▼Ư❈ ▲Ư❈

▼ư❝ ❧ư❝ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶
▲í✐ ♥â✐ ✤➛✉ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷
❈❤÷ì♥❣ ■✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣
Đ ỵ
Đ tử
Đ ỵ ⑩♥❤ ①↕ ♠ð ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶✼
❈❤÷ì♥❣ ■■✿ ✣↕✐ sè ❇❛♥❛❝❤ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✸
§ ✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✸
§ ✷✳ P❤ê ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✣↕✐ sè
ữỡ số
Đ ✶✳ P❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✸
§ ✷✳ P❤➨♣ ✤è✐ ❤đ♣ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✼
§ ✸✳ ❇✐➯♥ ❙❤✐❧♦✈ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✵
❑➳t ❧✉➟♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✺
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✻

✶


▲❮■ ◆➶■ ✣❺❯
❚r♦♥❣ ❧✉➟♥ →♥ ✤÷đ❝ ✈✐➳t ✈➔♦ ♥➠♠ ✶✾✷✵ ❝õ❛ ❙t❡❢❛♥ ❇❛♥❛❝❤✱ æ♥❣ ✤➣ ❤➻♥❤
t❤ù❝ ❤â❛ ❦❤→✐ ♥✐➺♠ ❜➙② ❣✐í ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤÷ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱
✈➔ ❝❤ù♥❣ ỵ ỡ s ừ t ❚r♦♥❣ ✤â✱ ✣↕✐ sè
❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
◗✉❛ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔
◆➤♥❣✱ t→❝ ❣✐↔ ✤➣ ✤÷đ❝ ❤å❝ q✉❛ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ❚ỉ♣ỉ ✤↕✐
❝÷ì♥❣✱ t→❝ ❣✐↔ ❜✐➳t r➡♥❣ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ❧➔ ♠ët ✈➜♥ ✤➲ ✤❛♥❣
✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❚æ♣æ ✣↕✐ sè q✉❛♥ t➙♠✳
◆❤➟♥ t❤➜② t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✈➔ sü

ữợ ừ t ữỡ ố t q✉②➳t ✤à♥❤ ❝❤å♥
♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✏✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✑ ✳
✷✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐✿ ❚➻♠ ❤✐➸✉ sì ❦❤❛✐ ✈➲ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦

❤♦→♥✳
✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿

✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♠ët sè

t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ tr♦♥❣ ✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✳
✹✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐✿

✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
✣↕✐ sè ❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐
t♦→♥ ●✐↔✐ t
ợ ử tr õ ữủ ❜❛ ❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ t➟♣ ❤ñ♣
♠ð✱ t➟♣ ❤ñ♣ ✤â♥❣✱ →♥❤ ①↕ ❧✐➯♥ tö❝✱ ✳✳✳❀ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ✤à♥❤ ỵ



q ữ ỵ ỵ ①↕ ♠ð✱ ✳✳✳

❈❤÷ì♥❣ ✷✿ ✣↕✐ sè ❇❛♥❛❝❤

❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì
❜↔♥ ♥❤÷✿ ✣↕✐ sè ♣❤ù❝✱ ✣↕✐ sè ❇❛♥❛❝❤✱ ✣↕✐ sè ❝♦♥✱ ỗ ự
õ ởt số ♥❤÷✿ P❤ê✱ ❣✐↔✐✱ ✳✳✳❈✉è✐ ❝ị♥❣ ❝❤ó♥❣ tỉ✐ tr➻♥❤
❜➔② ✈➔ ❝❤ù♥❣ ởt số ỵ q

ữỡ số
r ữỡ trữợ t ú tổ ữ r❛ ❦❤→✐ ♥✐➺♠ ✈➲ ✣↕✐ sè
❇❛♥❛❝❤ ❣✐❛♦ ❤♦→♥✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ●❡❧❢❛♥❞✱ ♣❤➨♣ ✤è✐ ❤đ♣✱ ❜✐➯♥ ❙❤✐❧♦✈✳
❈✉è✐ ❝ị♥❣ ❝❤ó♥❣ tỉ✐ tr ự ởt số ỵ q✉❛♥✳
❉♦ ❦❤✉æ♥ ❦❤ê ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ ❜➔✐ ❜→♦ ✤÷đ❝
❞ị♥❣ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❝❤ó♥❣ tỉ✐ ữủ tr ữợ ờ ổ
ự

❧➔ ♠ët sè ❦➼ ❤✐➺✉ ✤÷đ❝ ✈✐➳t tr♦♥❣ ❦❤â❛ ❧✉➟♥
●✐↔ sû A, B, E ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ✳ ❑❤✐ ✤â A ❧➔
E

❜❛♦ ✤â♥❣ ❝õ❛ A tr♦♥❣ X ✱ A ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ A tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥

E ✱ ✐♥tA ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ A tr♦♥❣ X ✱ A\B ❧➔ ❤✐➺✉ ❝õ❛ A ✈➔ B ✳
●✐↔ sû V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ x ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ V ✱ x ✤÷đ❝ ❣å✐
❧➔ ❝❤✉➞♥ ❝õ❛ x✳
◆❤➙♥ ❞à♣ ♥➔②✱ ❝❤♦ ♣❤➨♣ t→❝ ❣✐↔ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t
✤➳♥ t❤➛② ❣✐→♦ ❚❤✳❙ ữỡ ố t t ữợ t
tr sốt q tr ự ỗ tớ t ①✐♥ ❝❤➙♥ t❤➔♥❤
❝↔♠ ì♥ ❇❛♥ ❝❤õ ◆❤✐➺♠ ❑❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ✤➣
♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕②✱ tr✉②➲♥ t❤ö ❦✐➳♥ t❤ù❝ tr♦♥❣ s✉èt ✹ ♥➠♠ ❤å❝ q✉❛✳
✸



❈✉è✐ ❝ị♥❣ t→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ t➜t ❝↔ ❝→❝ ❜↕♥ ❜➧ tr♦♥❣ ❧ỵ♣ ✵✽❈❚❚✷ ✤➣
✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→
tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ✈➻ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥
❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ❝↔ ♥ë✐ ❞✉♥❣
❧➝♥ ❤➻♥❤ t❤ù❝✳ ❱➻ ✈➟② t→❝ rt ữủ ỳ ớ qỵ
ừ t ổ ỳ õ ỵ ừ ❜↕♥ ✤å❝✳

✣➔ ◆➤♥❣✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✷

❚→❝ ❣✐↔

✹


❈❍×❒◆● ■

▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❇❷◆ ❱➋ ❚➷P➷ ✣❸■ ì
Đ ị
ỡ ỷ ❝❤✉➞♥✳●✐↔ sû p : E → R ❧➔ →♥❤ ①↕✱ ợ E
ổ tỡ
p ữủ ồ

sỡ ♥➳✉
p (αx) = αp (x) , ∀α ≥ 0, ∀x ∈ E
p (x + y) ≤ p (x) + p (y)

❜✳ p ✤÷đ❝ ❣å✐ ❧➔

♥û❛ ❝❤✉➞♥ ♥➳✉




 p (x) ≥ 0, ∀x ∈ E
p (αx) = | α | p (x) , ∀α ≥ 0, ∀x ∈ E


 p (x + y) p (x) + p (y)

ỵ ❍❛❤♥ ✲ ❇❛♥❛❝❤ ✭❝❤♦ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ ✮✳
●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝✱ p ❧➔ ♠ët sì ❝❤✉➞♥ tr➯♥ X ✈➔ M ❧➔
❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ X ✳ ◆➳✉ f : M → R ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ①→❝
✤à♥❤ tr➯♥ M ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x) ≤ p (x) , ∀x ∈ M t tỗ t
t t : X R ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X
s❛♦ ❝❤♦
i✳ λ (x) = f (x) , ∀x ∈ M
ii✳ λ (x) ≤ p (x) , ∀x ∈ M

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❣å✐ ♠ët s✉② rë♥❣ ❝õ❛ f ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤
g ①→❝ ✤à♥❤ tr➯♥ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ Dg ⊃ M t❤ä❛ ♠➣♥
i✳ g (x) = f (x) , ∀x ∈ M
✺


ii✳ g (x) ≤ p (x) , ∀x ∈ M
●å✐ F ❧➔ t➟♣ t➜t ❝↔ ❝→❝ s✉② rë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ F = ∅ ❜ð✐ ✈➻ f ∈ F ✳
❚❛ ①→❝ ✤à♥❤ ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr♦♥❣ F ♥❤÷ s❛✉✱ ♥➳✉

g1 , g2 ∈ F t❤➻
g1 ≤ g2 ⇔ Dg1 ⊂ Dg2 ✈➔ g1 (x) = g2 (x) , ∀x ∈ Dg1

●✐↔ sû D ❧➔ ♠ët t➟♣ ❝♦♥ s➢♣ t❤➥♥❣ ❝õ❛ F ✳ ❑➼ ❤✐➺✉ D∗ =

Dg ✳
g∈D

❑❤✐ ✤â✱ D∗ ⊃ M ✈➔ D∗ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ ❳✳
❱ỵ✐ ♠å✐ x D tỗ t g D s x ∈ Dg ✳
❚❛ ①→❝ ✤à♥❤ ♣❤✐➳♠ ❤➔♠ g ∗ tr➯♥ D∗ ❜➡♥❣ ❝→❝❤ ✤➦t g ∗ (x) = g (x)✳
❚❛ ✤÷đ❝ g ∗ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ tr➯♥ D∗ ✈➔

g ∗ ≥ g, ∀g ∈ D
◆❤÷ ✈➟②✱ ♠å✐ t➟♣ ❝♦♥ s➢♣ t❤➥♥❣ ❤➔♥❣ D ❝õ❛ F ✤➲✉ ❝â ♠ët ❧➙♥ ❝➟♥ tr➯♥
tr♦♥❣ F ✳
❚❤❡♦ ❇ê ✤➲ r tỗ t ởt tỷ ỹ ừ F ✳ ❇➙② ❣✐í t❛
❝❤ù♥❣ ♠✐♥❤ ✤â ❧➔ ♣❤✐➳♠ ❤➔♠ ❝➛♥ t➻♠✳ ▼✉è♥ t❤➳✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣
♠✐♥❤ ♠✐➲♥ ①→❝ ✤à♥❤ D ❝õ❛ λ ❧➔ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X ✳

P❤↔♥ ❝❤ù♥❣✳ ●✐↔ sû D = X ✳ ❑❤✐ ✤â✱ tỗ t x0 X s x0 / D s✉②
r❛ x0 = 0✳
❑➼ ❤✐➺✉ [ x0 ] ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ♠ët ❝❤✐➲✉ ❝õ❛ X ❣➙② ♥➯♥ ❜ð✐

x0 ✳
❚❛ ❝â D ∩ [x0 ] = {0}
❑➼ ❤✐➺✉ Z = D + [x0 ]✳ ▼é✐ ✈❡❝tì z ∈ Z õ t t
ữợ

z = x + αx0 , x ∈ D.
▲➜② z, z ∈ D✱ t❛ ❝â

λ(x) − λ(x ) = λ(x − x ) ≤ p(x − x )

= p(x + x0 − x − x0 ) ≤ p(x + x0 + p(−x − x0 ))
❙✉② r❛
✻


sup [−p (−x − x0 ) − λ (x)] ≤ inf [p (x + x0 ) − λ (x)]
x∈D

x∈D

❉♦ ✤â✱ tỗ t số c p(x + x0 ) − λ(x), ∀x ∈ D s❛♦ ❝❤♦

−p (−x − x0 ) − λ (x) ≤ c, ∀x ∈ D
❱ỵ✐ z ∈ Z ✈➔ z = x + αx0 , x ∈ D t❛ ✤➦t G(z) = λ(x) + αc✳
❑❤✐ ✤â G ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr➯♥ Z ✈➔ ♥➳✉ x ∈ D t❤➻

G(x) = λ(x) ≤ p(x)
◆➳✉ z ∈ Z \ D✱ t❤➻ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ z = x + αx0 ✱ t❛ ❝â α = 0✳
✯ ❳➨t tr÷í♥❣ ❤đ♣ α > 0✳
❱➻ c ∈ p(x + x0 ) − λ(x) ✈ỵ✐ ♠å✐ x ∈ D✱ ♥➯♥

c≤p

x
x
+ x0 − λ
α
α

❙✉② r❛


αc ≤ αp

x
x
+ x0 − αλ
= p (x + αx0 ) − λ (x)
α
α

❉♦ ✤â

λ (x) + αc ≤ p (x + αx0 )
❙✉② r❛ G(z) ≤ p(z)✳
✯ ❳➨t tr÷í♥❣ ❤đ♣ α < 0✳ ❱➻ −p (−x − x0 ) − λ (x) ≤ c, ∀x ∈ D ✤ó♥❣ ✈ỵ✐
♠å✐ x ∈ D ♥➯♥

−p −

x
x
− x0 − λ
≤c
α
α

❙✉② r❛

− (−α) p −


x
x
− x0 + αλ
≤ −αc
α
α

❉♦ ✤â

−p (x + αx0 ) + λ (x) ≤ −αc

✼


❙✉② r❛ G(z) ≤ p(z)✳
❇ð✐ ✈➟② G ∈ F ✈➔ G ≥ λ, G = λ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ λ ❧➔ ♣❤➛♥ tû
❝ü❝ ✤↕✐ ❝õ❛ F ✳
❈❤ù♥❣ tä D = X

ỵ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝ ✮✳
●✐↔ ①û X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ ♣❤ù❝✱ p ❧➔ ♠ët ♥û❛ ❝❤✉➞♥
tr➯♥ X ✈➔ M ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ X ✳ ◆➳✉ f : X → C ❧➔ ♣❤✐➳♠ ❤➔♠
t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ tr➯♥ M ✈➔ t❤♦➣ ♠➣♥ ✤✐➲✉ ❦✐➺♥
|f (x)| p(x), x M

t tỗ t t t➼♥❤ λ : X → C ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë
❦❤æ♥❣ ❣✐❛♥ X s❛♦ ❝❤♦
i. λx = f (x), ∀x ∈ M.
ii. |λx| ≤ p(x), ∀x ∈ X.


❈❤ù♥❣ ♠✐♥❤✳ ❚❛ f (x) ữợ f (x) = u1 (x) + iu2 (x) ✈➔
♥❤➟♥ ✤÷đ❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ f1 (x) ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥
t✉②➳♥ t➼♥❤ t❤ü❝ M ỗ tớ

f1 (x) |f1 (x)| |f (x)| p(x), x M.
ỵ tỗ t↕✐ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ λ1 (x) ①→❝ ✤à♥❤
tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ X s❛♦ ❝❤♦
❛ λ1 (x) = f1 (x), ∀x ∈ M
❜ |λ1 (x)| ≤ p(x), ∀x ∈ X
❉♦ p ❧➔ ♠ët ♥ú❛ ❝❤✉➞♥✱ tø |λ1 (x)| ≤ p(x), ∀x ∈ X t❛ s✉② r❛

−λ1 (x) = λ1 (−x) ≤ p (−x) = p (x)
❙✉② r❛
✽


|λ1 (x)| ≤ p (x) , ∀x ∈ X
✣➦t

λ (x) = λ1 (x) − iλ2 (ix) , ∀x ∈ X
❑❤✐ ✤â✱ λ ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ♣❤ù❝ ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥
t✉②➳♥ t➼♥❤ ♣❤ù❝ X ✳
▼➔ t❛ ❝â f (x) = u1 (x) − iu1 (ix) ♥➯♥ s✉② r❛

λ (x) = f (x) , ∀x ∈ M
▼➦t ❦❤→❝✱ ♥➳✉ λ(x) = 0✱ t❤➻ λ(x) = λ(x)eiθ ✳
❱➻ ✈➟②

|λ (x)| = e−iθ λ (x) = λ xe−iθ
❱➻ λ xe−iθ ❧➔ sè t❤ü❝ ✈➔ λ (x) = λ1 (x) − iλ2 (ix) , ∀x ∈ X ♥➯♥ t❛ ❝â


λ xe−iθ = λ1 xe−iθ
❇ð✐ ✈➟② |λ (x)| = λ1 xe−iθ ≤ p xe−iθ = p (x)✳

✶✳✶✳✹ ❈→❝ ❤➺ q✉↔✳
●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ F ⊂ E ✳ ❑❤✐ ✤â✱
♥➳✉ f : F → K ❧➔ →♥❤ ①↕ t t tử t tỗ t
t t ❧✐➯♥ tö❝ f : E → K t❤ä❛

❛✳ ❍➺ q✉↔ ✶✳


 f |F = f
 f = f

●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✱F ⊂ E ✈➔ v ∈ E\F
s❛♦ ❝❤♦ d (v, F ) = δ > 0 õ tỗ t t t
tử f : E → K t❤ä❛

❜✳ ❍➺ q✉↔ ✷✳

✾


f |F = 0
f (v) = δ

❈❤ù♥❣ ♠✐♥❤✳
❛✳ ✣➦t p : E → R


x → p (x) = f . x
❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ p ❧➔ ♥û❛ ❝❤✉➞♥✳ ❍ì♥ ♥ú❛ ✈➻ f ❧➔ t✉②➳♥ t➼♥❤✱ ❧✐➯♥
tö❝ tr➯♥ F

| f (x) | ≤ f . x = p (x) , x F
ớ ỵ tỗ t ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ f : E → K
t❤ä❛ ♠➣♥


 f |F = f
 f (x) ≤ p (x) , ∀x ∈ E
❚❛ ❝â f (x) ≤ f . x = p (x) , ∀x ∈ E ✳ ❉♦ ✤â f t✉②➳♥ t➼♥❤ ✈➔ ❜à
❝❤➦♥✳❙✉② r❛ f ❧✐➯♥ tö❝ ✈➔

f

≤ f ✳

❍ì♥ ♥ú❛✱ t❛ ❝â

f

=

sup

f (x) ≥

x∈E, x =1


❉♦ ✈➟②

f

sup

f (x) =

x∈F, x =1

sup

| f (x) | = f

x∈E, x =1

= f ✳

❜✳ ✣➦t G = v, F = {λv + z : z ∈ F, λ ∈ K}✳ ❑❤✐ ✤â G ❧➔ ❦❤ỉ♥❣ ❣✐❛♥
✈❡❝tì ❝♦♥ ❝õ❛ E ✳ ✣➦t g : G → K ①→❝ ✤à♥❤ ❜ð✐

x = λv + z → g (x) = λ.δ
❙✉② r❛ g ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤✳
✯ ❈❤ù♥❣ ♠✐♥❤ g ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳
◆➳✉ λ = 0✱ t❤➻
✶✵


z
λ

⇒ x = λv + z ≥ | λ.δ | = | g (x) |
x = λv + z = | λ | . v − −

≥ | λ | inf v − z
z∈F

⇒ | g (x) | ≤ x , ∀x = λv + z ∈ G
◆➳✉ λ = 0 t❤➻ | g (x) | = | g (0.v + z) | = 0.δ = 0 ≤ x
❉♦ ✈➟② | g (x) | ≤ x , ∀x ∈ G✱ ❦➨♦ t❤❡♦ g ✲ ❜à ❝❤➦♥✳
❱➻ g ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ♥➯♥ g ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔
✯ ❈❤ù♥❣ ♠✐♥❤

g ≤ 1✳

g = 1✳

δ
− δ✳
r
❑❤✐ ✤â✱ tø ✤à♥❤ ♥❣❤➽❛ ừ t s r tỗ t z F s❛♦ ❝❤♦

❱ỵ✐ ♠å✐ r ∈ (0, 1) t❛ ❧➜② ε =

δ+ε> v−z
❚❛ ❝â

δ
≥ v−z
r
δ

≥r
⇒
v−z
| g (v − z) |
⇒
≥ r, ∀r ∈ (0, 1)
v−z
▼➔

| g (x) |
| g (v − z) |
≥ sup
≥ r, ∀r ∈ (0, 1)
x
v−z
z∈F
x=0,x∈G

g = sup
◆➯♥ t❛ ❝â

g ≥ sup r = 1
r∈(0,1)

❉♦ ✈➟②

g = 1✳

❚â♠ tỗ t G E t t➼♥❤ ❧✐➯♥ tö❝ g : G → K
t❤ä❛ ♠➣♥







g =1

g|F = 0


 g (v) = δ
✶✶


ớ q tỗ t t t ❧✐➯♥ tö❝ f : E → K✱ t❤ä❛
♠➣♥

f |G = g
f

= g =1

❙✉② r❛

f |G = g = 0
f (v) = g (v) = δ

✶✷



§

✷ ⑩◆❍ ❳❸ ▲■➊◆ ❚Ö❈

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ t➟♣ ❝♦♥ ❝õ❛ X ✳ ❑❤✐ ✤â✱ ❣✐❛♦ ❝õ❛ t➜t ❝↔
❝→❝ t➟♣ ❤đ♣ ✤â♥❣ ❝❤ù❛ A ✤÷đ❝ ❣å✐ ❧➔

❜❛♦ ✤â♥❣ ❝õ❛ A✳

❑➼ ❤✐➺✉ ❧➔ A✳

✶✳✷✳✷ ◆❤➟♥ ①➨t✳ ●✐↔ sû A, B ❧➔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✳ ❑❤✐ ✤â✱
✶✳ A ❧➔ ♠ët t➟♣ ❤ñ♣ ✤â♥❣ ✈➔ ✤â ❧➔ t➟♣ ✤â♥❣ ♥❤ä ♥❤➜t ❝❤ù❛ A✳
✷✳ A ❧➔ t➟♣ ❤ñ♣ ✤â♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = A✳
✸✳ ◆➳✉ A ⊂ B ✱ t❤➻ A ⊂ B ✳
✹✳ ∅ = ∅, A ⊂ A, A ∪ B = A ∪ B, A = A✳

✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ X ✳ ❑❤✐ ✤â✱ ❤ñ♣ ❝õ❛
t➜t ❝↔ ❝→❝ t➟♣ ❤đ♣ ♠ð tr♦♥❣ A ✤÷đ❝ ❣å✐ ❧➔

♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ ❤ñ♣ A✳

❑➼ ❤✐➺✉ ❧➔ ✐♥tA✳

✶✳✷✳✹ ◆❤➟♥ ①➨t✳ ●✐↔ sû A, B ❧➔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✳ ❑❤✐ ✤â
✶✳ ✐♥t A ❧➔ ♠ët t➟♣ ❤ñ♣ ♠ð ✈➔ ✤â ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t ❝❤ù❛ tr♦♥❣ A✳
✷✳ A ❧➔ t➟♣ ❤ñ♣ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = ✐♥tA✳
✸✳ x ∈ ✐♥tA ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x ❧➔ ♠ët ✤✐➸♠ tr♦♥❣ ❝õ❛ A✳
✹✳ ◆➳✉ A ⊂ B t❤➻ ✐♥tA ⊂ intB ✳


✶✳✷✳✺ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : (X, τX ) → (Y, τY ) ❧➔ →♥❤ ①↕✳
❑❤✐ ✤â✱
✶✳ ❚❛ ♥â✐

→♥❤ ①↕ f ❧✐➯♥ tö❝ t↕✐ ✤✐➸♠ x0 ∈ X ✱ ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ V

❝õ❛ f (x0 ) Y tỗ t ởt ❧➙♥ ❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦

f (U ) ⊂ V ✳
✶✸


✷✳ f

❣å✐ ❧➔ ❧✐➯♥ tö❝ ✭tr➯♥ X ✮ ♥➳✉ f tử t ồ ừ X

ỵ ●✐↔ sû f : X → Y ❧➔ ♠ët →♥❤ ①↕✳ ❑❤✐ ✤â✱ f ❧✐➯♥ tư❝
❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠é✐ V

∈ τY

t❛ ✤➲✉ ❝â f −1 (V ) ∈ τX ✳

❈❤ù♥❣ ♠✐♥❤✳
i) ✣✐➲✉ ❦✐➺♥ ❝➛♥✳
●✐↔ sû f ❧✐➯♥ tö❝ ✈➔ V ∈ τY ✳ ❑❤✐ ✤â✱ ♥➳✉ f −1 (V ) = ∅✱ t❤➻ f −1 (V ) ∈ τX ✳
●✐↔ sû f −1 (V ) = ∅ ✈➔ x ❧➔ ♠ët ✤✐➸♠ ❜➜t ❦➻ ❝õ❛ f −1 (V ) ✳❑❤✐ ✤â✱

f (x) ∈ V ✳

❱➻ f ❧✐➯♥ tö❝ t↕✐ x ✈➔ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ f (x) tỗ t ởt U
ừ x s U ⊂ f −1 (V )✳
❉♦ ✤â✱ f −1 (V ) ∈ τX ✳

ii) ✣✐➲✉ ❦✐➺♥ ✤õ✳
●✐↔ sû ✈ỵ✐ ♠å✐ V ∈ τY t❛ ✤➲✉ ❝â f −1 (V ) ∈ τX ✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣

f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳
❚❤➟t ✈➟②✱ ❣✐↔ sû x ∈ X ✈➔ V ❧➔ ❧➙♥ ❝➟♥ ❜➜t ❦➻ ❝õ❛ f (x)✳ ❑❤✐ õ tỗ t

G Y s f (x) G ⊂ V ✳
❉♦ ✤â x ∈ f −1 (G) ⊂ f −1 (V )✳
◆➳✉ ✤➦t U = f −1 (G) t❤➻ U ∈ τX ✈➔ x ∈ U ✳
❙✉② r❛ U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x ✈➔ f (U ) ⊂ V ✳
❇ð✐ ✈➟② f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳

✶✳✷✳✼ ❇ê ✤➲✳ ◆➳✉ A ⊂ X ✱ t❤➻ intA = X\X\A
ỵ sỷ f : X Y ❧➔ ♠ët →♥❤ ①↕✳ ❑❤✐ ✤â✱ ❝→❝ ♠➺♥❤ ✤➲
s❛✉ t÷ì♥❣ ✤÷ì♥❣
✶✳ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝✳

✷✳ f −1(F ) ✤â♥❣ tr♦♥❣ X ✈ỵ✐ ♠å✐ t➟♣ F ✤â♥❣ tr♦♥❣ Y ✳
✸✳ ❱ỵ✐ ♠é✐ t➟♣ A ⊂ X ✱ t❛ ❝â f (A) ⊂ f (A)✳
✶✹


✹✳ ❱ỵ✐ ♠é✐ t➟♣ B ⊂ Y ✱ t❛ ❝â f −1 (B) = f −1

B


✳

✺✳ ❱ỵ✐ ♠é✐ t➟♣ B ⊂ Y ✱ t❛ ❝â f −1 (intB) ⊂ intf −1 (B)✳
❈❤ù♥❣ ♠✐♥❤✳
(1) ⇒ (2)✳
●✐↔ sû F ❧➔ ♠ët t➟♣ ✤â♥❣ tr♦♥❣ Y ✳
❑❤✐ ✤â✱ Y \F ❧➔ t➟♣ ♠ð tr♦♥❣ Y ✳
❱➻ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ♥➯♥ t ỵ t f 1 (Y \F ) ❧➔ t➟♣ ♠ð
tr♦♥❣ X ✳
▼➦t ❦❤→❝✱ ✈➻ f −1 (Y \F ) = X\f −1 (F ) ♥➯♥ X\f −1 (F ) ❧➔ t➟♣ ♠ð tr♦♥❣

X✳
❉♦ ✤â f −1 (F ) ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ X ✳

(2) ⇒ (3)✳
❱➻ f (A) ✤â♥❣ tr♦♥❣ Y ♥➯♥ f −1 (f (A)) ✤â♥❣ tr♦♥❣ X ✳
▼➦t ❦❤→❝✱ ✈➻ A ∈ f −1 (f (A)) ♥➯♥ s✉② r❛ A ∈ f −1 (f (A))✳
❉♦ ✤â f (A) ⊂ f (A)✳

(3) ⇒ (4)✳
¯ ♥➯♥ s✉② r❛
❱➻ f f −1 (B) ⊂ f (f −1 (B)) ⊂ B
f −1 (B) ⊂ f −1 f (f −1 (B)) ⊂ f −1 B ✳
(4) ⇒ (5)✳
❱➻ B ⊂ Y ♥➯♥ ♥❤í ❇ê ✤➲ ✶✳✷✳✼ t❛ ❝â intB = Y \Y \B ✳
❉♦ ✤â f −1 (intB) = f −1 Y \Y \B = X\f −1 Y \B ✳
❙✉② r❛ f −1 Y \B ⊃ f −1 (Y \B) = X\f −1 (B)✳
❱➻ t❤➳ f −1 (intB) ⊂ X\X\f −1 (B) = intf −1 (B)✳
❇ð✐ ✈➟② f −1 (intB) ⊂ intf −1 (B)✳


(5) ⇒ (1)✳
●✐↔ sû V ❧➔ ♠ët t➟♣ ♠ð ❜➜t ❦➻ tr♦♥❣ Y ✳
❑❤✐ ✤â✱ ♥❤í ◆❤➟♥ ①➨t ✶✳✷✳✹ ✭✷✮ t❛ s✉② r❛ ✐♥tV = V ✳
❉♦ ✤â✱ f −1 (V ) = f −1 (intV ) ⊂ intf −1 (V )✳
✶✺


❙✉② r❛ intf −1 (V ) = f −1 (V )
ớ ỵ t s r f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳

✶✳✷✳✾ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X → Y
✶✳ ❚❛ ♥â✐ f ❧➔ ♠ët

→♥❤ ①↕ ♠ð

❧➔ ♠ët →♥❤ ①↕✳ ❑❤✐ ✤â✱

♥➳✉ ✈ỵ✐ ♠å✐ t➟♣ ♠ð A ⊂ X t❛ ✤➲✉ ❝â

f (A) ♠ð tr♦♥❣ Y ✳
✷✳ ❚❛ ♥â✐ f ❧➔ ♠ët

→♥❤ ①↕ ✤â♥❣ ♥➳✉ ✈ỵ✐ ♠å✐ t➟♣ ✤â♥❣ A ⊂ X t❛ ✤➲✉

❝â f (A) ✤â♥❣ tr♦♥❣ Y ✳

✶✻


§


✸ ✣➚◆❍ ▲Þ ⑩◆❍ ❳❸ ▼Ð

✶✳✸✳✶ ❇ê ✤➲✳ ●✐↔ sû f : E → F ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tø ❦❤ỉ♥❣
❣✐❛♥ ❇❛♥❛❝❤ E ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ F ✳
✣➦t B E = B E (0, 1) = {x E : x < 1}
õ tỗ t δ > 0 s❛♦ ❝❤♦

B F (0, δ) = {y ∈ F : y < δ} ⊂ f (δ)

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ n = 1, 2, ...✱ t❛ ✤➦t
Bn E = B E 0,

1
2n

=

x∈E: x <

1
2n

❙✉② r❛ B0 E = B E ✳
❑❤✐ ✤â
✯E=

∞

nB1 E ✳


n=1

❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥ r➡♥❣

∞
n=1

❇➙② ❣✐í ❣✐↔ sû x ∈ E ✳ ❑❤✐ ✤â
n
n < ✳
2
❉♦ ✤â
E

x ∈ nB1 = nB

E

1
0,
2

nB1 E ⊂ E
n < tỗ t n N s



x




E

nB1 E ⊂
n=1

nB1 E

n=1

✯ ❉♦ f ❧➔ t✉②➳♥ t➼♥❤ ✈➔ t♦➔♥ →♥❤ ♥➯♥
∞

F = f (E) = f

nB1 E

n=1

∞

=

nf B1 E

n=1

❱➻ F ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➯♥ F t❤✉ë❝ ♣❤↕♠ trị tự s r tỗ t


n0 s
int n0 f B1 E = ∅✳
▼➦t ❦❤→❝✱ ✈➻ →♥❤ ①↕ x → n0 ởt ỗ ổ



int f B1 E =
õ tỗ t v ∈ f B1 E ✈➔ δ > 0 s❛♦ ❝❤♦ B (v, F ) ⊂ int f B1 E ✳
❚❛ ❝â

v + B F (0, 2δ) ⊂ intf B1 E ⊂ f B1 E
⇒ B F (0, 2δ) ⊂ −v + intf B1 E ⊂ 2.f B1 E
⇒ B F (0, δ) ⊂ 21 B F (0, 2δ) ⊂ 12 .2.f B1 E = f B1 E
❇ð✐ ✈➟②✱ B F (0, δ) ⊂ f B1 E ✳ ❚❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❛ ❝â

BF

0,

δ
2n−1

⊂ f Bn E

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ B F (0, δ) ⊂ f Bn E ✳ ❚❤➟t ✈➟②✱ ❧➜②

y ∈ B F (0, δ)✳ ❱➻ B F (0, δ) ⊂ f B1 E ♥➯♥
BF
❙✉② r❛ ∃y1 ∈ B F


y,

δ
2

y,

δ
2

∩ f B1 E = ∅

⊂ f B2 E õ tỗ t x1 B1 E s❛♦ ❝❤♦

y1 = f (x1 ) ✈➔ y − y1 = y − f (x1 ) <
❚÷ì♥❣ tü t❛ ❧➜② y − f (x1 ) ∈ B F

BF

y − f (x1 ) ,

δ
22

0,

δ
2

⊂ f B2 E s✉② r❛


∩ f B2 E ⇒ ∃y2 ∈ B F

y − f (x1 ) ,

tỗ t x2 B2 E s y2 = f (x2 )✱ t❛ ❝â

y − f (x1 ) − f (x2 ) <

δ
22

❚✐➳♣ tö❝ q✉→ tr➻♥❤ tr➯♥✱ t❛ ❝❤å♥ ✤÷đ❝ xn ∈ Bn E s❛♦ ❝❤♦
n

y−

f (xi ) <
i=1

✶✽

δ
2

δ
2n

δ
22


∩ f B2 E


1
❚❛ ❝â xn ∈ Bn E ✱ s✉② r❛ xn < n
2
∞ 1
n
❱➻
❤ë✐ tö✱ ♥➯♥
xn ❤ë✐ tö✳
n
n=1 2
i=1
❉♦ ✤â✱
♥➯♥

∞

n

xn ❤ë✐ tư t✉②➺t ✤è✐✳ ▼➦t ❦❤→❝ ✈➻ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

i=1

∞

xn ❤ë✐ tư✱ s✉② r❛ x =


n=1

xn ∈ E ✳

n=1

❍ì♥ ♥ú❛✱ ✈➻

n

y−

f (xi ) <
i=1

♥➯♥

∞

δ
2n

f (xn ) = y ✳

n=1

❚❛ ❝â

f (x) = f


n

lim

n→∞ i=1

n

xi

= lim f
n→∞

∞

n

xi

= lim

i=1

n→∞ i=1

f (xn )

f (xi ) =
n=1


=y
❇ð✐ ✈➟② y = f (x)✳
❚❛ ❝â

xn

x =
n=1

∞

∞

∞

≤

xn <
n=1

1
=1
n
n=1 2

❙✉② r❛ x ∈ B E ✳ ❱➻ y = f (x) ♥➯♥ y ∈ f (B E )✳
❉♦ ✤â B F (0, δ) ⊂ f B E

ỵ sỷ f : E → F ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✱


❧✐➯♥ tö❝ tø ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ F ✳ ❑❤✐ ✤â✱ f
❧➔ →♥❤ ①↕ ♠ð✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû U ✲ ♠ð tr♦♥❣ E ✱ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f (U ) ❧➔ t➟♣
♠ð tr♦♥❣ F
❚❤➟t ✈➟②✱ ❣✐↔ sû v f (U ) õ tỗ t x U s❛♦ ❝❤♦ v ∈ f (u)✳
❱➻ U ❧➔ t➟♣ ủ x U tỗ t > 0 s❛♦ ❝❤♦

B E (u, ε) ⊂ U ✳
✶✾


ớ ờ tỗ t > 0 s ❝❤♦ B F (0, δ) ⊂ f B E ✳
✣➦t r = ε.δ ✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ B F (v, r) ⊂ f (U )✳❚❤➟t ✈➟②✱ ❣✐↔ sû

y ∈ B F (v, r)✱ t❛ ❝â
y − v < r = ε.δ ⇒ ε−1 (y − v) < δ ⇒ ε−1 (y − v) ∈ B F (0, δ) ⊂ f B E
r tỗ t x B E s❛♦ ❝❤♦ f (z) = ε−1 (y − v)✳
✣➦t x = εz + v ✳
❚❛ ❝â

x − u = ε z < ε ⇒ x ∈ B E (u, ε) ⊂ U ✳
❍ì♥ ♥ú❛

f (x) = f (εz + u) = ε.f (z) + f (u) = y − v + f (u) = y ✳
❙✉② r❛

y = f (x)
x∈U

⇒ y ∈ f (U )


❉♦ ✤â✱ B F (v, r) ⊂ f (U )✳ ❉♦ ✈➟②✱ f (U ) ❧➔ t➟♣ ❤ñ♣ ♠ð✳

✶✳✸✳✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû E, F ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐ ✤â✱
E × F ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳ ❚r➯♥ E × F t❛ ①→❝ ✤à♥❤ ❤➔♠
. :E×F →R
(x, y) → (x, y) = x + y
❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝

. ❧➔ ❝❤✉➞♥ tr➯♥ E × F ✳ ❙✉② r❛ (E × F, . )

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳
●✐↔ sû f : E → F ✳ ❑➼ ❤✐➺✉ Gf = {(x, f (x)) ∈ E × F : x ∈ E}✳
❑❤✐ ✤â✱ Gf ữủ ồ

ỗ t ừ f

ỵ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✱ t❤➻ Gf ❧➔ t➟♣ ✤â♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {(xn, f (xn))} ⊂ Gf ,
❝❤ù♥❣ ♠✐♥❤ r➡♥❣ (a, b) ∈ Gf ✳
✷✵

(xn , f (xn )) → (a, b)✳ ❚❛ ❝➛♥


❚❤➟t ✈➟②✱ ✈➻ (xn , f (xn )) → (a, b) ♥➯♥

xn − a + f (xn ) − b → 0 ❦❤✐

n→∞

❉♦ ✤â

xn − a → 0
f (xn ) − b → 0

⇒

xn → a
f (xn ) → b

⇒

f (xn ) → f (a)
f (xn ) → b

❙✉② r❛ b = f (a)✱ ❦➨♦ t❤❡♦ (a, b) = (a, f (a)) Gf
Gf õ

ỵ ỗ t õ sỷ f : E F ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤

❧✐➯♥ tư❝ tø ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ F ✳ ❑❤✐ ✤â✱ f
❧➔ ❤➔♠ ❧✐➯♥ tö❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Gf ✤â♥❣✳
❈❤ù♥❣ ♠✐♥❤✳
i) ỷ ử ỵ

ii) ✤õ✿ ●✐↔ sû Gf ❧➔ t➟♣ ✤â♥❣✳❚❛ ❝❤ù♥❣ ♠✐♥❤ f ❧➔ →♥❤ ①↕ ❧✐➯♥
tö❝✳❚❤➟t ✈➟②✱ ①➨t ❤➔♠ p : E → R ①→❝ ✤à♥❤ ❜ð✐

x → p (x) = x + f (x) ✳
❑❤✐ ✤â✱ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ p ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E ✈➔


. ≤ p✳ ❍ì♥

♥ú❛✱
✯ (E, P ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
❚❤➟t ✈➟②✱ ❣✐↔ sû {xn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ E ✳ ❑❤✐ ✤â✱ ợ ồ > 0
tỗ t n0 s p (xn − xm ) < ε, ∀m, n ≥ n0 ✳
❚❛ ❝â

xn − xm + f (xn ) − f (xm ) < ε, ∀m, n ≥ n0 ✳
❙✉② r❛

xn − xm < ε, ∀m, n ≥ n0
f (xn ) − f (xm ) < ε, ∀m, n ≥ n0
❇ð✐ t❤➳✱ {xn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ E ✈➔ {f (xn )} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣

F✳
❱➻ E, F ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➯♥
✷✶


xn → a ∈ E
f (xn ) → b ∈ F

⇒ (xn , f (xn )) → (a, b) ∈ E × F

▼➦t ❦❤→❝✱ ✈➻ Gf ✤â♥❣ ♥➯♥ (a, b) ∈ Gf ✳ ❑➨♦ t❤❡♦ b = f (a)✳
❙✉② r❛ p (xn − a) = xn − a + f (xn ) − f (a) → 0 ❦❤✐ n → ∞✳ ❉♦
✤â✱ xn → a✳ ❇ð✐ ✈➟② (E, P ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
❈✉è✐ ❝ị♥❣ ✈➻





 (E, . ) ❴Banach
(E, p) Banach


. p
tỗ t k > 0 s ❝❤♦ p (x) ≤ k x ∀x ∈ E ✳ ❙✉② r❛

f (x) ≤ p (x) ≤ k x , ∀x ∈ E ✱ ♥❣❤➽❛ ❧➔ f ❜à ❝❤➦♥✳
❇ð✐ ✈➟②✱ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳

✷✷


ì

Đ
✣à♥❤ ♥❣❤➽❛✳ ▼ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì A tr➯♥ tr÷í♥❣ C ✤÷đ❝ tr❛♥❣
❜à t❤➯♠ ♣❤➨♣ ♥❤➙♥ ✭tr♦♥❣✮ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
✶✳ x(yz) = (xy)z,

∀x, y, z ∈ A❀

✷✳ x(y + z) = xy + xz,

∀x, y, z ∈ A❀


(x + y)z = xz + yz,

∀x, y, z ∈ A❀

✸✳ (αx)y = x(αy) = αxy,
∀x, y ∈ A, ∀α ∈ C
✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤↕✐ sè ♣❤ù❝ ✭❤❛② ✤ì♥ ❣✐↔♥ ❧➔ ♠ët ✤↕✐ sè✮✳
▼ët ✤↕✐ sè ♣❤ù❝ A t❤♦↔ ♠➣♥ t❤➯♠ ❝→❝ ✤✐➲✉ ❦✐➺♥
✹✳ A ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐

. ♥➔♦ ✤â✳

✺✳ x . y ≤ x

∀x, y ∈ A❀

. y

,

✻✳ ỗ t e A s xe = ex = x,
✼✳

e = 1❀

✤÷đ❝ ❣å✐ ❧➔ ♠ët

∀x ∈ A❀

✤↕✐ sè ❇❛♥❛❝❤✳


✷✳✶✳✷ ◆❤➟♥ ①➨t✳
✶✳ ❚r♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❦❤æ♥❣ ✤á✐ ❤ä✐ ♣❤➨♣ ♥❤➙♥ ❣✐❛♦ ❤♦→♥✳
✷✳ P❤➛♥ tû e ❝õ❛ ✤↕✐ sè ❇❛♥❛❝❤ ❧➔ ❞✉② ♥❤➜t✳
✸✳ ▼ët ✤↕✐ sè ❜➜t ❦ý t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1) − (6) ❝õ❛ ✣à♥❤ ♥❣❤➽❛
✷✳✶✳✶ ❜❛♦ ❣✐í ❝ơ♥❣ ❝â t❤➸ ♥❤ó♥❣ ✤÷đ❝ ✈➔♦ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳

✷✸


✹✳ P❤➨♣ ♥❤➙♥ ✭tr♦♥❣✮ ❧✐➯♥ tö❝✱ ❧✐➯♥ tö❝ tr→✐✱ ❧✐➯♥ tö❝ ♣❤↔✐ tr➯♥ ✤↕✐ sè
❇❛♥❛❝❤✳

✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥
❇❛♥❛❝❤ B ⊂ A ❝❤ù❛ ✤ì♥ ✈à ❝õ❛ A ✈➔ ❦❤➨♣ ❦➼♥ ợ tr
ừ A t ữủ ồ

số ❝♦♥ ❝õ❛ ✤↕✐ sè ❇❛♥❛❝❤ A✳

◆❤➟♥ ①➨t✳ ▼ët ✤↕✐ sè ❝♦♥ ❜➜t ❦➻ ❝õ❛ B(X) ♠➔ ❝â ❝❤ù❛ ✤ì♥ ✈à
I ∈ B(X) ❧➔ ♠ët ✤↕✐ sè ❇❛♥❛❝❤✳

✷✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ⑩♥❤ ①↕ t✉②➳♥ t➼♥❤ h : A → B tø ✤↕✐ sè ✭❇❛♥❛❝❤✮
A ✈➔♦ ✤↕✐ sè ✭❇❛♥❛❝❤✮ B ✤÷đ❝ ❣å✐ ❧➔ ởt

ỗ

h(xy) = h(x).h(y), x, y A

♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ♠ët ✤↕✐ sè ♣❤ù❝ ✈➔ ϕ : A → C ❧➔ ♠ët

♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr A ổ ỗ t 0 õ

(xy) = ϕ(x).ϕ(y), ∀x, y ∈ A✱ t❤➻ ♣❤✐➳♥ ❤➔♠ ϕ ữủ ồ ởt

ự tr số A

ỗ

P tỷ x ∈ A ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ ✤↕✐ số õ ỡ A tỗ
t tỷ x1 ∈ A s❛♦ ❝❤♦ x−1 x = xx−1 = e✳

✷✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✱ ♣❤➛♥ tỷ f A ữủ ồ


tỗ t g ∈ A s❛♦ ❝❤♦ f.g = 1✳

▲ó❝ ✤â t❛ ❦➼ ❤✐➺✉ g = f −1 ❤♦➦❝ g =

1
✳
f

❑➼ ❤✐➺✉ A−1 = f ∈ A : ∃f −1 ✳

✷✳✶✳✼ ▼➺♥❤ ỗ ự tr số ♣❤ù❝ A ✈ỵ✐ ✤ì♥ ✈à

e✱

t❤➻ ϕ(e) = 1 ✈➔ ♥➳✉ x ❧➔ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A t❤➻ ϕ(x) = 0✳


❈❤ù♥❣ ♠✐♥❤✳
✯ ●✐↔ sû ϕ : A → C ỗ ự õ ỡ e



❱ỵ✐ ♠å✐ x ∈ A✱ t❛ ❝â

ϕ(x) = ϕ(x.e) = (x).(e)
ỗ ự tỗ t x0 ∈ A s❛♦ ❝❤♦ ϕ(x0 ) = 0 ❙✉② r❛

ϕ(e) = 1✳
✯ ●✐↔ sû x ∈ A ✈➔ x ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ϕ(x) = 0✳
❚❤➟t ✈➟②✱ ✈➻ x ❧➔ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ tr♦♥❣ A ♥➯♥ tø ✤➥♥❣ t❤ù❝

1 = ϕ(e) = ϕ(xx−1 ) = (x).(x1 )
s r (x) = 0

ỵ sû ❆ ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✱ x ∈ A ✈➔

x < 1✳

❑❤✐ ✤â✱

❛✳ P❤➛♥ tû e − x ❦❤↔ ♥❣❤à❝❤✳
❜✳

−1

(e − x)


−e−x

x 2
≤
1− x

✳

❝✳ | ϕ(x) | < 1 ✈ỵ✐ ồ ỗ ự tr A
ự
t ộ
2



n

e + x + x + ... + x + ... =

xn ✳

n=0

❱➻

xn ≤ x

n

✈➔


❙✉② r❛

xn ❤ë✐ tö t✉②➺t ✤è✐ tr♦♥❣

n=0

✤↕✐ sè ❇❛♥❛❝❤ A✳
∞

∞

x < 1✱ ♥➯♥ ❝❤✉é✐

xn = s ∈ A✳

n=0

❱ỵ✐ ♠é✐ n ≥ 1 t❛ ❦➼ ❤✐➺✉ Sn = e + x + x2 + ... + xn ✳❑❤✐ ✤â✱
Sn (e − x) = e + x + x2 + ... + xn (e − x)

= e + ex + ex2 + ... + exn − ex − x2 − ... − xn − xn+1
= e − xn+1
(e − x) Sn = (e − x) e + x + x2 + ... + xn
= e + ex + ex2 + ... + exn − ex − x2 − ... − xn − xn+1
= e − xn+1
✷✺