❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❍❷■ ❍⑨

✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❱⑨ ✣➮■ ◆●❼❯ ❈Õ❆
▼❐❚ ❙➮ ❑❍➷◆● ●■❆◆ ❍⑨▼
❈❤✉②➯♥ ♥❣➔♥❤✿

❚♦→♥ ●✐↔✐ t➼❝❤

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

❍⑨ ◆❐■✱ ✺✴✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❍❷■ ❍⑨

✣➚◆❍ ▲Þ ❍❆❍◆✲❇❆◆❆❈❍ ❱⑨ ✣➮■ ◆●❼❯ ❈Õ❆
▼❐❚ ❙➮ ❑❍➷◆● ●■❆◆ ❍⑨▼
❈❤✉②➯♥ ♥❣➔♥❤✿

❚♦→♥ ●✐↔✐ t➼❝❤

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

●✐↔♥❣ ✈✐➯♥ ❤÷î♥❣ ❞➝♥✿

❚❙✳❇Ò■ ❑■➊◆ ❈×❮◆●

❍⑨ ◆❐■✱ ✺✴✷✵✶✾



r q tr ự tỹ õ ợ sỹ ố ừ
t ụ ữ sỹ ữợ ú ù t t ừ t ổ
s t õ
tọ ỏ t ỡ t tợ t t t ổ ổ t
t rữớ ồ ữ P ở t ổ trỹ t
tr t ỳ tự qỵ ổ ụ ữ
ự tr tớ ứ q
t tọ ỏ t ỡ s s t t s ũ
ữớ ữớ t t ú ù ụ ữ ỳ
tự t t õ

t ỡ

ở t





▲❮■ ❈❆▼ ✣❖❆◆
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ ❇ò✐ ❑✐➯♥ ❈÷í♥❣ ❦❤â❛
❧✉➟♥ ❝õ❛ ❡♠ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❦❤æ♥❣ trò♥❣ ✈î✐ ❜➜t ❦➻ ✤➲ t➔✐ ♥➔♦ ❦❤→❝✱❝→❝ t❤æ♥❣

t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝ rã r➔♥❣✳
❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛
❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈î✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳

❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥

◆❣✉②➵♥ ❍↔✐ ❍➔


✐

▼ö❝ ❧ö❝
▼Ð ✣❺❯

✸

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✻

✶✳✶

❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷

❑❤æ♥❣ ❣✐❛♥ ❞➣② c0


✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸

❑❤æ♥❣ ❣✐❛♥ ❞➣② lp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹

❑❤æ♥❣ ❣✐❛♥ Lp

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ✣à♥❤ ❧þ ❍❛❤♥✲❇❛♥❛❝❤

✶✹

✷✶

✷✳✶

✣è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✷✶

✷✳✷

✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳


✷✹

✷✳✸

✣à♥❤ ❧➼ ❍❛❤♥✲❇❛♥❛❝❤ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✹

▼ët sè ❤➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✸ ✣è✐ ♥❣➝✉ tr♦♥❣ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠

✸✶

✸✳✶

✣è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ lp , 1 < p < ∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

✸✳✷

✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1 , ❦❤æ♥❣ ❣✐❛♥ l∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺


✸✳✷✳✶✳ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l1

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✸✳✷✳✷✳ ✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ l∞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻

✣è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Lp (1 < p < ∞) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

✸✳✸


✐✐

❑➌❚ ▲❯❾◆

✹✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✶


é

ỵ ồ t
t ởt ừ t t ồ ự
ổ tỡ ữủ tr t ởt trú tổổ ũ ủ
t tỷ t t tử ỳ ú t q ữỡ ừ
õ t ữ ỵ tt ữỡ tr
tữớ ữỡ tr r ỵ tt t ỹ
tr ữỡ t ỵ tt ớ
ỳ ừ t t ỗ tứ ổ tr ữỡ
tr t ừ rt r t t ụ
ữủ ỳ t tỹ q trồ õ tr t ỹ tr
ự tr tự t ồ r t
ỵ ởt ổ ử q trồ õ
rở ừ t t tr ởt
ổ ừ ởt ổ tỡ t ở ổ õ õ
ụ ự tọ r õ ừ tử tr ộ
ổ ự ổ ủ õ
t õ ữủ t t t s t ỳ ữớ
ở ự ỵ ỳ ợ ử t
t tự t trữợ r trữớ ữủ sỹ ú




ù t t ừ t ũ ữớ tỹ õ tốt
ợ t ố ừ ởt số ổ


ử ử ự
ự ố tr ổ ổ
tỹ ự ố

tr ởt số ổ

Pữỡ ự
Pữỡ ự tt
Pữỡ t

trú õ

ử ử t t t ử ử
õ ỗ ữỡ
ữỡ ởt số tự ữỡ s tr
ỡ ổ ổ
ổ c0 , lp , Lp
ữỡ ử ừ ữỡ tr
ố tr ổ ổ
ỵ tỹ ỵ ự ởt số q
ữỡ ố tr ởt số ổ ử ừ
ữỡ tr ố ừ ổ
ừ ổ



p

, 1 < p < ố

, 1 ố ừ ổ Lp , 1 < p <

õ ữủ tr tr ỡ s t t ữủ
t tr t õ õ ừ t

ộ ừ ố ỳ ữủ q ừ


✺

❍❛❤♥✲❇❛♥❛❝❤✳ ♥➢♠ ✤÷ñ❝ t➼♥❤ ✤è✐ ♥❣➝✉ ❝õ❛ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠✳
❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❦❤æ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥
❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❊♠ ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü
✤â♥❣ ❣â♣ þ ❦✐➳♥ ♣❤↔♥ ❜✐➺♥ tø q✉þ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳

❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦


❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì E tr➯♥ tr÷í♥❣ K ❧➔
♠ët →♥❤ ①↕

f :

E −→ [0, ∞)
x −→

x

t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t ❞÷î✐ ✤➙②✿
✭✐✮ ✭❇➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✮ x + y ≤ x + y ✭ ∀ ①✱② ∈ E ✮
✭✐✐✮ αx = |α| x (∀α ∈ K, x ∈ E)
✭✐✐✐✮ x = 0 ⇒ x = 0 ✭∀x∈ E ✮

❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì E tr➯♥ K ❝ò♥❣ ✈î✐ ❝❤✉➞♥ x tr♦♥❣ ♥â ✤÷ñ❝ ❣å✐ ❧➔ ♠ët
❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✭tr➯♥ K✮ ✈➔ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ ✭E, . ✮
❚r÷í♥❣ K ❧➔ tr÷í♥❣ R ❤♦➦❝ tr÷í♥❣ C

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼ët ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì E ❧➔ ♠ët →♥❤
①↕ p ✿ E → ❬ 0✱ ∞✮ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✭p✭x + y ✮ ≤ p✭x✮

+ p✭y ✮ ✈î✐ ∀ x✱ y ∈ ❊✮ ✈➔ t➼♥❤ ❝❤➜t t❤✉➛♥ ♥❤➜t ❞÷ì♥❣ ✭ p✭αx✮ = |α|p(x)
✈î✐ x ∈ E ✈➔ α ∈ K✮✳


✼

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ (E,

. ) ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣

❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ E ✈î✐ ♠❡tr✐❝ d(x, y) = x − y , x, y ∈ E ❧➔ ♠ët ❦❤æ♥❣
❣✐❛♥ ✤➛② ✤õ✳

❱➼ ❞ö ✶✳✶✳✶✳ ◆❤ú♥❣ ✈➼ ❞ö t❤æ♥❣ t❤÷í♥❣ ♥❤➜t ❝õ❛ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤
❝❤✉➞♥ ❧➔ Rn ✈➔ Cn

• ❊ ❂ Rn ❝ò♥❣ ✈î✐ ( ① 1 , ① 2 , ..., ① n ) ❂

Σni=1 x2i ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥

✤à♥❤ ❝❤✉➞♥ ✭ tr➯♥ tr÷í♥❣ R✮✳

• ❊ ❂ Cn ❝ò♥❣ ✈î✐ ( ③ 1 , ③ 2 , ..., ③ n ) ❂


Σnj=1 |zj |2 ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥

✤à♥❤ ❝❤✉➞♥ ✭tr➯♥ tr÷í♥❣ C✮

❱➼ ❞ö ✶✳✶✳✷✳ Kn ❝ò♥❣ ✈î✐ t✐➯✉ ❝❤✉➞♥ ❊✉❝❧✐❞❡❛♥ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝
✤➛② ✭✤â ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✮

❱➼ ❞ö ✶✳✶✳✸✳ ( ∞,

.

∞)

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✭ ✤â ❧➔ ✤➛② ✤õ✱ tr÷î❝

✤â ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t r➡♥❣ ✤â ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✮✳

▼➺♥❤ ✤➲ ✶✳✶✳ ◆➳✉ (E,

.

E)

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ F ⊆ E

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❣✐❛♥ ✈❡❝tì ❝♦♥✱ ❦❤✐ ✤â F trð t❤➔♥❤ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤
❝❤✉➞♥ ♥➳✉ t❛ ①→❝ ✤à♥❤ ❝❤✉➞♥ tr➯♥ F ❜ð✐ .

x

❚❛ ❣å✐ (F, .

F)

F

= x

E

F

❜ð✐

✈î✐ ① ∈ F.

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ (E, .

▼➺♥❤ ✤➲ ✶✳✷✳ ◆➳✉ (E,

.

E)

E )✳

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ (F, .

F)


❧➔ ❦❤æ♥❣

❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❝♦♥✱ ❦❤✐ ✤â ❋ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉
❋ ✤â♥❣ tr♦♥❣ ❊✳

▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ (E,

. ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐ ✤â ❊ ❧➔




ởt ổ ồ ộ ở tử tt ố ở
tử tr

ổ c0
ổ tỡ c0 ỗ tt số ở tử


c0 = {(xn )
n=1 l : lim xn = 0},
n

ừ x = (xn ) c0 ữủ

xn = sup |xn |
1n<

c0 ởt ổ õ ừ l ởt ổ
t .




ự ú t õ t c0 ữ ởt ổ ừ
tt {(xn )
n=1 } ũ ợ lim xn = 0 ở tử
n


tr K ú t s r {(xn )
n=1 } l

ổ tỹ sỹ tt
t t r c0 ởt ổ tỡ lim xn =

0 lim yn = 0 õ lim (xn + yn ) = 0 õ
n

n

(xn )
n=1

n
+ (yn )
n=1

c0



(xn )
n=1 , (yn )n=1 c0 ụ ổ õ ự

(xn )
n=1 c0 K, (xn )n=1 c0

ự trỹ t c0 õ tr l ởt út tt
tỷ ừ c0 ổ ữợ ự tọ c0 l
õ ú t ự r ởt (zn )
n=1 ừ zn c0 ở
tử tr l ởt ợ w l t w c0


✾

❚❛ ✈✐➳t r❛ ♠é✐ zn ∈ c0 ♥❤÷ ❧➔ ♠ët ❞➣② ✈æ ❤÷î♥❣ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣
♠ët ❝➦♣ ❝❤➾ sè ❞÷î✐✿

zn = (zn,1 , zn,2 , zn,3 , ...) = (zn,j )∞
j=1
✭tr♦♥❣ ✤â zn,j ∈ K ❧➔ ❝→❝ ✈æ ❤÷î♥❣✮✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t w = (wj )∞
j=1
✈➔ ❜➙② ❣✐í ❝❤ó♥❣ t❛ ✤❛♥❣ ❣✐↔ sû r➡♥❣ zn → w tr♦♥❣ (l∞ , .

lim zn − w

n→∞

∞


= lim

n→∞

sup |zn,j − wj |

∞ )✳

◆❣❤➽❛ ❧➔

= 0.

j≥1

✣➸ ❝❤ù♥❣ tä w ∈ c0 ✱ ❜➢t ✤➛✉ ✈î✐ ε > 0 ❝❤♦ tr÷î❝✳ ❑❤✐ ✤â t❛ ❝â t❤➸ t➻♠

N ≥ 0 s❛♦ ❝❤♦ zn − w

∞

< ε/2 ✈î✐ ♠å✐ n ≥ N ✳ ✣➦❝ ❜✐➺t zN − w

∞

<

ε/2✳ ❱➻ zN ∈ c0 ♥➯♥ limj→∞ zN j = 0✳ ◆❤÷ ✈➟② tç♥ t↕✐ j0 > 0 s❛♦ ❝❤♦
|zN,j | < ε/2 ❝è ✤à♥❤ ✈î✐ ♠å✐ j ≥ j0 ✳ ❈❤♦ j ≥ j0 ❦❤✐ ✤â t❛ ❝â✿
|wj | ≤ |wj − zN,j | + |zN,j | < ε/2 + ε/2 = ε.
✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ limj→∞ wj = 0 ✈➔ w ∈ c0 ✳

✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ c0 ✤â♥❣ tr♦♥❣ l∞ ✈➔ ❤♦➔♥ t❤➔♥❤ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤

c0 ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❞➣② lp
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ ✶ ≤ p < ∞ ✱ lp ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝õ❛ t➜t ❝↔ ❞➣② ♣❤➛♥
tû {an }∞
n=1 s❛♦ ❝❤♦✿

∞

|an |p < ∞.
n=1

▼➺♥❤ ✤➲ ✶✳✺✳ lp ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✤è✐ ✈î✐ ♣❤➨♣ ❝ë♥❣ ❝→❝ ❞➣②


✶✵

✈➔ ♥❤➙♥ ♠ët ❞➣② ✈î✐ ♠ët ✈æ ❤÷î♥❣ t❤æ♥❣ t❤÷í♥❣✳ ◆â ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥
∞

(an )n

p

1/p
p


|an |

=

.

n=1

✣➸ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ♥➔②✱ t❛ ❝➛♥ ♠ët sè ❦➳t q✉↔ s❛✉✳

❇ê ✤➲ ✶✳✶✳ ●✐↔ sû 1 < p < ∞ ✈➔ q ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ p1 + 1q = 1✳ ❑❤✐ ✤â✿
ap bq
+ , ✈î✐ ❛✱❜ ≥ 0.
ab ≤
p
q

❇ê ✤➲ ✶✳✷ ✭❇➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r✮✳ ●✐↔ sû 1 ≤ p < ∞ ✈➔ p1 + 1q = 1
✭♥➳✉ p = 1 t❤➻ ❤✐➸✉ ❧➔ q = ∞✮ ✈➔ ❣✐→ trà ❝õ❛ p ✈➔ q t❤ä❛ ♠➣♥ ♠è✐ q✉❛♥
❤➺ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❝➦♣ sè ♠ô ❧✐➯♥ ❤ñ♣✳ ❈❤♦ (an )n ∈ lp ✈➔ (bn )n ∈ lq ✱
∞

|an bn | ≤ (an )n

p

(bn )n q .

n=1


❈❤ù♥❣ ♠✐♥❤✳ ❇➜t ✤➥♥❣ t❤ù❝ ❤♦➔♥ t♦➔♥ ✤ó♥❣ ♥➳✉ p = 1 ✈➔ q = ∞✳
●✐↔ sû p > 1✳ ❈❤♦✿
∞

A = (an )n

p

1/p
p

|an |

=
n=1
∞

B = (bn )n

q

1/q
q

|bn |

=
n=1

◆➳✉ ♠ët tr♦♥❣ A = 0 ❤♦➦❝ B = 0✱ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ t➛♠ t❤÷í♥❣✳ ◆➳✉

❦❤æ♥❣ ♣❤↔✐ ♥❤÷ ✈➟②✱ ❞ò♥❣ ❇ê ✤➲ 1.1 ✈î✐ a = |an |/A ✈➔ b = |bn |/B ❝â ✤÷ñ❝✿

|an bn |
1 |an |p 1 |bn |q
≤
+
AB
p Ap
q Bq


✶✶
∞

n=1

1
|an bn |
≤
AB
p

|an |p 1
+
Ap
q

|bn |q
1 1
=

+ =1
Bq
p q

❉♦ ✤â✿

|an bn | ≤ AB

◆❤➟♥ ①➨t ✶✳✸✳✶✳ ❈❤♦ p = 2 ✈➔ q = 2✱ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r ✤÷❛ ✈➲ ❜➜t
✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③
∞

1/2

|an bn | ≤

2

|an |

|yn |

n

n=1

1/2

2


.

n

❇ê ✤➲ ✶✳✸ ✭❇➜t ✤➥♥❣ t❤ù❝ ▼✐♥❦♦✇s❦✐✮✳ ◆➳✉ x = (xn)n ✈➔ y = (yn)n
tr♦♥❣ lp (1 ≤ p ≤ ∞) (xn + yn )n ❝ô♥❣ t❤✉ë❝ lp ✈➔

(xn + yn )n

p

≤ (xn )n

p

+ (yn )n p .

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ♥➔② ❦❤→ t➛♠ t❤÷í♥❣ ❦❤✐ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ p = 1 ✈➔

p = ∞✳ ❇ð✐ ✈➟②✱ ❣✐↔ sû 1 < p < ∞✳
✣➛✉ t✐➯♥ t❛ ❝❤ó þ r➡♥❣✿

|xn + yn | ≤ |xn | + |yn | ≤ 2max(|xn |, |yn |)
|xn + yn |p ≤ 2p max(|xn |p , |yn |p ) ≤ 2p |xn |p + |yn |p
|xn + yn |p ≤ 2p
n

|xn |p +
n


✣✐➲✉ tr➯♥ ❝❤ù♥❣ tä r➡♥❣ (xn + yn )n ∈ lp ✳

|yn |p
n


✶✷

❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱

|xn + yn ||xn + yn |p−1

|xn + yn |p =
n

n

n

n

❱✐➳t

n |xn ||xn

|yn ||xn + yn |p−1

|xn ||xn + yn |p−1 +

=


+ yn |p−1 =

n an b n

tr♦♥❣ ✤â an = |xn | ✈➔ bn = |xn +

yn |p−1 .
❑❤✐ ✤â t❛ ❝â (an )n ∈ lp ✈➔ (bn )n ∈ lq ✈➻

bqn =

|xn + yn |(p−1)q

n

n

|xn + yn |p < ∞

=
n

1 1
+ = 1 ✤➸ ❝❤ù♥❣ ♠✐♥❤ (p − 1)q = p✮✳
p q
❚ø ❇ê ✤➲ ✶✳✷ t❛ ❦➳t ❧✉➟♥✿

✭sû ❞ö♥❣ ♠è✐ q✉❛♥ ❤➺


|xn ||xn + yn |p−1 ≤

|xn |p

n

1
p

|xn + yn |(p−1)q
n

n

= (xn )n

1
q

p

(xn + yn )n

p
q

p

❚÷ì♥❣ tü✿


|xn ||xn + yn |p−1 ≤ (yn )n

p

(xn + yn )n

p
q

p

n

❈ë♥❣ ❤❛✐ ✈➳ ❝õ❛ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ t❛ ❝â✿

(xn + yn )n

p
p

≤ (xn )n

p

(xn + yn )n

p
q

p


+ (yn )n

p

(xn + yn )n

p
q

p.


✶✸

❇➙② ❣✐í✱ ♥➳✉ (xn + yn )n

p

= 0 ❦❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤

❧➔ t❤ä❛ ♠➣♥✳ ◆➳✉ (xn + yn )n

= 0 t❛ ❝â t❤➸ ❝❤✐❛ ❝❤♦ (xn + yn )n

p

p
q


p

♥❤➟♥ ✤÷ñ❝✿
p− pq
p

(xn + yn )n
❚ø p −

p
q

≤ (xn )n

p

+ (yn )n p .

= 1✱ ❜➜t ✤➥♥❣ t❤ù❝ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ▼➺♥❤ ✤➲ ✶✳✺

❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ❇ê ✤➲ ✶✳✸ t❛ ❞➵ ❞➔♥❣ t❤➜② ✤÷ñ❝ lp ❧➔
♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✈➔ .

p

❧➔ ♠ët ❝❤✉➞♥ tr➯♥ ♥â✳

✣➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ lp ❧➔ ✤➛② ✤õ✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♠å✐ ❝❤✉é✐ Σk xk

❤ë✐ tö t✉②➺t ✤è✐ tr♦♥❣ lp ❧➔ ❤ë✐ tö✳
❱✐➳t xk = (xk,n )n = (xk,1 , xk,2 , ...) ✈î✐ ♠é✐ ❦✳ ❈❤ó þ r➡♥❣
1/p

|xk,n | ≤ xk

p

p

|xk,n |

=
n

❱➻ ✈➟②

k

|xk,n | ≤

k

xk

p

< ∞ ✈î✐ ♠é✐ ❦ ✈➔ ♥❤÷ ✈➟②
yn =


xk,n
k

❧➔ ❝â ♥❣❤➽❛ ✭✈➔ yn ∈ K✮✳


✶✹

❇➙② ❣✐í✱ ✈î✐ ❜➜t ❦➻ N ≥ 1✱
N

|yn |

K→∞

p

1/p

xk,n

= lim

K→∞

n=1

= lim

K


N

1/p
p

n=1

n=1

(x1,1 , x1,2 , ..., x1,N , 0, 0, ...) + (x2,1 , x2,2 , ..., x2,N , 0, 0, ...) + ...

+ (xK,1 , xK,2 , ..., xK,N , 0, 0, ...)

p

K

≤ lim

K→∞

(xk,1 , xk,2 , ..., xk,N , 0, 0, ...)
k=1
K

≤ lim

K→∞


p

xk

p

k=1

∞

=

xk

p

<∞

k=1

❈❤♦ N → ∞, t❛ ❝â y = (yn )n ∈ lp ✳ ⑩♣ ❞ö♥❣ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ❝❤♦

y−

K0
k=1 xk

✭K0 ≥ 0 ❜➜t ❦➻ ❝❤♦ tr÷î❝✮✱ t❛ ❝â
∞


K0

y−

≤

xk
k=1

p

xk

p

→ 0 ❦❤✐ K0 → ∞

K0 +1

◆â✐ ❝→❝❤ ❦❤→❝ ❝❤✉é✐ Σk xk ❤ë✐ tö ✤➳♥ y tr♦♥❣ lp ✳

✶✳✹ ❑❤æ♥❣ ❣✐❛♥ Lp
❈❤♦ (X, M, µ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤♦✱ ♥❣❤➽❛ ❧➔ X ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝
ré♥❣✱ M ❧➔ ♠ët σ−✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✈➔ µ ❧➔ ♠ët ✤ë ✤♦ tr➯♥ M✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❱î✐ F = R ❤♦➦❝ F = C ✈➔ 1 ≤ p < ∞✳ ❳➨t t➟♣
Lp (X, µ) =

f : X → F : f ❧➔ ✤♦ ✤÷ñ❝,


|f |p dµ < ∞
X


✶✺

▼ët t➼♥❤ ❝❤➜t ❧➔ ✤ó♥❣ ❤➛✉ ❦❤➢♣ ✭❤➛✉ ❦❤➢♣ ♥ì✐✮ ♥➳✉ t➟♣ ❤ñ♣ ♥❤ú♥❣ ♣❤➛♥
tû ð ✤â t➼♥❤ ❝❤➜t ❦❤æ♥❣ ✤ó♥❣ ❝❤ù❛ tr♦♥❣ ♠ët t➟♣ ❝â ✤ë ✤♦ ❦❤æ♥❣✳
✣➦t

L∞ (X, µ) =

f : X → F ✤♦ ✤÷ñ❝ |∃C > 0, f (x) ≤ ❈ ❤➛✉ ❦❤➢♣ tr➯♥ X

✈➔ ①→❝ ✤à♥❤ ❝❤✉➞♥ tr➯♥ L∞ (X, µ) ❜ð✐

f = ❡sss✉♣{|f (x)|, x ∈ X} = inf{C > 0, |f | ≤ C ❤➛✉ ❦❤➢♣ tr➯♥X}
◆➳✉ f ∈ Lp (X, µ), p ∈ [1, ∞)✱ ✈➔ f ❧➔ ❤➔♠ sè ✤♦ ✤÷ñ❝ tr➯♥ ❳✱ ①→❝ ✤à♥❤
❝❤✉➞♥ ❜ð✐

1/p

f

p

p

|f | dµ


=

.

X

❑❤✐ ✤â✱ Lp (X, µ) ❜❛♦ ❣ç♠ t➜t ❝↔ ❝→❝ f ♠➔

f
❈❤ó♥❣ t❛ ❣å✐ f

p

p

<∞

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

❱➼ ❞ö ✶✳✹✳✶✳ ◆➳✉ X = [0; 1] ✈➔ µ ❧➔ ♠ët ✤ë ✤♦ ▲❡❜❡s❣✉❡✱ ✈➔ f ❧✐➯♥ tö❝✱
t❤➻ f

∞

= sup{|f (x)||x ∈ [0; 1]}

❇ê ✤➲ ✶✳✹✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r✮ ◆➳✉ p ✈➔ q ❧➔ ❝→❝ sè ♠ô ❧✐➯♥ ❤ñ♣✱
1 < p < ∞ ✈➔ ♥➳✉ f ∈ Lp (X, µ) ✈➔ g ∈ Lq (X, µ) ❦❤✐ ✤â f g ∈ L1 (X, µ)✱
✈➔


fg

1

≤ f

p

g q.

❇ê ✤➲ ✶✳✺✳ ❱î✐ ❤❛✐ ❤➔♠ sè ❜➜t ❦➻ u = u(t), v = v(t) ❦❤æ♥❣ ➙♠ ✈➔ µ−


✶✻

✤♦ ✤÷ñ❝ tr➯♥ X t❛ ❝â ❜➜t ✤➠♥❣ t❤ù❝

up v q
+
≥ u.v
p
q
❞➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ up = v q
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè ϕ(t) =

tp t−q
+
, t > 0. ❚❛ ❝â
p
q


ϕ (t) = tp−1 − t−q−1 = t−q−1 (tp+q − 1),
ϕ (t) = 0 ⇔ t = 1(t > 0)
❚❛ ❝â min ϕ(t) = ϕ(1) = 1
0
1 −1
❉♦ ✤â ϕ(t) ≥ ϕ(1) = 1, ∀t ∈ (0, +∞). ❈❤å♥ t = u q v p t❛ ✤÷ñ❝
p
u q .v −1
p

+

q
u−1 .v p
q

≥1

up−1 .v −1 v q−1 .u−1
+
≥1
⇔
p
q
⇔

up v q
+

≥ uv
p
q

❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

1 −1
1
1
u q .v p = 1 ⇔ u q = v p ⇔ up = v q

⑩♣ ❞ö♥❣ ❜ê ✤➲ ✶✳✺ q✉❛② ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ ✶✳✹
❈❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✶✳✹✿


✶✼

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f = 0 ❤✳❦✳♥ tr➯♥ t➟♣ X ❤♦➦❝ g = 0 ❤✳❦✳♥ tr➯♥ t➟♣ X
t❤➻ ❜➜t ✤➥♥❣ t❤ù❝

fg

1

≤ f

p

g q.

❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳
◆➳✉ f = 0 ✈➔ g = 0 tr➯♥ t➟♣ E ⊂ E, µ(E ) > 0 t❤➻

|f |p dµ > 0,
X

✣➦t u =

|g|q dµ > 0
X

|g|
|f |
,v =
✈➔ →♣ ❞ö♥❣ ❜ê ✤➲ ✶✳✺ t❛ ✤÷ñ❝
f p
g q
|f |.|g|
f p. g

≤
q

|f |p
|g|q
+
p( X |f |p ) q( X |g|q )

▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ t❤❡♦ ✤ë ✤♦ µ tr➯♥ ❳ t❛ ♥❤➟♥ ✤÷ñ❝ ❜➜t ✤➥♥❣ t❤ù❝
❝➛♥ ❝❤ù♥❣ ♠✐♥❤✿

fg

1

≤ f

p

g q.

❇ê ✤➲ ✶✳✻✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ▼✐♥❦♦✇s❦✐✮ ❈❤♦ f, g ∈ Lp(X, µ)✱✈î✐ 1 ≤ p ≤
∞. f + g ∈ Lp (X, µ) ✈➔
f +g

p

≤ f

p

+ g

p

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ p = 1 ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳
1 1
❱î✐ p > 1 t❛ ❧➜② sè t❤ü❝ q s❛♦ ❝❤♦ + = 1, t❤➻ q > 1 ✈➔
p q

p + q = pq → p = q(p − 1), q = p(q − 1)


✶✽

❚❛ ❝â

|f + q|p = |f + g|p−1 |f + g| ≤ |f ||f + g|p−1 + |g||f + g|p−1
⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r✬s ❧➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ t❛ ✤÷ñ❝

|f + g|p ≤

(|f | + |g|)|f + g|p−1 =

X

X

|f |p

≤

|f ||f + g|p−1 +

|g||f + g|p−1

X

1
p

1
q

|f + g|q(p−1)

.

X

|g|p

+

X

X

1
p

|f + g|q(p−1)

.

X

1
q

X

❤❛②

|f + g|p ≤

1
p

|f |p

X

X

❚ø ✤â
p

X

1− 1q

|f + g|

≤

1
q

|f + g|p

1
q

X

p

|f |

X

✭❱➻ 1 −

|q|p

+

1
p

1
p

p

|g|

+

X

1
p

X

= p1 ✮✳ ❚❛ ✤÷ñ❝
f +g

p

≤ f

p

+ g p.

✣à♥❤ ❧þ ✶✳✶ ✭❋✐s❝❤❡r✲❘✐❡s③✮✳ ❑❤æ♥❣ ❣✐❛♥ Lp ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐
♠é✐ ♣✱ 1 ≤ p ≤ ∞
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤✐❛ ❤❛✐ tr÷í♥❣ ❤ñ♣ p = ∞ ✈➔ 1 ≤ p < ∞
❚r÷í♥❣ ❤ñ♣ ✶✿ p = ∞
●✐↔ sû fn ❧➔ ♠ët ❞➣② ❈❛✉❝❤✉② tr♦♥❣ L∞ ✳ ❱î✐ ♠é✐ sè ♥❣✉②➯♥ k ≥ 1 ❝â
♠ët sè ♥❣✉②➯♥ Nk ✤➸ fm − fn

∞

≤ k1 ∀m, n ≥ Nk ✳ ❉♦ ✤â ❝â ♠ët t➟♣ ❝â

✶✾

✤ë ✤♦ ❦❤æ♥❣ Ek s❛♦ ❝❤♦

|fm (x) − fn (x)| ≤

1
k

∀x ∈ X \ Ek ✱ ∀m, n ≥ Nk
✣➦t E = ∪k Ek t❤➻ ❊ ❧➔ t➟♣ ❝â ✤ë ✤♦ ❦❤æ♥❣ ✈➔ t❛ t❤➜② ∀x ∈ X \ E ✱ ❞➣②

fn (x) ❧➔ ❈❛✉❝❤② ✭tr♦♥❣ R✮✳❱➻ ✈➟②✱ fn (x) → f (x)∀x ∈ X \ E ✳ ❈❤✉②➸♥ q✉❛
❣✐î✐ ❤↕♥ ❦❤✐ m → ∞ t❛ ✤÷ñ❝

1
|f (x) − fn (x)| ≤ ∀x ∈ X \ E, ∀n ≥ Nk .
k
❚❛ ❦➳t ❧✉➟♥ r➡♥❣ f ∈ L∞ ✈➔ f − fn

∞

≤ k1 ∀n ≥ Nk , ❞♦ ✤â fn → f tr♦♥❣

L∞ ✳
❚r÷í♥❣ ❤ñ♣ ✷✿ 1 ≤ p < ∞
●✐↔ sû (fn ) ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ Lp ✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❞➣② ♥➔② ❝â
♠ët ❞➣② ❝♦♥ ❤ë✐ tö tr♦♥❣ Lp ✳
❈❤ó♥❣ t❛ ❧➜② r❛ ♠ët ❞➣② ❝♦♥ (fnk ) s❛♦ ❝❤♦

fnk+1 − fnk

p

≤

1
∀k ≥ 1.
2k

❈ö t❤➸ ❧➔♠ ♥❤÷ s❛✉✿ ❈❤å♥ n1 s❛♦ ❝❤♦ fm − fn p ≤ 12 ∀m, n ≥ n1 ✱ s❛✉ ✤â
1
❝❤å♥ n2 ≥ n1 s❛♦ ❝❤♦ fm − fn p ≤ 2 ∀m, n ≥ n2 ✱✳✳✳❚❛ ❦❤➥♥❣ ✤à♥❤ r➡♥❣
2
fnk ❤ë✐ tö tr♦♥❣ Lp ✳ ✣➸ ✤ì♥ ❣✐↔♥ t❛ ❦➼ ❤✐➺✉ fk t❤❛② ❝❤♦ fnk ✳ ◆❤÷ ✈➟②

fk+1 − fk

p

≤

1
∀k ≥ 1
2k


✷✵

✣➦t

n

|fk+1 (x) − fk (x)|,

gn (x) =
k=1

t❤➻

gn

p

≤ 1.

❚❤❡♦ ✤à♥❤ ❧➼ ✈➲ sü ❤ë✐ tö ✤ì♥ ✤✐➺✉✱ gn (x) t✐➳♥ tî✐ ❣✐î✐ ❤↕♥ ❤ú✉ ❤↕♥ ❤❛②
❝á♥ ❣å✐ ❧➔ g(x) ❤✳❦✳♥ tr➯♥ ❳ ✈î✐ g ∈ Lp ✳ ▼➦t ❦❤→❝ ✈î✐ m ≥ n ≥ 2 t❛ ❝â

|fm (x)−fn (x)| ≤ |fm (x)−fm−1 (x)|+...+|fn+1 (x)−fn (x)| ≤ g(x)−gn−1 (x).
❉♦ ✈➟② ✈î✐ ❤➛✉ ❤➳t x ∈ X ✱ fn (x) ❧➔ ❈❛✉❝❤② ✈➔ ❤ë✐ tö ❣✐î✐ ❤↕♥ ❤ú✉ ❤↕♥✱
❣å✐ ❧➔ f (x)✳
❱î✐ ❤➛✉ ❤➳t x ∈ X ❚❛ ❝â

|f (x) − fn (x)| ≤ g(x)∀n ≥ 2
❱➔ ✤➦❝ ❜✐➺t f ∈ Lp ✳ ❈✉è✐ ❝ò♥❣✱ t❤❡♦ ✤à♥❤ ❧þ ❤ë✐ tö trë✐✱ fn − f
❞♦ ✤â fn (x) − f (x)|p → 0 ❤✳❦✳♥ ✈➔ |fn − f |p ≤ g p ∈ L1

p

→ 0,


❈❤÷ì♥❣ ✷
✣à♥❤ ❧þ ❍❛❤♥✲❇❛♥❛❝❤
✷✳✶ ✣è✐ ♥❣➝✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ◆➳✉ ❊ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ tr➯♥ tr÷í♥❣ K✱
❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❊ ❧➔✿

E ∗ = B(E, K) ❂ ④ T : E → K : T ❧✐➯♥ tö❝ ✈➔ t✉②➳♥ t➼♥❤⑥
❝→❝ ♣❤➛♥ tû ❝õ❛ E ∗ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤✐➳♥ ❤➔♠ t✉②➳♥ t➼♥❤ ✭❧✐➯♥ tö❝✮ tr➯♥ E ✳

▼➺♥❤ ✤➲ ✷✳✶✳ ◆➳✉ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤➻ E ∗ ❧➔ ♠ët ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤✳
▼➺♥❤ ✤➲ tr➯♥ ❧➔ ❤➺ q✉↔ ❝õ❛ ✣à♥❤ ❧þ ❞÷î✐ ✤➙②

✣à♥❤ ❧þ ✷✳✶✳ ❈❤♦ ❊ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ ❋ ❧➔ ♠ët ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤✳ ❈❤♦ B(E, F ) ❧➔ ❦➼ ❤✐➺✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t➜t ❝↔ t♦→♥ tû t✉②➳♥
t➼♥❤ ❜à ❝❤➦♥ T : E → F ✳❈❤ó♥❣ t❛ ❝â t❤➸ ❧➟♣ B(E, F ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥
✈❡❝tì ①→❝ ✤à♥❤ ❜ð✐ ❚ ✰ ❙ ✈➔ λT (T, S ∈ B(E, F ), λ ∈ K) ♥❤÷ s❛✉✿

(T + S)(x) = T (x) + S(x), (λT )(x) = λ(T (x))(x ∈ E)