▼Ư❈ ▲Ư❈
▼Ư❈ ▲Ư❈✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶
▲❮■ ▼Ð ✣❺❯✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷
❈❤÷ì♥❣ ■✳ ▼ët sè tự ỡ
ữỡ ỵ
ỵ
ổPổ
t ỗ t


❚❍❆▼ ❑❍❷❖✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✹✸

✶




ỵ ồ t
ổ õ ỵ t trú ừ tr
t rt s ợ ữ t t ỵ
ởt tr ỵ q trå♥❣ ✈➔ ❝ì ❜↔♥ ♥❤➜t ❝õ❛ ●✐↔✐ t➼❝❤ ❤➔♠✱ ❧➔
♠ët tr ỳ ỵ ỡ s õ ỵ tt
r ỵ sỹ tỗ t↕✐ ♠➔ ❞↕♥❣ ❝õ❛ ♥â ✤➦❝ ❜✐➺t t❤➼❝❤ ❤ñ♣ ♥❤ú♥❣
✈➜♥ t t ợ ởt ữủ ợ ỳ ự ử tỹ t q
trồ ỵ ởt ỵ r➜t ✤÷đ❝ ❝→❝ ♥❤➔ ●✐↔✐ t➼❝❤
❤å❝ ÷✉ ❝❤✉ë♥❣✳ ▼ư❝ ✤➼❝❤ ừ sỹ tỗ
t ừ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❝ị♥❣ ✈ỵ✐ ♥❤ú♥❣ ✤➦❝ t➼♥❤ õ
t t q trồ ừ ỵ ữủ sỹ
ữợ ừ t ữỡ ố ❚✉②➸♥✱ t→❝ ❣✐↔ ✤➣ q✉②➳t ✤à♥❤ ❝❤å♥
♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐✿ ỵ
ử ố t s ỵ

ợ ử ự ữ tr t ữủ ữỡ
ợ ở ♥❤÷ s❛✉✿

❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥✳ ❚r➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦❤→✐

♥✐➺♠ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ❝õ❛ tỉ♣ỉ ✤↕✐ ❝÷ì♥❣ ✤➸ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ❝❤ù♥❣
♠✐♥❤ ỵ ờ q ữỡ s

ữỡ ỵ ổ

ủ sì ❝❤✉➞♥✱ ♥û❛ ❝❤✉➞♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✱ ♥❣♦➔✐ r ữ
r ở số ỵ ỡ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t
❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✤à♥❤ ỵ ổổ
ởt số t ❝❤➜t ❝õ❛ ♥â✳ ❈✉è✐ ❝ị♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✷✱ ♥➯✉ ❧↕✐ t t
ừ ỗ ự ữủ ởt số ỵ q



ố tữủ ự ỵ
P ✤➲ t➔✐✳ ●✐↔✐ t➼❝❤ ❤➔♠✳
✺✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐✳
❑❤♦→ ❧✉➟♥ s➩ ❧➔ ♠ët t➔✐ t s ỡ ỵ
✈➔ ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ♥â ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ỵ

tớ t❤ù❝ ♥➯♥ ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❦❤♦→
❧✉➟♥ ♥➔② ❝❤➾ tr➼❝❤ ð ❞↕♥❣ ❜ê ✤➲✱ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷♥❣ ❝â ú t
t t

ởt số ỵ ữủ q ữợ tr t ở t


r t ❜ë ❜➔✐ ✈✐➳t ❦❤✐ ❝❤♦ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ X t❤➻ t❛ ❤✐➸✉ ❧➔ ❝→❝
❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ t→❝ ❣✐↔ q✉✐ ÷ỵ❝ t➜t ❝↔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❧➔ τ1 ✈➔ ❝❤➼♥❤
q✉② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t❤✉➟t ♥❣ú ❦❤→❝ ♥➳✉ ❦❤æ♥❣ ♥â✐ ❣➻ t❤➯♠ t❤➻ ✤÷đ❝
❤✐➸✉ t❤ỉ♥❣ t❤÷í♥❣✳

●✐↔ sû E ❧➔ t➟♣ ỗ ừ ổ ỗ ữỡ X õ
õ ừ E ữủ ỵ E õ ỵ E
sỷ X ổ tổổ õ ổ ỗ ữỡ ỵ
Xw ố ừ Xw ữủ ỵ X ∗ ✳
✇

✣➸ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ♥➔② t→❝ ❣✐↔ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï r➜t ♥❤✐➲✉
❝õ❛ ❣✐❛ ✤➻♥❤✱ t❤➛② ❝æ ✈➔ ❜↕♥ ❜➧✳ ✣➛✉ t✐➯♥ ❝❤♦ ♣❤➨♣ t→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐
❝→♠ ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ❚❤✳❙ ▲÷ì♥❣ ◗✉è❝ ❚✉②➸♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ð
tr♦♥❣ s✉èt q✉→ tr➻♥❤ ✤➸ ❤♦➔♥ t❤➔♥❤ ❦❤♦→ ❧✉➟♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝❤➙♥ t❤➔♥❤
❝→♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥
tr÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕②✳
❈✉è✐ ❝ò♥❣ t→❝ ❣✐↔ ①✐♥ ❝→♠ ì♥ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ✈➔ t➜t ❝↔
✸


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


❚→❝ ❣✐↔✳

✹


❈❤÷ì♥❣ ■✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❇❷◆
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ♠ët t➟♣ ❤ñ♣✱ K ❧➔ ♠ët tr÷í♥❣✳ X ✤÷đ❝
❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥ tr÷í♥❣ K ✱ ♥➳✉
✶✳ ✣÷❛ ✤÷đ❝ ♣❤➨♣ ❝ë♥❣ ❝→❝ ♣❤➛♥ tû ✈➔♦ X ✱ tù❝ ❧➔ ù♥❣ ♠é✐ ❝➦♣
✭x, y) X ì X ợ ởt tỷ ừ X ỵ x + y s
a x + y = y + x, ∀ x, y ∈ X.
b✮ ✭x + y) + z = x + (y + z), x, y, z X
c ỗ t tỷ ổ ừ X ỵ s❛♦ ❝❤♦
x + 0 = x, ∀x ∈ X ✳
d✮

❱ỵ✐ ồ x X õ tỗ t tỷ ố ừ x ỵ x s

x + (x) = 0, ∀x ∈ X.

✷✳ ✣÷❛ ✤÷đ❝ ♣❤➨♣ ♥❤➙♥ ❝→❝ ♣❤➛♥ tû ❝õ❛ K ✈ỵ✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✱
tù❝ ự ợ ộ , x K ì X ợ ởt tỷ ừ X ỵ
αx✱ ❣å✐ ❧➔ t➼❝❤ ❝õ❛ α ✈ỵ✐ x✱ s❛♦ ❝❤♦
a✮ ✶✳x = x, ∀x ∈ X ✱ ✶ ❧➔ ✤ì♥ ✈à ❝õ❛ tr÷í♥❣ K ✳
b✮ α(βx) = (αβ)x, ∀α, β ∈ K, ∀x ∈ X ✳
c✮ (α + β)x = αx + βx, ∀α, β ∈ K, ∀x, y ∈ X ✳
d✮ α(x + y) = αx + αy, ∀α K, x, y X

ú ỵ t❛ ❝❤➾ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣
❧➔ tr÷í♥❣ R ❝→❝ sè t❤ü❝ ✈➔ ❦❤✐ ✤â X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥

t✉②➳♥ t➼♥❤ t❤ü❝✳

a. K

❧➔ tr÷í♥❣ C ❝→❝ sè ♣❤ù❝ ✈➔ ❦❤✐ ✤â X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥
t✉②➳♥ t➼♥❤ ♣❤ù❝✳
❱➻ t❤➳ t❛ s➩ ❣å✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ K ❧➔ ❝→❝ sè ✈➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ X
✤÷đ❝ ❣å✐ ❧➔ ✈❡❝tì ❤♦➦❝ ✤✐➸♠✳

b. K

✺


✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❝♦♥ M ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X ✤÷đ❝ ❣å✐
❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✱ ♥➳✉ ✈ỵ✐ ♠å✐ x,
sè α, β t❛ ✤➲✉ ❝â αx + βy ∈ M ✳

y ∈ M✱

✈➔ ♠å✐

✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ Φ ✭Φ ❧➔

t❤ü❝ ❤♦➦❝ ♣❤ù❝✮✳ ❚❛ ♥â✐ r➡♥❣ tỉ♣ỉ tr X tữỡ t ợ trú
số tr X ♥➳✉ ❝→❝ ♣❤➨♣ t♦→♥ ✤↕✐ sè tr➯♥ X ✭❝ë♥❣ ổ
ữợ tử t tổổ õ ❧➔
a.

b.


❱ỵ✐ ∀x, y ∈ X ✱ ✈➔ ♠å✐ ❧➙♥ ❝➟♥ W ừ x + y tỗ t ởt
U ❝õ❛ x ✈➔ V ❝õ❛ y s❛♦ ❝❤♦ U + V ⊂ W ✳
❱ỵ✐ ∀x ∈ X, ∀α ∈ ồ W ừ x tỗ t ởt ❧➙♥ ❝➟♥ U
❝õ❛ ✤✐➸♠ x ✈➔ sè r > 0 s❛♦ ❝❤♦ βU ⊂ W, ∀β ∈ Φ ♠➔ |β − α| < r✳

✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X tr➯♥ tr÷í♥❣ Φ ✤÷đ❝ ❣å✐ ❧➔

❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✭❤❛② ❦❤æ♥❣ ❣✐❛♥ tæ♣æ t✉②➳♥ t➼♥❤ ✮ ♥➳✉ tr➯♥ ✤â ✤➣
❝❤♦ ởt tổổ tữỡ t ợ trú số tr➯♥ X s❛♦ ❝❤♦ ♠é✐
✤✐➸♠ ❝õ❛ X ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣✳

✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛ ●✐↔ sû X, Y ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ tr➯♥

❝ị♥❣ ♠ët tr÷í♥❣ Φ✱ →♥❤ ①↕ Λ : X → Y ✤÷đ❝ ❣å✐ ❧➔ t✉②➳♥ t➼♥❤ ♥➳✉
Λ(αx + βy) = αΛ(x) + βΛ(y), ∀x, y ∈ X, ∀α, β ∈ Φ✳

✯ ▼ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ Λ : X → Φ ✤÷đ❝ ❣å✐ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✳

✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì
tỉ♣ỉ X ✳
a.

❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❤ót✱ ♥➳✉ ∀x ∈ X, ∃Λ > 0 s❛♦ ❝❤♦
x ∈ αA, ∀α ∈ Φ

b.

♠➔ |α| ≥ Λ.


❚➟♣ A ữủ ồ ỗ x, y A,


[0, 1]

t❛ ❝â


Λx + (1 − Λ)y ∈ A✳
c.

❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❝➙♥ ♥➳✉ ∀x ∈ A✱ t❛ ❝â αx ∈ A,

∀α ∈ Φ

♠➔

|α| ≤ 1.
d.
e.

❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ t✉②➺t ố ỗ A ứ ỗ ứ
A ữủ ❣å✐ ❧➔ ❜à ❝❤➦♥ ♥➳✉ ✈ỵ✐ ♠å✐ ❧➙♥ ❝➟♥ ❝õ❛ 0 X
tỗ t số > 0 s❛♦ ❝❤♦
A ⊂ tV, ∀t > s✳

✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t ♥➳✉
♠å✐ ♣❤õ tr X tỗ t ừ ỳ ✭♥❣❤➽❛ ❧➔✿ ◆➳✉
u = {uα , α ∈ λ} ❧➔ ởt ừ ỗ t ừ X t tỗ t
n

1 , 2 , ..., n s X ⊂
uαi ✮✳
i=1

✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ K ⊂ X ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➳✉ ❦❤ỉ♥❣

❣✐❛♥ ❝♦♥ K ❝ị♥❣ ✈ỵ✐ tỉ♣ỉ ❝↔♠ s✐♥❤ tr➯♥ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t✱
♥❣❤➽❛ ❧➔ ♠å✐ ừ ừ K ỗ t tr ổ K
tỗ t ừ ỳ

sỷ A t ỗ út ừ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì
tỉ♣ỉ X ✱ ❤➔♠ t❤ü❝ ❦❤ỉ♥❣ ➙♠ ✭t✉②➺t ố ỗ àA : X R ổ
tự
àA (x) = inf{t > 0 : t−1 x ∈ A}, ∀x ∈ X

✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❝ï ❤♦➦❝ ♣❤✐➳♠ ❤➔♠ ▼✐♥❦♦✇s❦✐ ❝õ❛ A✳

✶✳✶✳✶✵ ◆❤➟♥ ①➨t✳
a.

❚➟♣ A ⇔ ∀x1, x2, ..., xn X,

n N

tỗ t số > 0 s❛♦ ❝❤♦

{x1 , x2 , ..., xn } ⊂ αA, ∀α ∈ Φ
✼



♠➔ |α| ≥ Λ.
b.

❚➟♣ A ❝➙♥ ⇔ αA ⊂ A,

∀α ∈ Φ, |α| ≤ 1.

✶✳✶✳✶✶ ▼➺♥❤ ✤➲✳ ●✐↔ sû Λ : X → Y ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â
a. Λ(0) = 0✳
b.

c.

d.

◆➳✉ A ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ỗ ừ X t (A)
ụ
B ởt ổ ỗ ừ Y t❤➻
Λ−1 (B) ❝ô♥❣ ✈➟②✳
✣➦❝ ❜✐➺t✱ Λ−1(0) = {x ∈ X :
❣✐❛♥ ❝♦♥ ❝õ❛ X ✳

Λ(x) = 0} = N (Λ)

❧➔ ♠ët ❦❤æ♥❣

✶✳✶✳✶✷ ✣à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët

❝➦♣ ✭X, ρ✮✱ tr♦♥❣ ✤â X ❧➔ ♠ët t➟♣ ❤đ♣✱ ρ : X × X → R ❧➔ ♠ët ❤➔♠ sè
①→❝ ✤à♥❤ tr➯♥ X × X t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉

✶✳ ρ(x, y) ≥ 0✱ ✈ỵ✐ ♠å✐ x, y ∈ X ✱
ρ(x, y) = 0 ⇔ x = y ❀
✷✳ ρ(x, y) = ρ(y, x)✱ ✈ỵ✐ ♠å✐ x, y, z ∈ X ❀
✸✳ ρ(x, z) ∈ ρ(x, y) + ρ(y, z)✱ ✈ỵ✐ ♠å✐ x, y, z ∈ X.

✶✳✶✳✶✸ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ❧➔ ♠ët ❝➦♣ ✭X, τ ✮✱ tr♦♥❣ ✤â X ❧➔
♠ët t➟♣ ❤ñ♣✱ τ ❧➔ ♠ët ❤å ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X t❤♦↔ ♠➣♥
✶✳ ∅ ∈ τ, X ∈ τ ✳
✷✳ U1,

U2 ∈ τ ⇒ U1 ∩ U2 ∈ τ ✳

✸✳ ◆➳✉ {Uα}α∈I t❤➻ Uα ∈ τ .
α∈I
❚➟♣ ❤đ♣ X ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥✱ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ❣å✐ ❧➔ ❝→❝ ✤✐➸♠ ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ✈➔ ♠é✐ ♣❤➛♥ tû ❝õ❛ τ ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ❤đ♣ ♠ð ❝õ❛ X ✳
◆❤÷ ✈➟②✱
✽


❚➟♣ ❤đ♣ ré♥❣ ✈➔ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ ❧➔ ♥❤ú♥❣ t➟♣ ❤ñ♣ ♠ð✳
ii. ●✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ❤ñ♣ ♠ð ❧➔ ♠ët t➟♣ ❤ñ♣ ♠ð✳
iii. ❍ñ♣ ❝õ❛ ♠ët ❤å t✉ý þ ♥❤ú♥❣ t➟♣ ❤ñ♣ ♠ð ❧➔ ♠ët t➟♣ ❤ñ♣ ♠ð✳
✶✳✶✳✶✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✭X, τ ✮ ❣å✐ ❧➔ ♠ët T 1−❦❤ỉ♥❣
❣✐❛♥ ♥➳✉ ✈ỵ✐ ❤❛✐ ♣❤➛♥ tû ❦❤→❝ ♥❤❛✉ t ý x1 x2 ừ X tỗ t ♠ët
t➟♣ ❤đ♣ ♠ð U ❝❤ù❛ x1 ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤ù❛ x2.
i.

✶✳✶✳✶✺ ✣à♥❤ ♥❣❤➽❛✳ X ✤÷đ❝ ❣å✐ ❧➔ T 2−❦❤ỉ♥❣ ❣✐❛♥ ✭❤❛② ❧➔ ❦❤æ♥❣ ❣✐❛♥
❍❛✉s❞♦s❢❢ ✮ ♥➳✉ ∀x, y ∈ X,

❝õ❛ y s U V = 0

x = y

tỗ t U ừ x V

ú ỵ ộ T 2−❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ❧➔ ♠ët T 1−❦❤æ♥❣ ❣✐❛♥✳
✶✳✶✳✶✻ ✣à♥❤ sỷ A t õ tũ ỵ ừ ổ

tỡ tổổ X
a. ỗ ừ A ỵ convA tờ ủ tt tờ ❤ñ♣ t✉②➳♥
k
t➼♥❤ ❤ú✉ ❤↕♥ Λixi✱ tr♦♥❣ ✤â
i=1

∗

k

Λi ≥ 0, xi ∈ A, i = 1, 2, ..., k, k ∈ N ,

i = 1.
i=1

b.

tt ố ỗ ừ A t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤
k
❤ú✉ ❤↕♥ Λixi✱ tr♦♥❣ ✤â
i=1


∗

k

Λi ∈ Φ, xi ∈ A, i = 1, 2, ..., k, k ∈ N ,
i=1

Λ i ≤ 1

t ỗ ừ A ừ tt t ủ ỗ tr
X

ự A õ t ỗ ọ t ự A

sỷ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ Φ ✭Φ

t❤ü❝ ❤❛② ♣❤ù❝✮✳ ❍➔♠ p :
t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✳

X→R

✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ X ♥➳✉
✾


a. p(x + y) ≤ p(x) + p(y), ∀x, y ∈ X.
b. p(λx) = |λ| p(x), ∀λ ∈ Φ.

▼ët ♥û❛ ❝❤✉➞♥ p t❤♦↔ ♠➣♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥

c. p(x) = 0 ⇒ x = 0

t❤➻ p ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ tr X õ t ỵ
t ❝❤♦ p(x)✳ ◆❤÷ ✈➟② . t❤♦↔ ♠➣♥ ✸ ✤✐➲✉ ❦✐➺♥
✶✳

x ≥ 0, ∀x ∈ X

✷✳

λx = |λ| x , ∀x ∈ X, ∀λ ∈ Φ✳

✸✳

x + y ≤ x + y , ∀x, y ∈ X.

x

x = 0 ⇔ x = 0✳

✶✳✶✳✶✽ ✣à♥❤ ♥❣❤➽❛✳ ❍å P ❝→❝ ♥ú❛ ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì X

✤÷đ❝ ❣å✐ ❧➔ t→❝❤ ❝→❝ ✤✐➸♠ tr➯♥ X ♥➳✉ ✈ỵ✐ ♠é✐ x ∈ X ♠➔ x = 0 tỗ t
ỷ p P s p(x) = 0.

✶✳✶✳✶✾ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ ✭❤❛② ự ũ ợ

ởt tr X ữủ ồ ởt ổ
tỹ ự


ú ỵ r ổ ổ ỗ ✤à❛
♣❤÷ì♥❣✳

✶✳✶✳✷✵ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tổổ X tr trữớ


ữủ ồ ổ ỗ ♣❤÷ì♥❣ ♥➳✉ ♥â ❝â ♠ët ❝ì sð ✤à❛
♣❤÷ì♥❣ B s❛♦ ❝❤♦ ♠å✐ ♣❤➛♥ tû ❝õ❛ B ❧➔ ❝→❝ t➟♣ ❤ñ♣ ỗ

a. X

ữủ ồ ổ ữỡ ♥➳✉ ♥â ❝â ♠ët ❧➙♥
❝➟♥ U ❝õ❛ ✵ ❧➔ t➟♣ ❜à ❝❤➦♥✳

b. X

✶✵


✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣ ♥➳✉ ♥â ❝â ♠ët ❧➙♥
❝➟♥ U ❝õ❛ ✵ s❛♦ ❝❤♦ U ❧➔ t➟♣ ❝♦♠♣❛❝t✳

c. X

✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤↔ ♠❡tr✐❝✱ ♥➳✉ tỉ♣ỉ τ ð tr➯♥ X ✤÷đ❝
s✐♥❤ ❜ð✐ ♠ët ♠❡tr✐❝ d ❜➜t ❜✐➳♥ ♥➔♦ ✤â✳

d. X

✤÷đ❝ ❣å✐ ❧➔ F ✲❦❤ỉ♥❣ ❣✐❛♥✱ ♥➳✉ tỉ♣ỉ τ ð tr➯♥ X ✤÷đ❝ s✐♥❤ ❜ð✐

♠ët ♠❡tr✐❝ d ❜➜t ❜✐➳♥✱ ✤➛② ✤õ ♥➔♦ ✤â✳

e. X

✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r➨❝❤❡t ♥➳✉ X ❧➔ ♠ët F ✲❦❤æ♥❣ ❣✐❛♥✱ ỗ
ữỡ

f. X

ữủ ồ ổ ♥➳✉ tr➯♥ X ❝â ♠ët
❝❤✉➞♥ s❛♦ ❝❤♦ ♠❡tr✐❝ s✐♥❤ ❜ð✐ tữỡ t ợ tổổ
tr X

g. X

✤÷đ❝ ❣å✐ ❧➔ ❝â t➼♥❤ ❝❤➜t ❍❡✐♥❡ ✲ ❇♦r❡❧ ♥➳✉ ✈ỵ✐ ♠é✐ t➟♣ ❝♦♥ ✤â♥❣
❜à ❝❤➦♥ ❝õ❛ X ❧➔ t➟♣ ❝♦♠♣❛❝t✳

h. X

✶✳✶✳✷✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ⊂ X ✳ A ỗ ừ tt
t ủ ✤â♥❣ ❝❤ù❛ A✱ ♥❣❤➽❛ ❧➔
A=

{F : F

✤â♥❣,

A ⊂ F }.


▲ó❝ ✤â✱ t❛ ♥â✐ r➡♥❣ A ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ A✳
❈❤ó þ✳
✶✳ A ❧➔ ♠ët t➟♣ ✤â♥❣ ❝❤ù❛ A ✈➔ ♥â ❧➔ t➟♣ ✤â♥❣ ♥❤ä ♥❤➜t ❝❤ù❛ A✳
✷✳ A ✤â♥❣ ⇔ A = A✳
✸✳ ◆➳✉ A ⊂ B ⇔ A = B ✳

✶✳✶✳✷✷ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A, B ⊂ X; A ⊂ B t❛ ❝â ♥â✐ A trò ♠➟t

tr♦♥❣ B ♥➳✉ B ⊂ A✳ ❚❛ ♥â✐ A ❧➔ t➟♣ trò ♠➟t tr♦♥❣ X ♥➳✉ A = X ✳

✶✳✶✳✷✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû A ⊂ X ✱ ❦❤✐ ✤â
✶✶


✶✳ A ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❜à ❝❤➦♥ ♥➳✉ ♥â ❧➔ t➟♣ ❝♦♥ ❝õ❛ ♠ët ❤➻♥❤ ❝➛✉ ♥➔♦
✤â✱ ❦➼ ❤✐➺✉
d(A)❂

s✉♣{d(x, y) :

x, y ∈ A}✳

❑❤✐ ✤â✱ d(A) ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ A✳
✷✳ A ✤÷đ❝ ❣å✐ ❧➔ n❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ ♥➳✉ ∀ε > 0✱ ∃x1, x2, ..., xn ∈ X ✳
❙❛♦ ❝❤♦ A ⊂ S(xi, ε.)
i=1

✶✳✶✳✷✹ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ A ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ✳ ❑❤✐ ✤â✱
✶✳ ❚➟♣ ❤đ♣ U ⊂ X ✤÷đ❝ ❣å✐ ởt ừ A tỗ t t
V ∈ τ s❛♦ ❝❤♦

A ⊂ V ⊂ U✳

✷✳ ◆❤➟♥ ①➨t✳
a. ◆➳✉ U, V ❧➔ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ A t❤➻ U ∩ V ❝ô♥❣ ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛
A✳
b. A ❧➔ t➟♣ ❤ñ♣ ♠ð ⇔ A ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x A x A tỗ t
t V s❛♦ ❝❤♦ x ∈ V ⊂ A.
✶✳✶✳✷✺ ❇ê ✤➲ ❩♦r♥✳ ●✐↔ ①û X ❧➔ ♠ët t➟♣ ❤ñ♣ ✈➔ ≤ ❧➔ ♠ët t❤ù tü ✭❜ë
♣❤➟♥✮ tr➯♥ X ✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ x, y, z ∈ X t❛ ❝â x ∈ x❀ ♥➳✉ x ≤ y ✈➔ y ≤ x
t❤➻ x = y ✈➔ ♥➳✉ x ≤ y, y ≤ z t❤➻ x ≤ z ✭P❤↔♥ ①↕✱ ✤è✐ ①ù♥❣✱ ❜➢t ❝➛✉✮✳
▼ët t➟♣ ❝♦♥ A ⊂ X ✤÷đ❝ ❣å✐ ❧➔ s➢♣ t✉②➳♥ t➼♥❤ ♥➳✉ ✈ỵ✐ ♠å✐ x, y ∈ X
t❤➻ x ≤ y ❤♦➦❝ y ≤ x✳ P❤➛♥ tû a ∈ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❜✐➯♥ ✭❝➟♥✮ tr➯♥
❝õ❛ A ♥➳✉✿ x ≤ a ✈ỵ✐ ♠å✐ x ∈ A✳ P❤➛♥ tû a ∈ X ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû
❝ü❝ ✤↕✐ ♥➳✉ ♠å✐ x ∈ X ♠➔ a ≤ x t❤➻ a = x✳

✶✳✶✳✷✻ ✣à♥❤ ♥❣❤➽❛✳

✯ ❈❤♦ t➟♣ X ❀ F = {f : X → Yf ; Yf ❦❤æ♥❣ ❣✐❛♥ tæ♣æ}✱ ϑ ❧➔ ❤å t➼♥❤
❝❤➜t ❝→❝ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ t➟♣ ❝â ❞↕♥❣ f −1(V )✱ tr♦♥❣ ✤â V ♠ð
tr♦♥❣ Yf , f ∈ ✳
✶✷


✯ τ ❧➔ ❤å t➜t ❝↔ ❝→❝ ❤ñ♣ ♥➔♦ ✤â ❝→❝ t➟♣ t❤✉ë❝ ϑ✳ ❑❤✐ ✤â τ ❧➔ ♠ët tæ♣æ
tr➯♥ X ✈➔ ♥â ❧➔ tæ♣æ ②➳✉ ♥❤➜t ❧➔♠ ❝❤♦ ❝→❝ →♥❤ ①↕ f ∈ F ❧✐➯♥ tö❝✳
✯ τ ①➙② ❞ü♥❣ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ ✤➛✉ ❝õ❛ X s✐♥❤ ❜ð✐ F.

✶✳✶✳✷✼ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❝♦♥ E ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✤÷đ❝ ❣å✐
❧➔ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥ ♥➳✉ ♠é✐ ❧➙♥ ❝➟♥ V ❝õ❛ 0 ∈ X ❝â ♠ët t➟♣ ❝♦♥ ❤ú✉
❤↕♥ F ⊂ X s❛♦ ❝❤♦ E ⊂ F + V ✳


◆❤➟♥ ①➨t✳ ◆➳✉ K ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X t❤➻

❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✵ tr♦♥❣ X ✳ ❑❤✐ ✤â
{x + V, x ∈ K} ❧➔ ♣❤õ ♠ð ❝õ❛ K ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➯♥
K

n

∃x1 , .., xn ∈ K : K ⊂

(xi + V )
i=1

✣➦t
F = {xi : i = 1, n} ⇒ K ⊂ F + V.

❙✉② r❛ K ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥✳

✶✸


ữỡ ị
ỵ ✲ ❇❛♥❛❝❤
✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ tr➯♥ tr÷í♥❣ Φ

✭t❤ü❝ ❤❛② ♣❤ù❝✮✳ ❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝
tr➯♥ t➟♣ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ✭❤❛② ❦❤ỉ♥❣ ❣✐❛♥ ố
ừ X ỵ X ❚r➯♥ X ∗ t❛ ✤÷❛ ✈➔♦ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣

ổ ữợ ữ s
1 + 2)x = 1x + Λ2x, ∀Λ1, Λ2 ∈ X ∗, x ∈ X ✳
✭αΛ)x = αΛx,

∀Λ ∈ X ∗ , α ∈ Φ, x ∈ X

✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ♥➔② X ∗ ❧➟♣ t❤➔♥❤ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ tr➯♥
tr÷í♥❣ Φ✳

✷✳✶✳✷ ❇ê ✤➲✳ ❍➔♠ f : X −→ C ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ tr X
tỗ t ởt t✉②➳♥ t➼♥❤ t❤ü❝ u1 : X → R tr➯♥ X s❛♦
❝❤♦
f (x) = u1 (x) − iu1 (ix), ∀x ∈ X.

❈❤ù♥❣ ♠✐♥❤✳
❇➙② ❣✐í t❛ ❣✐↔ t❤✐➳t X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ♣❤ù❝✱ f (x) ❧➔ ♠ët
♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ♣❤ù❝ ①→❝ ✤à♥❤ tr➯♥ X ✳ ❑❤✐ ✤â✱
f (x) = u1 (x) + iu2 (x) ✭✶✮
tr♦♥❣ ✤â i ❧➔ sè ↔♦✱ u1(x) ❧➔ ♣❤➛♥ t❤ü❝ u2(x) ❧➔ ♣❤➛♥ ↔♦ ❝õ❛ f (x)
t ỵ ♣❤➛♥ tû ❝õ❛ x ✈ỵ✐ ❝→❝ sè t❤ü❝✱ t❤➻ t❛ ❝â
t❤➸ ❝♦✐ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝✳ ❚❛ ❝â u1(x) ✈➔ u2(x) ❧➔
♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ X.
❚ø ✭✶✮ s✉② r❛
f (ix) = u1 (ix) + iu2 (ix).
✶✹


▼➦t ❦❤→❝✱
f (ix) = if (x) = −u2 (x) + iu1 (x).


❙✉② r❛
u2 (x) = −u1 (ix), ∀x ∈ X.

t❤❛② ✈➔♦ ✭✶✮ t❛ ✤÷đ❝
f (x) = u1 (x) − iu1 (ix), ∀x ∈ X.

❱➟② u1 ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ ❝➛♥ t➻♠✳
◆❣÷đ❝ ❧↕✐✱ ✈➻ u1(x) ❧➔ t✉②➳♥ t➼♥❤ t❤ü❝ ♥➯♥ ❤➔♠ f (x) = u1(x) − iu1(ix)
✈ỵ✐ ♠å✐ x ∈ X t❤♦↔ ♠➣♥ f (x + y) = f (x) + f (y)✱ ✈ỵ✐ ♠å✐ x, y ∈ E. ❱ỵ✐
♠å✐ x ∈ E ✈➔ λ = α + iβ ∈ C t❛ ❝â
f (λx) = u1 (λx) − iu1 (iλx) = u1 ((α + iβ)x) − iu1 (i(α + iβ)x)
= αu1 (x) + βu1 (ix) − iαu1 (ix) + iβu1 (x)
= (α + iβ) u1 (x) − i (α + iβ) u1 (ix)
= λ (u1 (x) − iu1 (ix) ) = λf (x).

❱➟② u ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ♣❤ù❝✳

✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✱ p : X → R ❧➔ ❤➔♠
t❤ü❝✳ ❑❤✐ ✤â✱

✶✳ p ✤÷đ❝ ❣å✐ ❧➔ sì ❝❤✉➞♥ tr➯♥ X ♥➳✉
a. p(αx) = αp(x), ∀x ∈ X, α ≥ 0.
b. p(x + y) ≤ p(x) + p(y), ∀x, y ∈ X.

✷✳ p ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ X ♥➳✉
a. p(x) ≥ 0, ∀x ∈ X.
b. p(αx) = |α| p(x), ∀x ∈ X, α ∈ Φ.
c. p(x + y) ≤ p(x) + p(y), ∀x, y ∈ X.
✶✺



ỵ ổ t t tỹ

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✉②➳♥ t➼♥❤ Λ : X → R ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë
❦❤æ♥❣ ❣✐❛♥ X s❛♦ ❝❤♦
i. Λ(x) = f (x), ∀x ∈ M.
ii. Λx ≤ p(x), ∀x ∈ X.

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

●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 ỵ
D =
gD

Dg

õ D M ✈➔ D∗ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✳ t❛ ❝â
∀x ∈ D∗ , ∃g ∈ D : 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 ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤
✤â ❧➔ ♣❤✐➳♠ ❤➔♠ ❝➛♥ t➻♠✳ ▼✉è♥ t❤➳✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♠✐➲♥ ①→❝
✤à♥❤ D ❝õ❛ Λ ❧➔ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ X ✳
P❤↔♥ ❝❤ù♥❣✳ D = X ✳ ❑❤✐ ✤â✱ ∃x0 ∈ X, x0 ∈/ D s r x0 = 0 ỵ
[x0 ] ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ♠ët ❝❤✐➲✉ ❝õ❛ X ❣➙② x0
õ
D [x0 ] = {0}

ỵ ❤✐➺✉ 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 ) (x)].
xD

xD

r tỗ t↕✐ ❤➡♥❣ sè c s❛♦ ❝❤♦
−p(−x − x0 ) − Λ(x) ≤ c, ∀x ∈ D
c ≤ p(x + x0 ) − Λ(x), ∀x ∈ D

✈ỵ✐ z ∈ Z ✱ t❤❡♦ ✭✸✮✱ 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✳ ❈❤➾ ❝â ❤❛✐
❦❤↔ ♥➠♥❣ α > 0 ❤♦➦❝ α < 0✳
(a). ❱ỵ✐ α > 0✳ ❇ð✐ ✈➻ ✭✺✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ D✱ ❝❤♦ ♥➯♥
c ≤ p( αx + x0 ) − Λ( αx )✳

❙✉② r❛
αc ≤ αp( αx + x0 ) − αΛ( αx ) = p(x + αx0 ) − Λ(x)✳

❙✉② r❛
Λ(x) + αc ≤ p(x + αx0 ).

❙✉② r❛ G(z) ≤ p(z)✳
(b). ❱ỵ✐ α < 0✳ ❇ð✐ ✈➻ ✭✹✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ D✱ ❝❤♦ ♥➯♥
−p(− αx − x0 ) − Λ( αx ) ≤ c✳

❙✉② r❛
−(−α)p(− αx − x0 ) + αΛ( αx ) ≤ −αc

✳

❙✉② r❛
−p(x + αx0 ) + Λ(x) ≤ −αc✳


❙✉② r❛
Λ(x) + αc ≤ p(x + αx0 )

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


ỵ ổ t t ự
①û X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ ♣❤ù❝✱ p ❧➔ ♠ët ♥ú❛ ❝❤✉➞♥
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✉②➳♥ t➼♥❤ Λ : X → C ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë
❦❤æ♥❣ ❣✐❛♥ X s❛♦ ❝❤♦
i. Λx = f (x), ∀x ∈ M.
ii. |Λx| ≤ p(x), ∀x X.

ự f (x) ữợ ✭✶✮ ✈➔ ♥❤➟♥ ✤÷đ❝ ♣❤✐➳♠ ❤➔♠
t✉②➳♥ t➼♥❤ t❤ü❝ f1(x) ①→❝ tr ổ t t tỹ M
ỗ tớ
f1 (x) ≤ |f1 (x)| ≤ |f (x)| ≤ p(x), ∀x M.

ỵ tỗ t t t➼♥❤ t❤ü❝ Λ1(x) ①→❝ ✤à♥❤
tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝ X s❛♦ ❝❤♦
✭a✮ Λ1(x) = f1(x), ∀x ∈ M ✳
✭b✮ |Λ1(x)| ≤ p(x), ∀x ∈ X ✳

❉♦ p ❧➔ ♠ët ♥ú❛ ❝❤✉➞♥✱ tø b 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 ✳ ❚ø ✭✷✮ s✉② r❛
✶✾


Λ(x) = f (x), ∀x ∈ M ✳

▼➦t ❦❤→❝✱ ♥➳✉ Λ(x) = 0✱ t❤➻ Λ(x) =
✈➟②✱

Λ(x)eiθ (θ

♣❤ö t❤✉ë❝ ✈➔♦ x✮✳ ❱➻

|Λ(x)| = e−iθ Λ(x) = Λ(xe−iθ ).

❱➻ Λ(xe−iθ ) ❧➔ sè t❤ü❝✱ tø ✭✻✮ s✉② r❛
Λ(xe−iθ ) = Λ1 (xe−iθ )✳


s✉② r❛ |Λ(x)| = Λ1(xe−iθ ) ≤ p(xe−iθ ) = p(x).

✷✳✶✳✻ ❍➺ q✉↔✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ x0 X õ
tỗ t ởt ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ Λ ∈ X ∗ s❛♦ ❝❤♦
Λx0 = x0

✈➔ |Λx| ≤

x , ∀x ∈ X ✳

❈❤ù♥❣ ♠✐♥❤✳
+✮ ◆➳✉ x0 = 0✱ t❛ ❧➜② Λ = 0 ❧➔ ❤➔♠ t✉②➳♥ t➼♥❤ t❤✉ë❝ Λ ∈ X ∗ t❤ä❛ ♠➣♥
②➯✉ ❝➛✉ ❜✐➸✉ t❤ù❝✳
Λx0 = x0

✈➔ |Λx| ≤

x , ∀x ∈ X ✳

+✮

◆➳✉ x0 = 0 t❤➻ t❛ ❧➜② p(x) = x ✈➔ M =< x0 > ✈➔ f ❧➔ ❤➔♠ ①→❝
✤à♥❤ ❜ð✐ f (αx0) = α x0 ✳ ❑❤✐ ✤â t❛ ❝â
• f

❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ M → C.

• |f (αx0 )| = |α| x0 = αx0 = p(x0 ).

r t ỵ ổ t t

ự tỗ t t✉②➳♥ t➼♥❤ Λ : X → C s❛♦ ❝❤♦
✶✳ Λ(x0) = f (x0) =

x

✳

✷✳ |Λ(x)| ≤ p(x) =

x , ∀x ∈ X ✳

❘ã r➔♥❣ tø 2✮ t❛ s✉② r❛ Λ ∈ X ∗. ❉♦ ✈➟②✱
✷✵


∃Λ ∈ X ∗ : Λx0 = x0 , |Λx| ≤ x , ∀x ∈ X ✳

✷✳✶✳✼ ❇ê ✤➲ ✭❬✸❪✮✳ ●✐↔ sû K ✈➔ C ❧➔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X, K ❧➔ t➟♣

❝♦♠♣❛❝t✱ C ❧➔ t➟♣ ✤â♥❣ ✈➔ K C = õ tỗ t V ❝õ❛
✤✐➸♠ ✵ s❛♦ ❝❤♦
(K + V ) ∩ (C + V ) =

ỵ sỷ A, B t ỗ rộ tr ổ
tỡ tæ♣æ X ✈➔ A ∩ B = ∅✳
a.

◆➳✉ A ♠ð✱ t❤➻ ∃Λ ∈ X ∗ ✈➔ γ ∈ R s❛♦ ❝❤♦
ReΛx < γ ≤ ReΛy, ∀x ∈ A, ∀y ∈ B.


b.

◆➳✉ A ❝♦♠♣❛❝t✱ B ✤â♥❣ ✈➔ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ỗ ữỡ t
X 1 , γ2 ∈ R s❛♦ ❝❤♦
ReΛx < γ1 < γ2 < Rey, x A, y B.

ự rữợ t t❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ tr÷í♥❣ ❤đ♣ Λ ❧➔ ♣❤✐➳♠ ❤➔♠
t❤ü❝✳
✯ a✮ ▲➜② a0 ∈ A, b0 ∈ B ✳ ✣➦t x0 = b0 − a0, C = A − B + x0 õ C
ởt ỗ ừ ✵ tr♦♥❣ X ✳ ●✐↔ sû p ❧➔ ♣❤✐➳♠ ❤➔♠ ▼✐♥❝♦♣s❦②
❝õ❛ C. ❙✉② r❛ p t❤ä❛ ♠➣♥
p(x + y) ≤ p(x) + p(y)
p(tx) = tp(x), ∀t ≥ 0.

❉♦ ✤â✱ p tọ ừ ỵ
A ∩ B = ∅ ⇒ x0 ∈/ C ✱ ✈➻ ♥➳✉ x0 ∈ C ⇒ x0 = a − b + x0 ✈ỵ✐
a ∈ A, b ∈ B ⇒ a = b ∈ A ∩ B ✭♠➙✉ t❤✉➞♥✮✳ ❙✉② r❛
p(x0 ) ≥ 1.

✷✶


✣à♥❤ ♥❣❤➽❛✳ f (tx0) = t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M =< x0 >
◆➳✉ t ≥ 0✱ t❤➻ f (tx0) = t ≤ tp(x0) = p(tx0) ✭❞♦ p(x0) ≥ 1)
◆➳✉ t < 0✱ t❤➻ (tx0) < 0 ≤ p(tx0)
f ≤ p tr M
õ t ỵ tỗ t↕✐ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ü❝ Λ tr➯♥ X
t❤ä❛ ♠➣♥ Λ ≤ p, Λ |M = f.
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦t ❜✐➺t✱ Λ ≤ 1 tr➯♥ C s✉② r❛ Λ ≥ −1 tr➯♥ ✲C ✳ ❙✉②
r❛ Λ ≤ 1 tr➯♥ ❧➟♥ ❝➟♥ ❝õ❛ C ∩ (−C) ❝õ❛ ✵✳ ❙✉② r❛ Λ ❜à ❝❤➦♥ tr➯♥ ♠ët

❧➙♥ ❝➟♥ ❝õ❛ ✵ ♥➯♥ Λ ∈ X ∗✳
❇➙② ❣✐í ♥➳✉ a ∈ A, b ∈ B t❛ ❝â
Λa − Λb + 1 = Λ(a − b + x0 ) ≤ p(a − b + x0 ) < 1✱
✈➻ Λ(x0) = f (x0) = 1, a − b + x0 ∈ C ✈➔ C ❧➔ t➟♣ ♠ð✳ ❉♦ ✤â✱ Λa < Λb✳
✣✐➲✉ ✤â ❝❤ù♥❣ tä Λ(A) ✈➔ (B) t ỗ rớ (A)
tr ❝õ❛ Λ(B) tr♦♥❣ R✳ ❇ð✐ ✈➻ Λ ❧➔ →♥❤ ①↕ ♠ð ✈➔ A ♠ð ♥➯♥ Λ(A) ♠ð✳
❉♦ ✤â✱ ❧➜② γ ❧➔ ♠ót ♣❤↔✐ ❝õ❛ Λ(A).
❙✉② r❛
ReΛx < γ ≤ ReΛy, ∀x ∈ A, ∀y ∈ B.

✯ b✮ ❉♦ A ❝♦♠♣❛❝t✱ B õ X ỗ ữỡ tỗ t ỗ
ừ s
(A + V ) (B + V ) = ∅.

❙✉② r❛ (A + V ) ∩ B = ∅ ✭❇ê ✤➲ ✷✳✶✳✼✮
❚❤❡♦ a✮ t❤❛② A A + V ự tọ r tỗ t Λ ∈ X ∗ s❛♦
❝❤♦ Λ(A + V ) ✈➔ (B) rớ ỗ tr R (A + V ) ♠ð
♥➡♠ ❜➯♥ tr→✐ Λ(B)✳ ❇ð✐ ✈➻ Λ(A) ❝♦♠♣❛❝t tr♦♥❣ R ✈➔ ♥➡♠ tr♦♥❣
Λ(A + V ). ❙✉② r❛ ❧➜② γ1 ❧➔ ♠ót ♣❤↔✐ ❝õ❛ Λ(A), γ2 ❧➔ ♥➡♠ ❣✐ú❛ γ2 ✈➔
♠ót ♣❤↔✐ ❝õ❛ Λ(A + V )✳ ❙✉② r❛ ❝â
ReΛx < γ1 < γ2 < ReΛy, ∀x ∈ A, ∀y ∈ B.
✷✷


q X ổ ỗ ♣❤÷ì♥❣✱ t❤➻ X ∗ t→❝❤ ❝→❝
✤✐➸♠ ❝õ❛ X ✳

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ x1, x2 ∈ X, x1 = x2✳ ❳➨t A = {x1}✱ B = {x2}✳ ❑❤✐
✤â A, B ❧➔ ❝→❝ t ỗ rớ A t B õ tr ổ
ỗ ữỡ X õ ử ỵ b s r tỗ t X

s ❝❤♦
Λ(x1 ) = Λ(x2 ✮✳

❱➟② X ∗ t→❝❤ ❝→❝ ✤✐➸♠ ừ X

ỵ sỷ M ổ ừ ổ ỗ

ữỡ X x0 X x0 / M t tỗ t↕✐ Λ ∈ X ∗ s❛♦ ❝❤♦ Λx0 = 1
✈➔ Λx = 0✱ ∀x ∈ M ✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = {x0}, B = M ✳ ❑❤✐ ✤â A, B t ỗ rớ
tr ổ ỗ ữỡ A t B õ õ
ử ỵ ✷✳✶✳✽b✱ t❤➻ ∃Λ ∈ X ∗ s❛♦ ❝❤♦ Λx0 ✈➔ Λ(M ) rí✐ ♥❤❛✉✳ ❇ð✐ ✈➻
M ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ (M ) ổ ừ trữớ ổ
ữợ ❙✉② r❛ Λ(M ) = {0} ✈➔ Λx0 = 0. Λ ð ✤➙② t❛ ❝â t❤➸ ❝❤å♥ s❛♦ ❝❤♦
Λx0 = 1.

✭①➨t Λ

=

Λ
Λ(x0 )

✳ ❑❤✐ ✤â✱ Λ (x0) = 1✱ Λ /M = 0

ỵ sỷ M ổ ừ ổ ỗ

ữỡ X f ✿ M → Φ ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ M ✳ ❑❤✐
✤â✱ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ Λ ∈ X ∗ s❛♦ ❝❤♦
Λ(x) = f (x), ∀x ∈ M ✳


❈❤ù♥❣ ♠✐♥❤✳
+✮ ◆➳✉ f = 0 tr➯♥ M ✱ t❛ ❧➜② Λ = 0 tr➯♥ X ♥➯♥ Λ ∈ X ∗ ❧➔ ♣❤✐➳♠ ❤➔♠
❝➛♥ t➻♠✳
+✮ ◆➳✉ f = 0 tr➯♥ M t❤➻ ✤➦t
✷✸


M0 = {x ∈ M : f (x) = 0}

❝❤å♥ x0 ∈ M s❛♦ ❝❤♦ f (x0)❂ ✶✳
✭❧➜② x ∈ M : f (x) = 0✱ ✤➦t x0 = f (xx ) ⇒ f (x0) = 1 ❧➔ ❣✐→ trà ❝➛♥ t➻♠✮✳
❇ð✐ ✈➻ f ❧✐➯♥ tö❝ ♥➯♥ x0 ∈/ M0M ✈➔ ❜ð✐ ✈➻ tæ♣æ tr➯♥ M ❧➔ ❝↔♠ s✐♥❤ tø
tæ♣æ tr X s r
0

X

x0
/ M0 .

ỵ tỗ t X s x0 = 1, Λ |M = 0 ✳
◆➳✉ x ∈ M ✱ t❤➻ x − f (x)x0 ∈ M0✱ ✈➻ t❤➳ f (x0) = 1✳
❙✉② r❛
0 = Λ(x − f (x)x0 ) = Λx − f (x)Λx0 ❂ Λx − f (x)
❙✉② r❛ Λ = f tr➯♥ M ✳ ◆➯♥ t❛ s✉② r (x) = f (x), x M.
0

ỵ sỷ B t ỗ õ tr ổ ỗ
ữỡ X x0 X ữ x0 / B õ tỗ t

∈ X ∗ s❛♦ ❝❤♦ |Λ(x)| ≤ 1, ∀x ∈ B ✳ ❑❤✐ ✤â Λ ∈ X ∗ s❛♦ ❝❤♦
|Λ(x)| ≤ 1, ∀x ∈ B

♥❤÷♥❣ Λ(x0) > 1.
❈❤ù♥❣ ♠✐♥❤✳ ❇ð✐ B õ ỗ t A = {x0} s r A t
ỗ A, B t tọ ỵ b õ tỗ
t Λ1 ∈ X ∗ s❛♦ ❝❤♦
Λ1 x0 ❂reiθ
♥➡♠ ♣❤➼❛ ♥❣♦➔✐ ❝õ❛ K = Λ1(B) ✳ ❇ð✐ ✈➻ B ❝➙♥ ♥➯♥ K ❝ô♥❣ ❝➙♥✱ tù❝ ❧➔
∃S : 0 < s < r : z ≤ S, ∀z ∈ K.

❑❤✐ ✤â
Λ = S −1 e−iθ Λ1

❝â t➼♥❤ ❝❤➜t t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳
✷✹


✷✳✷ ❚æ♣æ ②➳✉
✷✳✷✳✶ ❇ê ✤➲✳ ●✐↔ sû F = {f : X → Yf ; Yf ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦s❢❢ }✳
◆➳✉ ❤å F t→❝❤ ❝→❝ ✤✐➸♠ tr➯♥ X ✱ t❤➻ tæ♣æ ✤➛✉ tr➯♥ X ❧➔ ❍❛✉s❞♦s❢❢✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû p ✈➔ q ❧➔ ❤❛✐ ✤✐➸♠ rí✐ ♥❤❛✉ tr♦♥❣ X ✱ F t
tr X tỗ t f ∈ F s❛♦ ❝❤♦ f (p) = f (q)✳ ❱➻ Yf ổ
ss tỗ t U1✱ U2 ❝õ❛ f (p) ✈➔ f (q) s❛♦ ❝❤♦
U1 ∩ U2 ❂ ∅ tr♦♥❣ Yf ✳ ▼➦t ❦❤→❝✱ ❞♦ f ❧✐➯♥ tö❝ ♥➯♥
f −1 (U1 ) ∩ f −1 (U2 ) = ∅.

❱➟② X ❧➔ T 2−❦❤æ♥❣ ❣✐❛♥✳


✷✳✷✳✷ ❇ê ✤➲✳ ◆➳✉ ✭X ✱ τ1✮ ❧➔ ♠ët T2−❦❤æ♥❣ ❣✐❛♥✱ ❝á♥ ✭X ✱ τ2✮ ❧➔ ❦❤æ♥❣
❣✐❛♥ ❝♦♠♣❛❝t ✈➔ τ1 ⊂ τ2 t❤➻ τ1 ❂ τ2✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû F ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❜➜t ❦ý tr♦♥❣ ✭X ✱ τ2✮✳ ❑❤✐ ✤â F
❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ ✭X ✱ τ2✮✳ ❉♦ τ1 ⊂ τ2 s✉② r❛ F ❧➔ ❝♦♠♣❛❝t tr♦♥❣ τ1
✭✈➻ ❣✐↔ sû u = {Uα}α∈I ❧➔ ♣❤õ ♠ð ❜➜t ❦ý ❝õ❛ F tr♦♥❣ τ1✳ ❙✉② r❛ u ❧➔
♣❤õ ♠ð ❝õ❛ F tr♦♥❣ ✭X ✱ τ2✮✮✳
❉♦ ✭X ✱τ1✮ ❧➔ T 2−❦❤æ♥❣ ❣✐❛♥ ♥➯♥ F ✤â♥❣ tr♦♥❣ ✭X ✱ τ1✮✳
❚ø ✤â s✉② r❛ τ1 = τ2✳

✷✳✷✳✸ ❇ê ✤➲✳ ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ❝♦♠♣❛❝t ✈➔ {fn⑥ ❧➔ ♠ët ❞➣②

❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tö❝ t→❝❤ ❝→❝ ✤✐➸♠ tr➯♥ X ✱ t❤➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❦❤↔
♠❡tr✐❝✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû τ ❧➔ tæ♣æ tr➯♥ X ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛
❣✐↔ sû r➡♥❣ |fn| ≤ 1✱ ✈ỵ✐ ♠å✐ n✳ ✣➦t
∞

d(p, q) =

2−n |fn (p) − fn (q)| (∗)

n=1

❚❛ ❝â
i✮ d(p, q) ≥ 0
✷✺