Không gian các dãy nhận giá trị trong không gian lồi địa phương xác định bởi hàm Orlicz

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍
✲ ✲ ✲ ✲ ✲ ✲

✲ ✲ ✲ ✲ ✲ ✲

❚❘❺◆ ❚❍➚ ❍➀◆●

❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚
❚❘❖◆● ❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❳⑩❈
✣➚◆❍ ❇Ð■ ❍⑨▼ ❖❘▲■❈❩✳
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣❤➺ ❆♥ ✲ ✷✵✶✺


❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍
✲ ✲ ✲ ✲ ✲ ✲

✲ ✲ ✲ ✲ ✲ ✲

❚❘❺◆ ❚❍➚ ❍➀◆●

❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚
❚❘❖◆● ❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❳⑩❈
✣➚◆❍ ❇Ð■ ❍⑨▼ ❖❘▲■❈❩✳
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

❚❙✳ ❑■➋❯ P❍×❒◆● ❈❍■

◆❣❤➺ ❆♥ ✲ ✷✵✶✺


✶

▼Ö❈ ▲Ö❈
▼ö❝ ❧ö❝
✶
▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✶ ❑❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥
❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
✺
✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣

✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✶✺

✷ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛

♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ❖r❧✐❝③
✶✽
✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ❖r❧✐❝③

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ lM (E)✳ ✳ ✷✽

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼




é

r t ợ ổ t t õ trỏ
q trồ ợ ổ ổ ờ ữủ
t ợ tr tr trữớ ổ ữợ t t ừ ổ
ỳ ử ừ t ờ
r sỷ ử ỵ tữ ừ r t strss
rr ỹ ổ t t
tr ổ ữợ tứ ợ tỹ t ú ữủ ồ
r t t ừ ổ r ụ ữủ
ự s s tổ q trú ừ r strss
rr r ỹ tr ởt số t q ừ strss

rr ổ r tr ổ ữợ t
ỹ ợ ổ tr tr ổ
t ữủ ởt số t t ừ ú ử ừ
ỹ ổ tr tr ổ ỗ
ữỡ r ú tổ ỹ ồ t

ổ tr tr ổ ỗ ữỡ
r
ở ừ tr ởt số t q t ổ
ỗ ữỡ ỹ ổ tr tr ổ
ỗ ữỡ r ữ r ởt số t
t ừ ú ở ừ ữủ tr tr
ữỡ




ổ ỗ ữỡ ổ
tr tr ổ ỗ ữỡ
ữỡ

ữỡ tr ỳ tự ỡ s ũ s
t ỳ t q ổ ỗ ữỡ ởt số
ợ ổ tr tr ổ ỗ ữỡ

ổ tr tr ổ ỗ
ữỡ r
ữỡ

ở ữỡ ú tổ t ữỡ ỹ

trú tổổ ỗ ữỡ ổ tr tr ổ
ỗ ữỡ r ỹ tr ỵ tữ tỹ
tr trữớ ủ tr ổ ữợ tr tr
ổ tr r ú tổ ự
ởt số t t t ởt số ố q ỳ ợ ổ ợ
ỹ ợ ởt số ổ tr tr ổ ỗ
ữỡ õ
ữủ t t trữớ ồ ữợ sỹ ữợ
ừ Pữỡ tọ ỏ t
ỡ s s t t t t ỡ
ừ ữ ồ Pỏ
ồ q ổ tr tờ t ữ ồrữớ
ồ ú ù tr sốt q tr ồ t t
ố ũ ỷ ớ ỡ tợ ỗ
t ồ ồ õ t t rữớ ồ
t t ủ ú t t ử tr
sốt q tr ồ t ũ õ rt ố ữ
ỹ ỏ ổ t tr ọ ỳ t sõt
rt ữủ ỳ ớ qỵ ừ t ổ
ỳ õ ỵ ừ ồ ữủ t ỡ

t


✹

❚r➛♥ ❚❤à ❍➡♥❣


✺

❈❍×❒◆● ✶

❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❱⑨ ❑❍➷◆● ●■❆◆
❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▲➬■
✣➚❆ P❍×❒◆●
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ ❞ò♥❣ ✈➲ s❛✉✱ ✤➦❝
❜✐➺t ❧➔ ♥❤ú♥❣ ❦➳t q✉↔ ❝➠♥ ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈➔ ♠ët sè
❧î♣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣✳

✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ❣✐↔✐ t➼❝❤ ❝ê ✤✐➸♥ ✈➔
❣✐↔✐ t➼❝❤ ❤➔♠ ❝➛♥ ❞ò♥❣ ✈➲ s❛✉✳ ❙❛✉ ✤➙②✱ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✈➲ ❤➔♠
❧ç✐✳ ❈→❝ ❦➳t q✉↔ s❛✉ ❝â t❤➸ t➻♠ t❤➜② ð tr♦♥❣ ❬✶❪ ✈➔ ❬✸❪✳

✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ❤➔♠ t❤ü❝ f : (a, b) → R✳ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐
♥➳✉

f λx + (1 − λ)y
✈î✐ ♠å✐ x, y ∈ (a, b) ✈➔ 0

λ

λf (x) + (1 − λ)f (y)

✭✶✳✶✮

1✳

✶✳✶✳✷ ◆❤➟♥ ①➨t✳ ✣✐➲✉ ❦✐➺♥ ✭✶✳✶✮ t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ✤✐➲✉ ❦✐➺♥ s❛✉✿

f (t) − f (s)
t−s

f (u) − f (t)
u−t

✭✶✳✷✮

✈î✐ ♠å✐ a < s < t < u < b✳
✶✳✶✳✸ ▼➺♥❤ ✤➲✳ ❈❤♦ f : (a, b) → R ❧➔ ❤➔♠ ❧ç✐ ✈➔ c ∈ (a, b)✳ ❑❤✐ ✤â✱
− f (c)
❤➔♠ p : (a, b) \ {c} → R ①→❝ ✤à♥❤ ❜ð✐ p(x) = f (x)x −
❧➔ ❦❤æ♥❣
c
❣✐↔♠✳


✻

◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ ✈î✐ ♠å✐ c ∈ (a, b) ❤➔♠ p ❦❤æ♥❣ ❣✐↔♠ t❤➻ f ❧➔ ❤➔♠ ❧ç✐✳

✶✳✶✳✹ ❍➺ q✉↔✳ ●✐↔ sû f ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ (a, b)✳ ❑❤✐ ✤â✱ f ❧➔ ❧ç✐ ❦❤✐
✈➔ ❝❤➾ ❦❤✐ f ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ (a, b)✳

✶✳✶✳✺ ❍➺ q✉↔✳ ◆➳✉ f
f

❝â ✤↕♦ ❤➔♠ ❝➜♣ ✷ tr➯♥
(x) > 0 ✈î✐ ♠å✐ x ∈ (a, b) t❤➻ f ❧➔ ❤➔♠ ❧ç✐✳
: (a, b) → R

(a, b)

✈➔

✶✳✶✳✻ ❱➼ ❞ö✳ ❚ø ❤➺ q✉↔ tr➯♥ t❛ t❤➜② ❤➔♠ f (x) = ex ❧ç✐ tr➯♥ R ✈➔ y = xp
❧➔ ❝→❝ ❤➔♠ ❧ç✐ tr➯♥ (0, ∞) ✈î✐ p

1✳

✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✼❪✮ ❍➔♠ M : [0, +∞) → R ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❖r❧✐❝③
♥➳✉
✶✮ M ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠✱ ❧✐➯♥ tö❝❀
✷✮ M (0) = 0 ✈➔ lim M (t) = ∞❀
t→∞

✸✮ M ❧➔ ❤➔♠ ❧ç✐✳

✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ❖r❧✐❝③ M ❣å✐ ❧➔ s✉② ❜✐➳♥ ♥➳✉ tç♥ t↕✐ t > 0 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 ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

❝❤✉➞♥ tr➯♥ E ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉

❦✐➺♥ s❛✉✿
✶✮ x

0✱ ✈î✐ ♠å✐ x ∈ E ✈➔ x = 0 ⇔ x = 0❀

✷✮ λx = |λ| x ✱ ✈î✐ ♠å✐ λ ∈ K ✈➔ ✈î✐ ♠å✐ x ∈ E ❀
✸✮ x + y

x + y , ✈î✐ ♠å✐ x, y ∈ E ✳

❑❤✐ ✤â (E, . ) ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳
❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈î✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥

d(x, y) = x−y , ∀x, y ∈ E ✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣

❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ E ✤➛② ✤õ ✈î✐ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥✳ ❱î✐ tæ♣æ s✐♥❤ ❜ð✐


✼

♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ✤â✱ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♥❤➙♥ ✈æ ❤÷î♥❣ tr➯♥ E
❧➔ ❧✐➯♥ tö❝✳
❈❤♦ E, F ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑þ ❤✐➺✉ L(E, F ) ❧➔ t➟♣ ❤ñ♣
❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tø E ✈➔♦ F ✳ ❚❛ ✤➣ ❜✐➳t L(E, F ) ❧➔ ❦❤æ♥❣
❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈î✐ ❝❤✉➞♥

f = sup

f (x) , ∀f ∈ L(E, F ).

x =1

◆➳✉ F ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤➻ L(E, F ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣➦❝ ❜✐➺t✱

L(E, K) := E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧✐➯♥ ❤ñ♣ t❤ù ♥❤➜t ❝õ❛ E ❝ô♥❣ ❧➔ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤✳
❈→❝ ❧î♣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ q✉❡♥ t❤✉ë❝ s❛✉ ✤÷ñ❝ q✉❛♥ t➙♠ ♥❤✐➲✉
tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝õ❛ ❝❤ó♥❣ tæ✐✳

✶✳✶✳✶✶ ❱➼ ❞ö✳ ●✐↔ sû K ❧➔ tr÷í♥❣ ❝→❝ sè t❤ü❝ ❤♦➦❝ ❝→❝ sè ♣❤ù❝✳ ❑þ ❤✐➺✉
l∞ = x = (xn ) ⊂ K : (xn ) ❧➔ ❞➣② ❜à ❝❤➦♥ ;
C = x = (xn ) ⊂ K : (xn ) ❧➔ ❞➣② ❤ë✐ tö ;
C0 = x = (xn ) ⊂ K : lim xn = 0 ;
n→∞

✈➔

∞

|xn |p < ∞ , p

lp = x = (xn ) ⊂ K :

1.

n=1

❱î✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ❝→❝ ❞➣② ✈➔ ♥❤➙♥ ♠ët sè ✈î✐ ♠ët ❞➣② t❤æ♥❣ t❤÷í♥❣
t❛ ❝â l∞ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈➔ C ✱ C0 ✈➔ lp ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥
❝õ❛ l∞ ✳ ❍ì♥ ♥ú❛

lp ⊂ C0 ⊂ C ⊂ l∞ .

❚❛ ✤➣ ❜✐➳t l∞ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐

x = sup |xn |, ∀x ∈ l∞ .
n 1

✭✶✳✸✮


✽

✣➦❝ ❜✐➺t C0 , C ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ l∞ ✱ ✈➻ t❤➳ ❝❤ó♥❣ ❝ô♥❣ ❧➔
❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥ tr➯♥✳ ❚✉② ♥❤✐➯♥ lp ❦❤æ♥❣ ✤â♥❣ tr♦♥❣

l∞ ✳
✣è✐ ✈î✐ lp ✱ ♥❣÷í✐ t❛ ①➨t ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
∞

x

p

|xn |p

=

1/p

, ∀x ∈ lp .

✭✶✳✹✮

n=1

❑❤✐ ✤â✱ lp ❝ô♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
▼ö❝ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣✳ ❈→❝ ❦➳t q✉↔ ❝➠♥ ❜↔♥ ✤÷ñ❝ tê♥❣ ❤ñ♣ ✈➔ tr➼❝❤ r❛ tø
❬✸❪✳

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝ò♥❣
✈î✐ ♠ët tæ♣æ tr➯♥ ✤â s❛♦ ❝❤♦ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♥❤➙♥ ✈æ ❤÷î♥❣ ❧➔
❧✐➯♥ tö❝✭ ❤♦➦❝ ❝ô♥❣ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ❧➔✿ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì X ✤÷ñ❝ ❣å✐ ❧➔

❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ♥➳✉ tr➯♥ ✤â ✤➣ ❝❤♦ ♠ët tæ♣æ t÷ì♥❣ t❤➼❝❤ ✈î✐ ❝➜✉
tró❝ ✤↕✐ sè tr➯♥ ❳ s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ tr➯♥ X ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣✮✳

✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû ❆ ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ X ✳
❛✮ ❚➟♣ A ✤÷ñ❝ ❣å✐ ❧➔

❧ç✐ ♥➳✉ ✈î✐ ♠å✐ x, y ∈ A ✈➔ ✈î✐ ♠å✐ t ∈ [0; 1]✱ t❛

❝â t.x + (1 − t).y ∈ A❀

❝➙♥ ♥➳✉ αA ⊂ A ✈î✐ ♠å✐ α ∈ K ✈➔ |α| < 1❀
❝✮ ❚➟♣ A ✤÷ñ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ ♥➳✉ ✈î✐ ♠é✐ ❧➙♥ ❝➟♥ V ❝õ❛ 0 tç♥ t↕✐ sè

❜✮ ❚➟♣ ❝♦♥ A ✤÷ñ❝ ❣å✐ ❧➔

s > 0 s❛♦ ❝❤♦ A ⊂ tV ✈î✐ ♠å✐ t > s✳

✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ ✈➨❝tì tæ♣æ ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
♥➳✉ ♥â ❝ì sð ❧➙♥ ❝➟♥ U ❝õ❛ 0 ❣ç♠ ❝→❝ t➟♣ ❧ç✐✳

✶✳✷✳✹ ▼➺♥❤ ✤➲✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣✳ ❑❤✐ ✤â 0 ∈ X

❝â ❝ì sð ❧➙♥ ❝➟♥ U t❤♦↔ ♠➣♥✿




U, V U t õ W U s W U V
U U ợ ồ K, = 0 ợ ồ U U;
ồ U U ỗ út
ỡ ỳ ổ t t tổổ X õ ồ t U t
t õ ổ ỗ ữỡ

t ộ ổ (E,

. ) ổ ỗ

ữỡ s t ỗ ừ

1
Bn = {x E : x < }, n = 1, 2, ...
n

X ổ tỡ tổổ
X ữủ ồ ổ

tr X õ ởt

s tr s s r tổổ tr X
X ữủ ồ

ữỡ tỗ t ừ 0 t


ữớ t ự ữủ t q s

ỵ ổ tỡ tổổ
õ ỗ ữỡ ữỡ

s r sỹ tỗ t tổổ ỗ ữỡ tứ ồ t ỗ
út

ổ tỡ E õ ồ U ỗ t ỗ

út t tr E tỗ t tổổ t s t ở
ổ ữợ tr E tử E tr t ổ ỗ ữỡ
ỡ ỳ ỡ s ừ 0 tr E ồ t
n

Vi , > 0, Vi U, 1

U =
i=1

i

n.


✶✵

✶✳✷✳✾ ▼➺♥❤ ✤➲✳ ◆➳✉ tæ♣æ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ T tr➯♥ X ♥❤➟♥ U ❧➔♠ ❝ì sð

❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ 0 ∈ X t❤➻ tæ♣æ ♥➔② ❧➔ ❍❛✉s❞♦r❢❢ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
εU = 0.
U ∈U;ε>0

❙❛✉ ✤➙② t❛ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦➳t q✉↔ ❝èt ②➳✉ ✈➲ sü ①→❝ ✤à♥❤ ❝õ❛ tæ♣æ
❧ç✐ ✤à❛ ♣❤÷ì♥❣ t❤æ♥❣ q✉❛ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥✳ ✣➛✉ t✐➯♥ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐
♥✐➺♠ ♥û❛ ❝❤✉➞♥✳

✶✳✷✳✶✵ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì✳ ❍➔♠ p ①→❝ ✤à♥❤
tr➯♥ X ✈➔ ♥❤➟♥ ❣✐→ trà t❤ü❝ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

♥û❛ ❝❤✉➞♥ tr➯♥ X ♥➳✉ ✈î✐

♠å✐ x, y ∈ X ✈➔ ✈î✐ ♠å✐ λ ∈ K t❛ ❝â

N1 ✮ p(x)

0;

N2 ✮ p(x + y)

p(x) + p(y);

N3 ✮ p(λx) = |λ|p(x).
◆û❛ ❝❤✉➞♥ p tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì X ❧➔ ❝❤✉➞♥ tr➯♥ X ♥➳✉ p(x) = 0
s✉② r❛ x = 0✳ ◆➳✉ p ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ X ✈➔ x ∈ X t❤➻ sè p(x) t❤÷í♥❣
✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ ||x||.

✶✳✷✳✶✶ ▼➺♥❤ ✤➲✳ ◆➳✉ p ❧➔ ♠ët ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì X t❤➻
✈î✐ ♠å✐ α > 0 ❝→❝ t➟♣ A = {x ∈ E : p(x) < α} ✈➔ B = {x ∈ E : p(x)
α} ❧➔ ❧ç✐✱ ❝➙♥ ✈➔ ❤ót✳

✶✳✷✳✶✷ ◆❤➟♥ ①➨t✳ ●✐↔ sû P ❧➔ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì
X ✳ ❑❤✐ ✤â✱ ❦➳t ❤ñ♣ ❝→❝ ▼➺♥❤ ✤➲ ✶✳✷✳✽ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✶✶ t❛ ❝â✿ ❚r➯♥ X
tç♥ t↕✐ ♠ët tæ♣æ ②➳✉ ♥❤➜t s❛♦ ❝❤♦ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ E ✈➔ ❝→❝ p ∈ P
❧✐➯♥ tö❝✳ ❍ì♥ ♥ú❛✱ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❝ì sð ❧➙♥ ❝➟♥ t↕✐ 0
❧➔ ❤å ❝→❝ t➟♣ ❧ç✐ ❝â ❞↕♥❣

U = {x ∈ E : sup pi (x) < ε, i = 1, 2..., n},
tr♦♥❣ ✤â ε > 0✱ pi ∈ P ✱ n ∈ N✳




sỷ A t ỗ út ừ ổ tỡ
tổổ X tỹ ổ àA : X R+

àA (x) = inf{t > 0 : x tA} ợ ồ x X
ữủ ồ

s ừ t ủ A

ỵ A t ỗ út ừ ổ tỡ tổổ

X

t àA := p ỷ tr X ỡ ỳ

{x X : p(x) < 1} A {x X : p(x)

1}.

t X ổ ỗ ữỡ t X õ ỡ s
t ỗ út õ ỡ s tữỡ ự ợ
ồ ỷ s tữỡ ự t ủ ợ
t s r r ộ tổổ ỗ ữỡ t ữủ
ởt ồ ỷ ữủ

t X ỗ ữỡ t ồ
ỷ P õ t ồ ỗ ởt tỷ ởt

t sỷ P ồ ỷ s r tổổ ỗ
ữỡ tr E õ E sr p(x) = 0 ợ ồ

p P t x = 0

ỵ E ổ sr ỗ ữỡ E

ữủ ồ ữủ ỷ t E tr tự
tr E tỗ t ởt tr s r tổổ trũ ợ tổổ ỗ ữỡ
ừ õ
ự sỷ {pn} ồ ỷ s r tổổ ỗ ữỡ
tr E ợ ộ x, y E t t

d(x, y) =
n=1

1 pn (x y)
.
2n 1 + pn (x y)


✶✷

❑❤✐ ✤â✱ rã r➔♥❣ d(x, y) ①→❝ ✤à♥❤ ✈➔ ❤ì♥ ♥ú❛ d ❧➔ ♠➯tr✐❝ tr➯♥ E ✳ ❚❛ ❝❤ù♥❣
♠✐♥❤ tæ♣æ s✐♥❤ ❜ð✐ d trò♥❣ ✈î✐ tæ♣æ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ s✐♥❤ ❜ð✐ {pn }✳
❱î✐ ε > 0 t❛ ①➨t

Bd (0, ε) = {x ∈ E : d(x, 0) < ε}
❧➔ ❤➻♥❤ ❝➛✉ tr♦♥❣ tæ♣æ ❞♦ ♠➯tr✐❝ d s✐♥❤ r❛✳ ❈❤å♥ n0 ✤õ ❧î♥ s❛♦ ❝❤♦
1
ε
<
.
2n
2
n>n0

❱î✐ U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ 0 tr♦♥❣ tæ♣æ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐
ε
U = {x ∈ E : pi (x) < , 1 i n0 }.
2
❑❤✐ ✤â✱ t❛ ❝â U ⊂ B(0, ε)✳ ❚❤➟t ✈➟②✱ ♥➳✉ x ∈ U t❤➻

n0

d(x, 0) =
i=1
n0

<
i=1
n0

1 pi (x)
+
2i 1 + pi (x)
1
pi (x) +
2i

n>n0

n>n0

1 pn (x)
2n 1 + pn (x)

1
2n

1ε ε
+
2i 2 2

i=1
ε ε
< + .
2 2

<

❉♦ ✤â x ∈ B(0, ε)
◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ t❛ ❧➜②

V = {x ∈ E : pi (x) < ε, i ∈ I, I ❤ú✉ ❤↕♥ }
❧➔ ❧➙♥ ❝➟♥ ❝õ❛ 0 ∈ E tr♦♥❣ tæ♣æ s✐♥❤ ❜ð✐ ❤å {pn }✳ ▲➜② ε1 > 0 s❛♦ ❝❤♦
ε
2i ε 1
, i ∈ I.
1+ε
❑❤✐ ✤â Bd (0, ε1 ) ⊂ V. ❚❤➟t ✈➟②✱ ❣✐↔ sû ❝â x ∈ Bd (0, ε1 ) ✈➔ i ∈ I s❛♦ ❝❤♦

pi (x)

ε. ❑❤✐ ✤â
ε1 > d(x, 0)
i∈I

1 pi (x)
2i 1 + pi (x)

i∈I

1 ε

2i 1 + ε

1 i
2 ε1 = ε1 .
2i




õ ổ r tỗ t tr d s r tổổ trũ ợ tổổ
ỗ ữỡ ữủ ồ ữủ ỷ E
tr
ữ ỵ tr t õ t s

t sỷ E ổ sr ỗ ữỡ E
ữủ ồ ữủ ỷ pn õ tổổ tr E
s tr



d(x, y) =
n=1

1 pn (x y)
.
2n 1 + pn (x y)

õ
(xk ) E ở tử tợ x E pn (xk x)

0 k ợ ồ n
(xk ) E pn (xk xl ) 0 k, l
ợ ồ n
ổ ỗ ữỡ tr ồ F ổ



ổ rt
sỷ E ổ ỗ ữỡ ồ
ồ ỷ P = {pa : I} õ t A E
õ ợ ộ p tự ợ ộ I p(x) < r < ợ
ồ x A
õ ừ t ồ

ởt số ử ổ rt

ử sỷ
R := {x = {xn } : xn R, n

1}

ợ ở ổ ữợ tổ tữớ t tứ số t
ồ Q = {pn } ồ ữủ ỷ tr R

Pn (x) = |xn |; x = {xn }, n = 1, 2, ...


✶✹

❑❤✐ ✤â R∞ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣✳ ❉♦ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ ❧➔ ✤➳♠

✤÷ñ❝ ♥➯♥ R∞ ❝á♥ ❦❤↔ ♠➯tr✐❝ ✈î✐ ♠➯tr✐❝✿
∞

d(x, y) =
n=1

1 |xn − yn |
2n 1 + |xn − yn |

✈î✐ ♠å✐ x, y ∈ R∞ ✳ ❚✉② ♥❤✐➯♥✱ R∞ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❜à ❝❤➦♥ ✤à❛
♣❤÷ì♥❣✳ ❚❤➟t ✈➟②✱ ♥➳✉ ♥❣÷ñ❝ ❧↕✐ t❤➻ ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐
✤â✱ tç♥ t↕✐ ❝❤✉➞♥ tr➯♥ R∞ s❛♦ ❝❤♦ tæ♣æ s✐♥❤ r❛ ❜ð✐ ❝❤✉➞♥ trò♥❣ ✈î✐ tæ♣æ
s✐♥❤ r❛ ❜ð✐ {pn }✳ ❳➨t B(0, 1) = {x ∈ R∞ : x < 1}✳ ❑❤✐ ✤â✱ tç♥ t↕✐

V = {x ∈ R∞ : pi (x) = |xi | < δ, i ∈ I}
tr♦♥❣ ✤â I ❧➔ t➟♣ ❤ú✉ ❤↕♥ s❛♦ ❝❤♦ V ⊂ B(0, 1)✳ ▲➜② x0 = {x0n } ∈ R∞

/ I ✳ ❑❤✐ ✤â✱ x0 = 0 ✈➔ s✉② r❛
s❛♦ ❝❤♦ x0n = 0 ♥➳✉ n ∈ I ✈➔ x0n = 0 ✈î✐ n ∈
x0 = r > 0✳ ❱î✐ ♠å✐ sè tü ♥❤✐➯♥ k ❞♦ ❝→❝❤ ①→❝ ✤à♥❤ ❝õ❛ x0 ✈➔ V t❛ ❝â
kx0 ∈ V ✳ ❉♦ ✤â kx0 ∈ B(0, 1) ✈î✐ ♠å✐ k ✳ ❙✉② r❛ kx0 = kr < 1 ✈î✐ ♠å✐
k ✳ ❚❛ ♥❤➟♥ ✤÷ñ❝ sü ♠➙✉ t❤✉➝♥✳

✶✳✷✳✷✷ ❱➼ ❞ö✳ ●å✐ C(R) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tö❝ tr➯♥
R✳ ❱î✐ ♠é✐ n = 1, 2, ... ✤➦t
pn (f ) = sup{|f (x)| : x ∈ [−n, n]},
✈î✐ ♠å✐ f ∈ C(R)✳ ❑❤✐ ✤â✱ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷ñ❝ pn ❧➔ ❝→❝ ♥û❛ ❝❤✉➞♥
tr➯♥ C(R)✳ ❉♦ ✤â✱ C(R) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ s✐♥❤ ❜ð✐ ❤å ❝→❝
♥û❛ ❝❤✉➞♥ {pn }✳ ❍ì♥ ♥û❛✱ C(R) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❋r❡❝❤❡t ✈î✐ ❦❤♦↔♥❣ ❝→❝❤
∞

d(f, g) =
n=1

1 pn (f − g)
,
2n 1 + pn (f − g)

✈î✐ ♠å✐ f, g ∈ C(R)✳

✶✳✷✳✷✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû E, F ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ✳ ❚❛ ❣å✐
E ✤➥♥❣ ❝➜✉ ✈î✐ F ♥➳✉ tç♥ t↕✐ →♥❤ ①↕ ϕ : E → F ❧➔ ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤
✈➔ ϕ, ϕ−1 ❧➔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝✳




õ ữủ t q s

ỵ E F ổ sr ỗ ữỡ

s ữủt ồ ỷ P = {p : I} Q = {q : I}
sỷ f : E F ởt t t tọ
r p (x)

q (f (x))

k p (x)

ợ ộ I ợ ộ x E õ f tứ E F

ổ tr tr ổ ỗ
ữỡ
ử tr ởt số t q ổ tr
tr ổ ỗ ữỡ t q ỡ ữủ t
ự tr
sỷ E ổ sr ỗ ữỡ tr trữớ K E
ữủ s ồ ỷ P = (p ), ; I ỵ

l (E) = x = (xn ) E : (p (xn )) : số ợ ồ ;
C(E) = x = (xn ) E : (xn ) ở tử ;
C0 (E) = x = (xn ) E : lim xn = 0 ;
n




(p (xn ))q < , ợ ồ I , q

lq (E) = x = (xn ) E :

1.

n=1

ợ t ở ởt số ợ ởt tổ tữớ
t õ l (E) ổ t t C(E) C0 (E) lp (E)
ổ ừ l (E) ỡ ỳ

lq (E) C0 (E) C(E) l (E).

✶✻

✶✳✸✳✶ ✣à♥❤ ❧þ✳ l∞(E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈î✐ ❤å ❝→❝
♥û❛ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐

✭✶✳✺✮

bα (x) = sup pα (xn )
n 1

✈î✐ ♠å✐ x = (xn) ∈ l∞(E) ✈➔ ✈î✐ ♠é✐ α ∈ I ✳

✶✳✸✳✷ ◆❤➟♥ ①➨t✳ ✶✮ ❱➻ C0(E), C(E) ✈➔ lp(E) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥
t➼♥❤ ❝♦♥ ❝õ❛ l∞ (E) ♥➯♥ ❝❤ó♥❣ ❝ô♥❣ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ❦❤✐ E
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈î✐ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ ①→❝ ✤à♥❤ tr♦♥❣ ✭✶✳✺✮✳
✷✮ ◆➳✉ E ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❤å ✤➳♠ ✤÷ñ❝ ❝❤✉➞♥ t❤➻ ❝→❝ ❦❤æ♥❣ ❣✐❛♥

C0 (E), C(E) ✈➔ lp (E) ❝ô♥❣ ✈➟②✳ ❉♦ ✤â✱ ♥➳✉ E ❧➔ F ✲❦❤æ♥❣ ❣✐❛♥ t❤➻
C0 (E), C(E) ❝ô♥❣ ❧➔ ❝→❝ F ✲❦❤æ♥❣ ❣✐❛♥✳
◆❣♦➔✐ r❛✱ tr➯♥ lq (E) ✈î✐ q

1 t❛ ❝á♥ ❝â ❦➳t q✉↔ s❛✉✿

✶✳✸✳✸ ✣à♥❤ ❧þ✳ lq (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✈î✐ ❤å ❝→❝
♥û❛ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐

∞
1

pqα (xn )] q

cα (x) = [

✭✶✳✻✮

n=1

✈î✐ ♠å✐ x = (xn) ∈ lq (E) ✈➔ α ∈ I ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ E ❧➔ F ✲❦❤æ♥❣ ❣✐❛♥ t❤➻
lq (E) ✈î✐ q 1 ❝ô♥❣ ❧➔ ❝→❝ F ✲❦❤æ♥❣ ❣✐❛♥✳
◆➳✉ (E, . ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤➻ ❝→❝ ❦❤æ♥❣ ❣✐❛♥

l∞ (E) = x = (xn ) ⊂ E : ( xn ) : ❧➔ ❞➣② sè ❜à ❝❤➦♥ ;
C(E) = x = (xn ) ⊂ E : (xn ) ❤ë✐ tö ;
C0 (E) = x = (xn ) ⊂ E : lim xn = 0 ;
n→∞

trð t❤➔♥❤ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈î✐ ❝❤✉➞♥

x = sup xn ,
n 1

✭✶✳✼✮


✶✼

✈➔

∞

lq (E) = x = (xn ) ⊂ E :

xn

q

< ∞ ,q

1

n=1

❝ô♥❣ trð t❤➔♥❤ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥
∞

x

q

=

xn

q

1/q

, ∀x ∈ lq (E).

✭✶✳✽✮

n=1

◆➳✉ E = K t❤➻ t❛ ♥❤➟♥ ✤÷ñ❝ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤➣ tr➻♥❤ ❜➔② ð ❱➼ ❞ö ✶✳✶✳✶✶✳


✶✽

❈❍×❒◆● ✷

❑❍➷◆● ●■❆◆ ❈⑩❈ ❉❶❨ ◆❍❾◆ ●■⑩ ❚❘➚ ❚❘❖◆●
❑❍➷◆● ●■❆◆ ▲➬■ ✣➚❆ P❍×❒◆● ❳⑩❈ ✣➚◆❍ ❇Ð■ ❍⑨▼
❖❘▲■❈❩
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ①➙② ❞ü♥❣ ❝➜✉ tró❝ tæ♣æ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣
①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❤➔♠ ❖r❧✐❝③ ❞ü❛ tr➯♥ þ t÷ð♥❣ ✤➣ t❤ü❝ ❤✐➺♥ tr♦♥❣ tr÷í♥❣
❤ñ♣ ❞➣② ♥❤➟♥ ❣✐→ trà ✈æ ❤÷î♥❣ ✈➔ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
✤à♥❤ ❝❤✉➞♥ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ❬✼❪ ✈➔ ❬✺❪✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tæ✐ ❝❤ù♥❣ ♠✐♥❤
♠ët sè t➼♥❤ ❝❤➜t✱ ①➨t ♠ët sè ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❧î♣ ❦❤æ♥❣ ❣✐❛♥ ♠î✐ ①➙②
❞ü♥❣ ✈î✐ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛
♣❤÷ì♥❣ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ▼ö❝ ✶✳✷✳

✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛
♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ❖r❧✐❝③
❈❤♦ M ❧➔ ❤➔♠ ❖r❧✐❝③✳ ❑þ ❤✐➺✉✿
∞

lM = {x = (xn ) ⊂ C :

M
n=1

|xn |
) < ∞; ✈î✐ ρ > 0 ♥➔♦ ✤â}
ρ

❑❤✐ ✤â✱ lM ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈î✐ ❝→❝ ♣❤➨♣ t♦→♥✿
❈ë♥❣ x + y = (xn + yn ) ✈î✐ ♠å✐ x = (xn ), y = (yn );
◆❤➙♥ ✈æ ❤÷î♥❣ α.x = (α.xn ) ✈î✐ ♠å✐ x = (xn ), α ∈ C✳ ❍ì♥ ♥ú❛✱ lM
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
∞

x = inf ρ > 0 :

M
n=1

|xn |
ρ

1 ,




ợ ồ x lM . ổ lM

ổ r ờ ỵ

hM = x = (xn ) C :

M
n=1

|xn |
< ợ ồ > 0 .


õ hM ổ õ ừ lM . ỳ t q tr ữủ
tr tr r ỹ t q t ổ

lM t ỹ ợ ổ r tr tr
ổ ự ởt số t t ừ ú r
ử ú tổ ự ữỡ ỹ ợ ổ
tr tr ổ ỗ ữỡ
r
sỷ M r E ởt ổ ỗ ữỡ
ồ ỷ P = {p : I} ỵ


lM (E) = (xn ) E :

p (xn )

M
n=1

< , p P > 0 õ

ờ s ũ tr ởt số ừ ự
t q s

ờ q(x) = 0 t


M
n=1

p (xn )
q (x)



1.

ự ợ ồ > 0 tứ ừ q(x) tỗ t > 0 s


q (x) +


M
n=1

p (xn )


1.

t ổ ừ M t s r

n=1

p (xn )
M
q (x) +



M
n=1

p (xn )


1.


✷✵

❈❤♦ ε → 0 t❛ ♥❤➟♥ ✤÷ñ❝
∞

M

n=1

pα (xn )
qα (x)

1.

✷✳✶✳✷ ❇ê ✤➲✳ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ t❤➻ lM (E) ⊂ l∞(E)✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû lM (E)

l∞ (E)✳ ❑❤✐ ✤â tç♥ t↕✐ x = (xn ) ∈ lM (E)

❦❤æ♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ tç♥ t↕✐ pα s❛♦ ❝❤♦ x = (xn ) ❦❤æ♥❣ ❜à ❝❤➦♥ t❤❡♦

pα ✳ ❉♦ ✤â✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t pα (xn ) > n ✈î✐ ♠å✐ n✳ ❱➻ x ∈ lM (E) ♥➯♥
tç♥ t↕✐ ρα > 0 s❛♦ ❝❤♦
∞

M
n=1

pα (xn )
< ∞.
ρα

❙✉② r❛ tç♥ t↕✐ k s❛♦ ❝❤♦

M

pα (xn )

ρα

✈î✐ ♠å✐ n✳ ❉♦ M ❧✐➯♥ tö❝ tr➯♥ [0, ∞)✱ M (0) = 0 ✈➔ lim M (t) = ∞ ♥➯♥
t→∞

tç♥ t↕✐ t0 ∈ (0, ∞) s❛♦ ❝❤♦ M (t0 ) = k ✳ ❱➻ lim pα (xn ) = ∞ ♥➯♥ tç♥ t↕✐
n→∞

pα (xn0 )
n0 s❛♦ ❝❤♦
> t0 ✳ ❑➨♦ t❤❡♦
ρα
M

pα (xn0 )
ρα

M (t0 ) = k.

✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐

M

pα (xn )
ρα

✈î✐ ♠å✐ n✳ ❱➟② lM (E) ⊂ l∞ (E)✳

✷✳✶✳✸ ✣à♥❤ ❧þ✳ lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈î✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣

❝→❝ ❞➣② ✈➔ ♥❤➙♥ ♠ët sè ✈î✐ ♠ët ❞➣② t❤æ♥❣ t❤÷í♥❣✳


✷✶

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x = (xn), y = (yn) ∈ lM (E)✳ ❑❤✐ ✤â✱ ✈î✐ ♠é✐ α ∈ I
tç♥ t↕✐ ρ1α , ρ2α > 0 s❛♦ ❝❤♦
∞
n=1

pα (xn )
M
< ∞,
ρ1α

∞

M
n=1

pα (yn )
< ∞.
ρ2α

▲➜② ρα = ρ1α + ρ2α t❛ ❝â

pα (xn + yn )
ρα

M

pα (xn + yn )
ρ1α + ρ2α
pα (xn ) + pα (yn )
≤M
ρ1α + ρ2α
ρ1α pα (xn )
ρ1α pα (xn )
=M
+ 1
ρ1α + ρ2α ρ1α
ρα + ρ2α ρ2α
pα (xn )
pα (yn )
ρ1α
ρ2α
M
M
≤ 1
+
.
ρα + ρ2α
ρ1α
ρ1α + ρ2α
ρ2α
=M

❙✉② r❛

∞

M
n=1

pα (xn + yn )
ρα

ρ1α
ρ1α + ρ2α

∞

M
n=1

pα (xn )
ρ2
+
ρ1α
ρ1α + ρ2α

∞

M
n=1

pα (yn )
ρ2α

< ∞,

tù❝ ❧➔ x + y ∈ lM (E)✳
◆➳✉ λ = 0 t❤➻ λx = (0, 0, ..., 0, ...) ∈ lM (E)✳ ◆➳✉ λ = 0 t❤➻ ✈î✐

ρα = |λ|ρ1α t❛ ❝â
∞

M
n=1

pα (λxn )
ρα

∞

=

M
n=1

|λ| .pα (xn )
ρα

∞

=

M
n=1

pα (xn )
ρ1α

❙✉② r❛ λx ∈ lM (E)✳ ❱➻ ✈➟② lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳

✣à♥❤ ❧þ s❛✉ ✤➙② tr❛♥❣ ❜à tæ♣æ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ❝❤♦ lM (E)✳

< ∞.


✷✷

✷✳✶✳✹ ✣à♥❤ ❧þ✳ lM (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❤å ♥û❛
❝❤✉➞♥ Q = {qα : α ∈ I} ①→❝ ✤à♥❤ ❜ð✐

∞

qα (x) = inf ρα > 0 :

M
n=1

pα (xn )
ρα

✭✷✳✷✮

1

✈î✐ ♠å✐ x = (xn) ∈ lM (E) ✈➔ ✈î✐ ♠é✐ α ∈ I ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ E ❧➔ ❍❛✉s❞♦r❢❢
t❤➻ lM (E) ❧➔ ❍❛✉s❞♦r❢❢✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠é✐ qα ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ lM (E) ✈î✐ ♠é✐ α✳
✐✮ ❚r÷î❝ ❤➳t tø ✷✳✷ s✉② r❛ ✈î✐ ♠é✐ α ∈ I t❛ ❝â qα (x)

0 ✈î✐ ♠é✐

x ∈ lM (E)✳ ❍ì♥ ♥ú❛✱ ♥➳✉ x = 0 t❤➻
qα (0) = inf{ρα > 0} = 0.
✐✐✮ ❚✐➳♣ t❤❡♦ t❛ ❝❤➾ r❛ qα (λx) = |λ|qα (x) ✈î✐ ♠å✐ x ∈ lM (E) ✈➔ ✈î✐
♠å✐ λ ∈ K✳ ❚r÷í♥❣ ❤ñ♣ λ = 0 ❤♦➦❝ x = 0 ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ◆➳✉ λ = 0 ✈➔

x = 0 t❤➻
∞

qα (λx) = inf

ρα > 0 :

M
n=1
∞

= inf

ρα > 0 :

M
n=1

✣➦t ρα =

pα (λxn )
≤1
ρα
|λ| pα (xn )
ρα

≤1

ρα
. ❑❤✐ ✤â t❛ ❝â
|λ|
∞

qα (λx) = inf

ρα |λ| :

M

pα (xn )
ρα

≤1

M

pα (xn )
ρα

≤1

n=1
∞

= |λ| inf

ρα :
n=1

= |λ|qα (x).
✐✐✐✮ ❱î✐ ♠é✐ α ∈ I ✈➔ ✈î✐ x, y ∈ lM (E) t❛ ✤➦t
∞

u = qα (x) = inf

ρα :

M
n=1

pα (xn )
ρα

≤1

.

✷✸

✈➔

∞

v = qα (y) = inf

ρα :

M
n=1

❑❤✐ ✤â

∞
n=1

●✐↔ sû t, s ∈ R s❛♦ ❝❤♦ s
♥➯♥ t❛ ❝â

∞

≤ 1 ✈➔

n=1
∞

u ✈➔ t

n=1

≤1

pα (yn )
qα (y)

.

≤ 1.

v ✳ ❑❤✐ ✤â✱ ❞♦ M ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠
∞

≤

M

pα (xn )
qα (x)

1

M

pα (yn )
qα (y)

≤ 1.

n=1
∞

pα (xn )
t

M

M
n=1

pα (xn )
s

M
✈➔

∞

pα (xn )
qα (x)

M

pα (xn )
ρα

≤
n=1

▼➦t ❦❤→❝✱ ✈î✐ ♠é✐ n = 1, 2, ... t❛ ❝â

pα (xn ) + pα (yn )
s pα (xn )
t pα (yn )
=
+
.
t+s
s+t s
s+t t
❚ø ▼ ❧➔ ❤➔♠ ❧ç✐ s✉② r❛

M

pα (xn + yn )
s+t

pα (xn ) + pα (yn )
s+t
s
pα (xn )
t
M
+
M
s+t
s
s+t
s

t
≤
+
= 1.
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♥❣ ✈î✐ ♠å✐ s

pα (xn + yn )
ρα

qα (x) ✈➔ t

≤1

≤ s + t.

qα (y) ♥➯♥ s✉② r❛

✭✷✳✸✮