϕ(f )| < α − ϕ(f0 ), ∀f ∈ V − f0 • ❚r÷í♥❣ ❤đ♣ ✷✿ ϕ(f0 ) > α ❍ì♥ ♥ú❛✱ t❛ t❤➜② r➡♥❣ ♥➳✉ ✶✺ ❞♦ ✤â |ϕ(f )| < ϕ(f0 ) − α, ∀f ∈ V − f0 ◆❤÷ ✈➟②✱ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ f0 ❜➜t ❦➻ t❤✉ë❝ X \H tỗ t ởt ✵✱ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ t➟♣ ↔♥❤ ❝õ❛ →♥❤ ①↕ ϕ ❝õ❛ ❦❤♦↔♥❣ ♠ð (−|ϕ(f0 ) − α|, |ϕ(f0 ) − α|)✳ ❙✉② r❛✱ ϕ ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ✤è✐ ✈ỵ✐ tỉ♣ỉ σ(X , X)✱ t❤❡♦ ♠➺♥❤ ✤➲ tỗ t x X s (f ) = f, x , ∀f ∈ X ✳ ❚ø ✤â t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ s✉② r❛ ✤÷đ❝ ❚❤❡ s♣❛❝❡ C[a,b] x = max |x(t)| ✇✐t❤ ♥♦r♠ ❝♦♠♣❧❡t❡ s♣❛❝❡ ✇✐t❤ ♠❡tr✐❝ 1≤i≤n ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ s✐♥❝❡ C[a,b] , d ✐s ❛ d(x, y) = x − y = max|x(t) − y(t)| ✣à♥❤ ỵ ỵ r ❝➛✉ ✤ì♥ ✈à ✤â♥❣ BX = {f ∈ X : f ≤ 1} ✶✻ ❧➔ ❝♦♠♣❛❝t t❤❡♦ tæ♣æ σ(X , X)✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♥❤➜t ✈➲ ❝→❝ ❧♦↕✐ tỉ♣ỉ ②➳✉ ❝ò♥❣ ♠ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣✳ ❚❤ỉ♥❣ q✉❛ ✈✐➺❝ ♣❤➙♥ t➼❝❤ ♠ët sè ✈➼ ❞ö ✤✐➸♥ ❤➻♥❤✱ t➔✐ ❧✐➺✉ ❝ơ♥❣ ✤➣ ❝❤➾ r❛ ✤÷đ❝ sü ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ✤â♥❣ ✈➔ ♠ð tr♦♥❣ tæ♣æ ②➳✉ ✈➔ tæ♣æ s✐♥❤ ❜ð✐ ❝❤✉➞♥✳ ✶✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣ ✲ ✣ë ✤♦ ✈➔ t➼❝❤ ♣❤➙♥✳ ◆❳❇ ●✐→♦ ❞ö❝ ✭✶✾✾✹✮✳ ❬✷❪ ❍♦➔♥❣ ❚ö②✱ ●✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐ ✲ t➟♣ ✸✳ ◆❳❇ ●✐→♦ ❞ư❝ ✭✶✾✼✽✮✳ ❬✸❪ ❇ò✐ ❳✉➙♥ ❍↔✐✱ ❚r➛♥ ◆❣å❝ ❍ë✐✱ ❚rà♥❤ ❚❤❛♥❤ ✣➧♦✱ ▲➯ ❱➠♥ ▲✉②➺♥✱ ✣↕✐ sè t✉②➳♥ t➼♥❤ ✈➔ ù♥❣ ❞ö♥❣ ✲ ❚➟♣ ✶ ✭✷✵✶✵✮✳ ❬✶❪ ◆❣✉②➵♥ ❳✉➙♥ ▲✐➯♠✱ ✶✽