✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
◆●❯❨➍◆ ◗❯❆◆● ❑❍❯➊
❳❻P ❳➓ ◆●❍■➏▼ ❈Õ❆
❇⑨■ ❚❖⑩◆ ❑❍➷◆● ✣■➎▼ ❈❍❯◆● ❚⑩❈❍
❚❘❖◆● ❑❍➷◆● ●■❆◆ ❇❆◆❆❈❍
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✷
◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
❚❙✳ ❚r÷ì♥❣ ▼✐♥❤ ❚✉②➯♥
❚❤→✐ ◆❣✉②➯♥ ✕ ✷✵✶✽
✐✐
▲í✐ ❝↔♠ ì♥
❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❙✳ ❚r÷ì♥❣ ▼✐♥❤ ❚✉②➯♥✱ ♥❣÷í✐ ✤➣
t➟♥ t➻♥❤ ❤÷î♥❣ ❞➝♥✱ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥
t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛
❚♦→♥ ✕ ❚✐♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✕✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï
tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❙ð ●✐→♦ ❞ö❝ ✈➔ ✣➔♦ t↕♦ t➾♥❤ ❍➔ ●✐❛♥❣✱ ❇❛♥ ●✐→♠
✤è❝ ❚r✉♥❣ t➙♠ ●✐→♦ ❞ö❝ t❤÷í♥❣ ①✉②➯♥ ✲ ❍÷î♥❣ ♥❣❤✐➺♣ t➾♥❤ ❍➔ ●✐❛♥❣✱ ❝ô♥❣ ♥❤÷
t♦➔♥ t❤➸ ❝→❝ ✤ç♥❣ ♥❣❤✐➺♣✱ ✤➣ q✉❛♥ t➙♠ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ tæ✐ t❤ü❝
❤✐➺♥ ✤ó♥❣ ❦➳ ❤♦↕❝❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
✐✐✐
▼ö❝ ❧ö❝
▲í✐ ❝↔♠ ì♥
✐✐
▼ët sè ❦þ ❤✐➺✉ ✈➔ ✈✐➳t t➢t
✐✈
▼ð ✤➛✉
✶
❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✸
✶✳✶✳
▼ët sè ✈➜♥ ✤➲ ✈➲ ❤➻♥❤ ❤å❝ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸
✶✳✷✳
⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✵
✶✳✸✳
P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✈➔ ♣❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✹
✶✳✸✳✶✳
P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✹
✶✳✸✳✷✳
P❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✻
❚♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✾
✶✳✹✳
❈❤÷ì♥❣ ✷ ❳➜♣ ①➾ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ t→❝❤
✷✳✶✳
✷✳✷✳
✷✷
❳➜♣ ①➾ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ t→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✷
✷✳✶✳✶✳
P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❝♦ ❤➭♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✷
✷✳✶✳✷✳
P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❝❤✐➳✉
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✺
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✶
Ù♥❣ ❞ö♥❣
✷✳✷✳✶✳
❇➔✐ t♦→♥ ✤✐➸♠ ❝ü❝ t✐➸✉ t→❝❤
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✶
✷✳✷✳✷✳
❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✸
❑➳t ❧✉➟♥
✸✺
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✸✻
ởt số ỵ t tt
E
ổ
E
ổ ố ừ
R
t ủ số tỹ
R+
t số tỹ ổ
inf M
ữợ ú ừ t ủ số
M
sup M
tr ú ừ t ủ số
M
max M
số ợ t tr t ủ số
min M
số ọ t tr t ủ số
rxX F (x)
t ỹ t ừ
t rộ
x
ợ ồ
D(A)
ừ t tỷ
R(A)
ừ t tỷ
A1
t tỷ ữủ ừ t tỷ
I
t tỷ ỗ t
Lp ()
ổ t
lp
ổ số tờ
E
M
X
A
A
{xn }
lim inf xn
ợ ữợ ừ số
{xn }
xn x0
{xn }
ở tử
xn
{xn }
ở tử
x0
tr
A
n
n
F
x
ợ tr ừ số
lim sup xn
M
x0
x0
p
tr
p
✈
JE
→♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ tr➯♥
jE
→♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà tr➯♥
δE (ε)
♠æ ✤✉♥ ❧ç✐ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
ρE (τ )
♠æ ✤✉♥ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
F ix(T )
❤♦➦❝
F (T )
t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
∂f
❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐
M
❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ ❤ñ♣
PC
♣❤➨♣ ♠➯tr✐❝ ❧➯♥
ΠC
♣❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t ❧➯♥
iC
❤➔♠ ❝❤➾ ❝õ❛ t➟♣ ❧ç✐
f
M
C
C
C
E
T
E
E
E
H1
C
H2
Q t ỗ õ rộ ừ ổ rt
tữỡ ự
T : H2 H1
T : H1 H2
t tỷ ủ ừ
ởt t tỷ t t
T
t t P õ
ữ s
ởt tỷ
x S = C T 1 (Q) = .
P
ổ t P t ữủ ợ t ự sr
ổ t ữủ t õ trỏ q
trồ tr ổ ử tr ồ ữớ ở tr tr
tr tữ ổ ử t õ t ử
t tr t ỵ tt trỏ ỡ
sỷ
C
ởt t ỗ õ ừ ổ rt
H1
t r
t ỹ t ừ
iC (x) =
0,
,
r minH1 iC (x)
ừ
iC
r
A = I PC
= C
x C,
x
/C
õ t ữủ
r r r
C
C = (iC )1 (0)
iC
ợ
iC
ữợ
ởt t tỷ ỡ ỹ
ụ t ổ ừ t tỷ ỡ
A
õ t õ t t t P trữớ ủ
r ừ t ổ t
t ổ t ữủ t s
B : H2 2H2
t tỷ ỡ ỹ
A : H1 2H1
T : H1 H2
ởt
t tỷ t t
ởt tỷ
x S = A1 (0) T 1 B 1 (0) = .
PP
t PP ừ t út ữớ
t tr ữợ q t ự ử ừ
tr t q ừ s tr t ữỡ
ữỡ t PP tr
ổ
ở ừ ữủ ữỡ
ữỡ tự
r ữỡ ởt số trú ồ
ừ ổ ữ ổ ỗ ổ
trỡ ố t tr tờ qt
t tỷ ỡ tr ổ t tỷ tr t tỷ
tờ qt
ữỡ ừ t ổ t
r ữỡ t tr tr ởt tt t
q ừ s ữỡ ữỡ
t ổ t tr ổ
r tr ữỡ ụ ự ử ừ ữỡ
ỵ t ỹ t t t
t
✸
❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❈❤÷ì♥❣ ♥➔② ❜❛♦ ❜ç♠ ✹ ♠ö❝✳ ▼ö❝ ✶✳✶ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ✈➲ ♠ët sè t➼♥❤
❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧ç✐ ✤➲✉✱ trì♥ ✤➲✉✳ ▼ö❝
✶✳✷ ❣✐î✐ t❤✐➺✉ ✈➲ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✳ ▼ö❝ ✶✳✸ tr➻♥❤ ❜➔② ✈➲ ♣❤➨♣ ❝❤✐➳✉
♠➯tr✐❝ ✈➔ ♣❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t ❝ò♥❣ ✈î✐ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤ó♥❣✳ ▼ö❝
✶✳✹ tr➻♥❤ ❜➔② ✈➲ t♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ t♦→♥ tû ❣✐↔✐ tê♥❣
q✉→t ✈➔ t♦→♥ tû ❣✐↔✐ ♠➯tr✐❝✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝
t➔✐ ❧✐➺✉ ❬✶✱ ✺✱ ✻✱ ✽✱ ✾✱ ✶✵❪✳
✶✳✶✳
▼ët sè ✈➜♥ ✤➲ ✈➲ ❤➻♥❤ ❤å❝ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
❈❤♦
E
❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔
E∗
❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ♥â✳ ✣➸
.
❝❤♦ ✤ì♥ ❣✐↔♥ ✈➔ t❤✉➟♥ t✐➺♥ ❤ì♥✱ ❝❤ó♥❣ tæ✐ t❤è♥❣ ♥❤➜t sû ❞ö♥❣ ❦➼ ❤✐➺✉
❝❤✉➞♥ tr➯♥
E
✈➔
E ∗❀
❙ü ❤ë✐ tö ♠↕♥❤ ✈➔ ②➳✉ ❝õ❛ ❞➣②
❧➛♥ ❧÷ñt ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔
xn → x
✈➔
xn
x
{xn }
✈➲ ♣❤➛♥ tû
x
✤➸ ❝❤➾
tr♦♥❣
E
tr♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ t❤÷í♥❣ ①✉②➯♥ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❞÷î✐ ✤➙② ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✳
▼➺♥❤ ✤➲ ✶✳✶✳
✭①❡♠ ❬✶❪ tr❛♥❣ ✹✶✮
❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱
❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✐✮
✐✐✮
E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳
▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✱ ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳
▼➺♥❤ ✤➲ ❞÷î✐ ✤➙② ❝❤♦ t❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ t➟♣ ✤â♥❣ ✈➔ t➟♣ ✤â♥❣ ②➳✉ tr♦♥❣ ❦❤æ♥❣
❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳
C t ỗ õ rộ ừ ổ ổ
t t X t C t õ
ự
xn
t
x
ự ự sỷ tỗ t
x
/ C.
x
ữ
C
tự tỗ t
ỵ t t ỗ tỗ t
>0
{xn } C
x X
s
t
s
y, x x, x ,
ợ ồ
y C
t t õ
xn , x x, x ,
ợ ồ
n 1
r
tự tr
n
xn
x
xn , x x, x
õ tr t
t ữủ
x, x x, x ,
ổ ỵ õ sỷ s
C
t õ
ữủ ự
ú ỵ
C
t õ t
C
t õ
ữợ t ởt sỹ tỗ t ỹ t ừ ởt
ỗ tữớ ỷ tử ữợ tr ổ
C t ỗ õ rộ ừ ổ
E f : C (, ] ởt ỗ tữớ ỷ tử
ữợ tr C s f (xn ) xn õ tỗ t x0 dom(f )
s
f (x0 ) = inf{f (x) : x C}.
ự
f (xn ) m
{xnk }
ợ
t
ừ
{xn }
m = inf{f (x) : x C}
n
s
{xn }
xnk
õ tỗ t
{xn } C
s
ổ t tỗ t ởt
tt
f (xnk )
t
m = õ {xn } tỗ t
{xnj }
{xn }
ừ
s
x nj
x0 C
f
ỷ tử ữợ tr tổổ
t õ
m f (x0 ) lim inf f (xnj ) = lim f (xn ) = m.
n
j
õ
m = f (x0 )
ữủ ự
t tr ử ú tổ ởt số ỡ
trú ồ ổ ữ t ỗ t trỡ ổ ỗ ổ
trỡ
ổ E ữủ ồ ỗ t ợ ồ x, y
E, x = y x = 1,
ú ỵ
y = 1 t õ
x+y
< 1.
2
ỏ õ t t ữợ tữỡ ữỡ
E ữủ ồ ỗ t ợ ồ x, y SE tọ
x+y
= 1 s r x = y ợ ồ x, y SE x = y t õ tx+(1t)y < 1
2
ợ ồ t (0, 1) tr õ
s ổ
SE = {x E :
E
x = 1}.
ởt ổ ỗ t õ ợ ộ
f E \ {0} tỗ t t tỷ x E s x = 1 x, f = f
ự
sỷ tỗ t
x, y E
tọ
x = y =1
x=y
s
x, f = y, f = f .
õ ợ
t (0, 1)
tứ t ỗ t ừ
E
t õ
f = t x, f + (1 t) y, f
= tx + (1 t)y, f
tx + (1 t)y
f
< f .
r t tỗ t t tỷ
x, f = f
x E
s
x = 1
✻
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
tç♥ t↕✐
δ(ε) > 0
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
s❛♦ ❝❤♦ ✈î✐ ♠å✐
E
✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ ✤➲✉ ♥➳✉ ✈î✐ ♠å✐
x, y ∈ E
ε > 0✱
x = 1✱ y = 1, x − y ≥ ε
♠➔
t❛
❧✉æ♥ ❝â
x+y
≤ 1 − δ(ε).
2
E
❉➵ t❤➜② r➡♥❣ ♥➳✉
❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧ç✐ ✤➲✉ t❤➻ ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ❧ç✐ ❝❤➦t✳ ❚✉② ♥❤✐➯♥ ✤✐➲✉ ♥❣÷ñ❝ ❧↕✐ ❦❤æ♥❣ ✤ó♥❣✱ ✈➼ ❞ö ❞÷î✐ ✤➙② ❝❤➾ r❛ ✤✐➲✉
✤â✳
❱➼ ❞ö ✶✳✶✳
✭①❡♠ ❬✶❪ tr❛♥❣ ✺✹✮ ❳➨t
.
❦❤æ♥❣✮ ✈î✐ ❝❤✉➞♥
E = c0
β ①→❝ ✤à♥❤ ❜ð✐
∞
x
β
= x
c0
+β
i=1
❑❤✐ ✤â✱
✭❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❤ë✐ tö ✈➲
(E, . β ), β > 0
|xi |2
i2
1/2
, x = (xi ) ∈ c0 .
❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t ♥❤÷♥❣ ❦❤æ♥❣ ❧➔ ❦❤æ♥❣ ❣✐❛♥
❧ç✐ ✤➲✉✳
✣➸ ✤♦ t➼♥❤ ❧ç✐ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✤✉♥ ❧ç✐ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
❧➔ ❤➔♠ sè
x+y
: x ≤ 1, y ≤ 1, x − y ≥ ε .
2
δE (ε) = inf 1 −
◆❤➟♥ ①➨t ✶✳✶✳
E
E ✱ ♥❣÷í✐ t❛ ✤÷❛ ✈➔♦ ❦❤→✐ ♥✐➺♠ s❛✉✿ ▼æ
▼æ ✤✉♥ ❧ç✐ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✈➔ t➠♥❣ tr➯♥ ✤♦↕♥
[0; 2]✳
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
E
E
❧ç✐ ❝❤➦t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
✭①❡♠ ❬✶❪ tr❛♥❣ ✺✾✮✳ ◆❣♦➔✐ r❛✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
δE (ε) > 0, ∀ε > 0
▼➺♥❤ ✤➲ ✶✳✺✳
❧➔ ❤➔♠ sè ①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝
E
δE (2) = 1
❧➔ ❧ç✐ ✤➲✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
✭①❡♠ ❬✶❪ tr❛♥❣ ✻✵✮✳
✭①❡♠ ❬✶❪ tr❛♥❣ ✺✻✮
▼å✐ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧ç✐ ✤➲✉ ❜➜t ❦➻ ❧➔ ❦❤æ♥❣
❣✐❛♥ ♣❤↔♥ ①↕✳
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳
♥➳✉ ♠å✐ ❞➣②
❱➼ ❞ö ✶✳✷✳
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
{xn } ⊂ E
xn
▼å✐ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❚❤➟t ✈➟②✱ ❣✐↔ sû
xn → x✳
t❤ä❛ ♠➣♥
{xn }
E
x
H
✤÷ñ❝ ❣å✐ ❧➔ ❝â t➼♥❤ ❝❤➜t ❑❛❞❡❝✲❑❧❡❡
✈➔
xn → x✱
❑❤✐ ✤â✱ t❛ ❝â
2
xn → x✳
✤➲✉ ❝â t➼♥❤ ❝❤➜t ❑❛❞❡❝✲❑❧❡❡✳
❧➔ ♠ët ❞➣② ❜➜t ❦ý tr♦♥❣
xn − x
t❤➻
= xn − x, xn − x
H
t❤ä❛ ♠➣♥
xn
x
✈➔
2
= xn
x
õ
2
2 xn , x + x
2 x
2
+ x
2
2
= 0.
xn x
ữợ t t ợ ổ rở ỡ õ t t
ồ ổ ỗ õ t t
ự
t ý tr
xn
xn
sỷ
H
E
ởt ổ ỗ
tọ
xn
x
x = 0 t xn 0 sỷ x = 0 xn
x õ
x
õ tỗ t > 0 {xnk } ừ {xn } s
x
k 1
E
xn
x
xn x
x
x
1=
s r t
x SE
k
E
1
2
x
xnk
+
xnk
x
E
s
fx E
1 ,
õ t t
ổ
>0
x
xnk
+
1 .
xnk
x
x
xn
r
xn
x
lim inf
xn x
tỗ t t
E
t õ
t õ
,
ổ ỗ tỗ t
1
2
ứ
ởt
xn x
x
xnk
xnk
x
ợ ồ
{xn }
s
E
ữủ ồ trỡ ợ ộ
x, fx = x
fx = 1
ởt ổ t t tr
ữủ ồ t t
x SE
ợ ộ
y SE
tỗ t ợ
tr
x SE
d
x + ty x
( x + ty )t=0 = lim
.
t0
dt
t
E
E
ởt ổ t t õ
ữủ ồ t õ t t ồ
✽
❜✮ ❈❤✉➞♥ tr➯♥
E
✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈î✐ ♠å✐
✭✶✳✶✮ tç♥ t↕✐ ✤➲✉ ✈î✐ ♠å✐
❝✮ ❈❤✉➞♥ tr➯♥
E
❞✮ ❈❤✉➞♥ tr➯♥
✈î✐ ♠å✐
E
❣✐î✐ ❤↕♥
x ∈ SE .
✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥➳✉ ✈î✐ ♠å✐
tç♥ t↕✐ ✤➲✉ ✈î✐ ♠å✐
y ∈ SE
x ∈ SE ✱ ❣✐î✐ ❤↕♥ ✭✶✳✶✮
y ∈ SE .
✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✤➲✉ ♥➳✉ ❣✐î✐ ❤↕♥ ✭✶✳✶✮ tç♥ t↕✐ ✤➲✉
x, y ∈ SE .
✣à♥❤ ❧þ ✶✳✶✳
✭①❡♠ ❬✶❪ tr❛♥❣ ✾✷✮
❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ t❛
❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿
❛✮
◆➳✉ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t t❤➻ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ trì♥✳
❜✮
◆➳✉ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ trì♥ t❤➻ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t✳
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ▼æ ✤✉♥ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ ❜ð✐
ρE (τ ) = sup{2−1 x + y + x − y
◆❤➟♥ ①➨t ✶✳✷✳
▼æ ✤✉♥ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
tö❝ ✈➔ t➠♥❣ tr➯♥ ❦❤♦↔♥❣
❱➼ ❞ö ✶✳✸✳
−1:
❬✶✵❪ ◆➳✉
E
[0; +∞)
x = 1,
E
y = τ }.
❧➔ ❤➔♠ sè ①→❝ ✤à♥❤✱ ❧✐➯♥
✭①❡♠ ❬✶❪ tr❛♥❣ ✾✺✮✳
❧➔ ❦❤æ♥❣ ❣✐❛♥
lp
❤♦➦❝
Lp (Ω)✱
t❤➻ t❛ ❝â
1
(1 + τ p )1/p − 1 < τ p , 1 < p < 2,
p
ρE (τ ) =
p
−
1
p−1 2
τ 2 + o(τ 2 ) <
τ , p ≥ 2.
2
2
✣à♥❤ ❧➼ ❞÷î✐ ✤➙② ❝❤♦ t❛ ❜✐➳t ✈➲ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♠æ ✤✉♥ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤
E
✈î✐ ♠æ ✤✉♥ ❧ç✐ ❝õ❛
✣à♥❤ ❧þ ✶✳✷✳
❛✮
❜✮
E∗
✭①❡♠ ❬✻❪ tr❛♥❣ ✼✵✮
✈➔ ♥❣÷ñ❝ ❧↕✐✳
❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛ ❝â
τε
− δE (ε) : ε ∈ [0, 2]}, τ > 0✳
2
τε
ρE (τ ) = sup{ − δE ∗ (ε) : ε ∈ [0, 2]}, τ > 0✳
2
ρE ∗ (τ ) = sup{
◆❤➟♥ ①➨t ✶✳✸✳
❚ø ✣à♥❤ ❧➼ ✶✳✷✱ s✉② r❛
tr♦♥❣ ✤â
ε0 (E ∗ )
2
ε0 (E)
,
2
ρE (τ )
ε0 (E) = sup{ε : δE (ε) = 0}, ρ0 (E) = limτ →0
.
τ
ρ0 (E) =
✈➔
ρ0 (E ∗ ) =
✣à♥❤ ♥❣❤➽❛ ✶✳✽✳
✾
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
E
✤÷ñ❝ ❣å✐ ❧➔ trì♥ ✤➲✉ ♥➳✉
ρE (τ )
= 0.
τ →0
τ
lim
❚ø ◆❤➟♥ ①➨t ✶✳✸✱ t❛ ❝â ✤à♥❤ ❧þ ❞÷î✐ ✤➙②✿
✣à♥❤ ❧þ ✶✳✸✳
❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛
✭①❡♠ ❬✻❪ tr❛♥❣ ✼✵✮
❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿
❛✮
◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ trì♥ ✤➲✉ t❤➻ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤➲✉❀
❜✮
◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤➲✉ t❤➻ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ trì♥ ✤➲✉✳
❱➼ ❞ö ✶✳✹✳
▼å✐
1 < p < +∞
✤➲✉ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧ç✐ ✤➲✉ ✈➔ trì♥ ✤➲✉ ✭①❡♠ ❬✺❪ tr❛♥❣
❦❤æ♥❣
❣✐❛♥
❍✐❧❜❡rt✱
❦❤æ♥❣
lp
❣✐❛♥
Lp (Ω)
❤❛②
✈î✐
✺✹✮✳
❈✉è✐ ❝ò♥❣ tr♦♥❣ ♠ö❝ ♥➔② ❧✉➟♥ ✈➠♥ ❣✐î✐ t❤✐➺✉ ✈➲ ❣✐î✐ ❤↕♥ ❝õ❛ ❞➣② t➟♣ ❤ñ♣
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤❡♦ ♥❣❤➽❛ ❝õ❛ ▼♦s❝♦ ❬✾❪✳
{Cn }
❈❤♦
❧➔ ♠ët ❞➣② ❝→❝ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ♣❤↔♥ ①↕
x∈
s✲▲✐n Cn ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ ❞➣②
✈î✐ ♠å✐
❞➣②
E ✳ ❚❛ ①→❝ ✤à♥❤ ❝→❝ t➟♣ ❝♦♥ s✲▲✐n Cn
n ≥ 1❀ x ∈
{yk } ⊂ E
✇✲▲sn Cn
= C0 ✱
E
♥❤÷ s❛✉✿
x
✈➔
x n ∈ Cn
❤ë✐ tö ♠↕♥❤ ✈➲
✇✲▲sn Cn ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ ❞➣② ❝♦♥
s❛♦ ❝❤♦
t❤➻
{xn } ⊂ E
✈➔ ✇✲▲sn Cn ❝õ❛
C0
yk
x
✈➔
yk ∈ Cnk
✈î✐ ♠å✐
✤÷ñ❝ ❣å✐ ❧➔ ❣✐î✐ ❤↕♥ ❝õ❛ ❞➣②
{Cnk }
k ≥ 1✳
{Cn }
❝õ❛{Cn } ✈➔
◆➳✉ s✲▲✐n Cn
=
t❤❡♦ ♥❣❤➽❛ ❝õ❛ ▼♦s❝♦
❬✾❪ ✈➔ ❣✐î✐ ❤↕♥ ♥➔② ✤÷ñ❝ ❦þ ❤✐➺✉ ❜ð✐
C0 = ▼✲ limn→∞ Cn ✳
❈❤ó þ ✶✳✸✳
❧➔ ♠ët ❞➣② ❣✐↔♠ ❝→❝ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ❝õ❛
❚❛ ❜✐➳t r➡♥❣✱ ♥➳✉
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕
❚❤➟t ✈➟②✱ rã r➔♥❣ ♥➳✉
✈î✐
xn = x
✈î✐ ♠å✐
n≥1
E
{Cn }
C 0 = ∩∞
n=1 Cn = ∅✱
✈➔
x ∈ C0
t❤➻
x∈
❤ë✐ tö ♠↕♥❤ ✈➲
t❤➻
s✲▲✐n Cn ✈➔
x✳
C0 = ▼✲ limn→∞ Cn ✳
x∈
❉♦ ✤â✱ t❛ ❝â
✇✲▲sn Cn ✱ ✈➻ ❞➣②
{xn }
C0 ⊂
C0 ⊂
s✲▲✐n Cn ✈➔
✇✲▲sn Cn ✳
❇➙② ❣✐í t❛ s➩ ❝❤➾ r❛ r➡♥❣
C0 ⊇
s✲▲✐n Cn ✈➔
tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ s✲▲✐n Cn ✱ tç♥ t↕✐ ❞➣②
xn → x✱
✈➔ ♠å✐
❦❤✐
n → ∞✳
k ≥ 0✳
✈î✐ ♠å✐
❱➻
❉♦ ✤â✱ ❝❤♦
n ≥ 1✳
❙✉② r❛
{Cn }
✇✲▲sn Cn ✳ ▲➜②
{xn } ⊂ E ✱ xn ∈ Cn
❧➔ ♠ët ❞➣② ❣✐↔♠✱ ♥➯♥
k→∞
x ∈ C0
C0 ⊇
✈➔ tø t➼♥❤ ✤â♥❣ ❝õ❛
✈➔ ❞♦ ✈➟②
C0 ⊇
✈î✐ ♠å✐
xn+k ∈ Cn
Cn ✱
x∈
s✲▲✐n Cn ✱
n ≥ 1 s❛♦ ❝❤♦
✈î✐ ♠å✐
t❛ ♥❤➟♥ ✤÷ñ❝
n≥1
x ∈ Cn
s✲▲✐n Cn ✳ ❚✐➳♣ t❤❡♦✱ ❧➜② ❜➜t ❦ý
y
sn Cn tứ ừ sn Cn tỗ t ởt
{yk } E
{Cn }
yk
s
x
yk C n k
ợ ồ
k 1
{Cnk }
ừ
ứ t ừ
t õ
yk+p Cnk
ợ ồ
k 1
k1
p 0
Cnk
õ tr
Ck Cnk
y Ck
ợ ồ
p
k 1
=
C nk
t ữủ
r
y C0
sn Cn
y C nk
õ
= C0
E
õ
ợ ồ
C0
ợ ồ
k 1
sn Cn
C0 = limn Cn
ố t
tr
ỗ õ
õ t t ữủ sn Cn
{Cn }
J : X 2X
X
ởt ổ t t
J(x) = {f X : x, f = x 2 , x = f }
ữủ ồ ố t ừ
ú ỵ
r ổ rt ố t trũ ợ
ỗ t
I
ố t
J
ỡ tr t t ỵ õ
t
J(x) =
X
õ ởt tr
x X
j
r ổ t t t
ợ ồ
J
X
t ổ õ
s r trỹ t tứ q ừ ỵ
ữợ ởt số t t ỡ ừ ố
t
J
ừ ổ t t
tr
X
X ởt ổ t t
J ố t ừ õ õ
J ởt tự J(x) = J(x), x X
J t t ữỡ tự J(x) = J(x), > 0, x X
✶✶
✐✐✐✮
J ❜à ❝❤➦♥✱ tù❝ ❧➔ ♥➳✉ D ❧➔ ♠ët t➟♣ ❝♦♥ ❜à ❝❤➦♥ ❝õ❛ X t❤➻ J(D) ❧➔ ♠ët t➟♣
❤ñ♣ ❜à ❝❤➦♥ tr♦♥❣ X ∗ ❀
✐✈✮
✈✮
◆➳✉ X ∗ ❧➔ ❧ç✐ ❝❤➦t t❤➻ J ❧➔ ✤ì♥ trà❀
J ❧➔ ✤ì♥ trà ✈➔ ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ ♠é✐ t➟♣ ❝♦♥ ❜à ❝❤➦♥ ❝õ❛ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉✳
❱➼ ❞ö ✶✳✺✳
❣✐❛♥
lp
❳➨t ❦❤æ♥❣ ❣✐❛♥
lp ✱
✈î✐
p > 1✳
❱➻ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉
❧➔ ❧ç✐ ✤➲✉✱ ♥➯♥ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝
J
❝õ❛
lp
lq
❝õ❛ ❦❤æ♥❣
❧➔ ✤ì♥ trà ✈➔ ❞➵ t❤➜②
♥â ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
J(x) =
θ
♥➳✉
x=θ
{ηn } ∈ lq
tr♦♥❣ ✤â
ηk = |ξk |p−1 s❣♥(ξk ) x
2−p
♥➳✉
✈î✐ ♠å✐
x = {ξn } = θ,
k ≥ 1✳
▼➺♥❤ ✤➲ ✶✳✽✳ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐ ✤â✱ t❛
❝â
❛✮
x+y
2
≤ y
2
+ 2 x, j(x + y) , ✈î✐ ♠å✐ j(x + y) ∈ J(x + y)✱
❜✮
x+y
2
≥ x
2
+ 2 y, j(x) , ✈î✐ ♠å✐ j(x) ∈ J(x)✱
✈î✐ ♠å✐ x, y ∈ E
❈❤ù♥❣ ♠✐♥❤✳
❚r÷î❝ ❤➳t✱ t❛ ❝❤➾ r❛
y
✈î✐ ♠å✐
2
− x
2
≥ 2 y − x, j(x) ,
✭✶✳✸✮
x, y ∈ E ✳
❚❤➟t ✈➟②✱ t❛ ❝â
y
2
− x
2
− 2 y − x, j(x) = x
2
+ y
2
− 2 y, j(x)
≥ x
2
+ y
2
−2 x
y
= ( x − y )2 ≥ 0.
❙✉② r❛✱ ✭✶✳✸✮ ✤ó♥❣✳
❛✮ ❚r♦♥❣ ✭✶✳✸✮ t❤❛②
x
❜ð✐
x + y✱
t❛ ♥❤➟♥ ✤÷ñ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❜✮ ❚r♦♥❣ ✭✶✳✸✮ t❤❛②
y
❜ð✐
x + y✱
t❛ ♥❤➟♥ ✤÷ñ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
E ởt ổ trỡ õ
x y, j(x)
j(y) 0 ợ ồ x, y E ỡ ỳ E ổ ỗ t x
y, j(x) j(y) = 0 t x = y
ự
ợ ồ
x, y E
t õ
x y, j(x) j(y) = x
2
x, j(y) y, j(x) + y
x
2
2 x
y + y
2
2
= ( x y )2 0.
õ t ữủ
x y, j(x) j(y) 0
ợ ồ
x, y E
sỷ
E
ổ ỗ t
x y, j(x) j(y) = 0
õ
tứ tr t ữủ
x, j(y) = y, j(x) = x
õ
x = 0
t
y=0
2
ữủ sỷ
= y 2.
x = y = d > 0
õ t
õ
t
x
y
, j(x) = , j(x) = j(x) .
d
d
x
y
ữủ
= x = y
d
d
s > 0 E ởt ổ õ E ỗ
tỗ t ởt ỗ tử t t g : [0, ) [0, )
g(0) = 0 s
x+y
ợ ồ x, y {z E :
g
x
2
+ 2 y, j(x) + g( y )
z s} ồ j(x) J(x)
õ ữợ ừ
2
g : X (, ]
t
x0
ỵ
ởt ỗ
g(x0 )
x0 dom(g)
ữủ
g(x0 ) = {f X : g(x) g(x0 ) x x0 , f }.
õ
g
ữợ t
x0
g(x0 ) = .
✶✸
❱➼ ❞ö ✶✳✻✳
♠å✐
x ∈ X✳
❈❤♦
X
❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✱
f ∈ ∂g(0)
y
λy
❜ð✐
✈î✐
λ > 0✱
λ → 0✱
y, f ≥ 0✳
●✐↔ sû
t❛ ♥❤➟♥ ✤÷ñ❝
❙✉② r❛✱
x = 0✱
2
2
= f
y, f ≤ 0
}, x = 0.
y, f = 0
✈î✐ ♠å✐
y ∈ X✳
❚❤❛②
y ∈ X✳
❉♦ ✤â✱
f = 0✳
✈î✐ ♠å✐
y
❜ð✐
❱➟②
−y
t❛ t❤✉ ✤÷ñ❝
∂g(0) = {0}✳
❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷ñ❝ r➡♥❣
f ∈ X∗
2
y − x, f = y, f − x
2
t❤ä❛ ♠➣♥
2
= f
= f
2
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû
f ∈ ∂g(x)✳
} ⊂ ∂g(x).
2
y ∈ X✱
t❛ ❝â
= g(y) − g(x).
❑❤✐ ✤â✱ t❛ ❝â
1
y − x, f ≤ ( y
2
❚❤❛②
2
✳ ❑❤✐ ✤â✱ ✈î✐ ♠å✐
≤ y . x − x 2
1
≤ ( y 2 + x 2) − x
2
y ∈ X✳
2
≥ y, f , ∀y ∈ X.
x
❚❤➟t ✈➟②✱ ❣✐↔ sû
2
≥ y, f , ∀y ∈ X.
{f ∈ X ∗ : x, f = x
✈î✐ ♠å✐
✈î✐
t❛ ♥❤➟♥ ✤÷ñ❝
λ
y
2
❈❤♦
2
❦❤✐ ✈➔ ❝❤➾ ❦❤✐
1
y
2
❚❤❛②
1
x
2
❑❤✐ ✤â✱
0, x = 0,
∂g(x) =
{f ∈ X ∗ : x, f = x
❚❤➟t ✈➟②✱
g(x) =
y = x + λz
✈î✐
λ∈R
2
− x 2)
✈➔
z ∈ X✱
t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✽ ❛✮✱ t❛
♥❤➟♥ ✤÷ñ❝
1
λ z, f ≤ ( x + λz
2
❑❤✐
λ > 0✱
2
1
− x 2 ) ≤ (λ2 z
2
2
+ 2|λ| x
tø ✭✶✳✹✮✱ t❛ ♥❤➟♥ ✤÷ñ❝
1
z, f ≤ (λ xz
2
2
+ 2 x z ).
z ).
✭✶✳✹✮
0+
x z
t t ữủ
ợ ồ
z X
ợ
z, f x
z = x
z
ợ ồ
z X
r
| z, f |
t ữủ
| x, f | x 2 , f x .
r t tự t ừ ợ
x, f
0
x=z
< 0
t ữủ
+2
x 2.
2
t ữủ
x, f x 2 .
ứ t ữủ
x, f = x
2
= f
2
.
ứ ử t õ ữợ
X ởt ổ t t g(x) = 12
x
2
ợ ồ x X õ J(x) = g(x)
ú ỵ
ứ
1
g(x) ợ g(x) =
x 2 ợ
2
t J ỡ tr
E
ồ
ởt ổ trỡ t
x E
J(x) =
tr trữớ ủ ố
P tr tờ qt
P tr
sỷ C ởt t ỗ õ rộ ừ ổ
ỗ t E õ t C 0 = x C : x = inf{ y : y
C} ỗ t ởt tỷ
ự
xn d
r
t
n
{xnk } {xn }
x C
d = inf{ y : y C}
õ tỗ t
ứ t ừ
s
xnk
x
{xn }
{xn } C
tỗ t
ứ t õ ừ
C
s
õ tứ t ỷ tử ữợ ừ t õ
x lim xn = d.
n
s
✶✺
x = d = inf{ y : y ∈ C}
❙✉② r❛
❤❛②
x ∈ C 0✳
❚❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t✳ ●✐↔ sû tç♥ t↕✐
C✱
❝❤➦t ❝õ❛
t❛ ❝â
tx + (1 − t)y < d
✈î✐ ♠å✐
y=x
t ∈ (0, 1)✱
✈➔
y ∈ C 0✳
❚ø t➼♥❤ ❧ç✐
✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐
d = inf{ y : y ∈ C}✳
❍➺ q✉↔ ✶✳✶✳ ●✐↔ sû C ❧➔ ♠ët t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ❧ç✐ ❝❤➦t ✈➔ ♣❤↔♥ ①↕ E ✳ ❑❤✐ ✤â✱ ✈î✐ ♠é✐ x ∈ E tç♥ t↕✐ ❞✉② ♥❤➜t ♣❤➛♥ tû
PC x ∈ C s❛♦ ❝❤♦
x − PC x = inf x − y .
y∈C
❈❤ù♥❣ ♠✐♥❤✳
⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✶✷ ❝❤♦ t➟♣
x−C
t❛ ♥❤➟♥ ✤÷ñ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
♠✐♥❤✳
❚ø ❍➺ q✉↔ ✶✳✶✱ ♥➳✉
C
❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ❝❤➦t
❧➔ ♠ët t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
E✱
t❤➻ t❛ ❝â →♥❤ ①↕
PC : E −→ C
①→❝ ✤à♥❤ ❜ð✐
x − PC x = inf x − y ,
y∈C
✈î✐ ♠å✐
x ∈ E✳
⑩♥❤ ①↕
PC
♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø
✣➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝
PC
E
❧➯♥
C✳
✤÷ñ❝ ❝❤♦ ❜ð✐ ♠➺♥❤ ✤➲ ❞÷î✐ ✤➙②✳
▼➺♥❤ ✤➲ ✶✳✶✸✳ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ❝❤➦t ✈➔ trì♥✳
❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ E ✱ x ∈ E ✈➔ z ∈ C ✳ ❑❤✐ ✤â✱
❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮
❜✮
z = PC x❀
y − z, j(x − z) ≤ 0 ✈î✐ ♠å✐ y ∈ C ✳
❈❤ù♥❣ ♠✐♥❤✳
●✐↔ sû ❜✮ ✤ó♥❣✱ ❦❤✐ ✤â t❛ ❝â
(y − x) − (x − z), j(x − z) ≤ 0.
❙✉② r❛
x−z
✈î✐ ♠å✐
y ∈ C✳
z = PC x✳
2
≤ y − x, j(x − z) ≤ y − x
❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷ñ❝
x−z ,
x−z ≤ x−y
✈î✐ ♠å✐
y ∈ C
❤❛②
ữủ sỷ
z = PC x
tự
1
xz
2
ợ ồ
y C
C
t ỗ
2
y, z C
1
x y 2,
2
ty + (1 t)z C
ợ ồ
t (0, 1)
õ t õ
ợ ồ
yC
1
1
x z 2 x [ty + (1 t)z] 2 ,
2
2
ồ t (0, 1)
õ tứ ú ỵ t ữủ
0 x z [x ty + (1 t)z], j(x [ty + (1 t)z) ,
ợ ồ
yC
ồ
t (0, 1)
r
y z, j(x [ty + (1 t)z) 0.
t 0+
t ữủ
y z, j(x z) 0.
ữủ ự
P tờ qt
E
ởt ổ trỡ t
: E ì E R
(x, y) = x
ợ ồ
2
2 x, j(y) + y 2 ,
x, y E
t
ứ ừ
t õ
( x y 2 ) (x, y)
ợ ồ
x, y E
ợ ộ
y
ố t
(x, y)
ỗ t
x
E ởt ổ ỗ trỡ {yn}
{zn } tr E (yn , zn ) 0 {yn } {zn } t
yn zn 0
✶✼
❈❤ù♥❣ ♠✐♥❤✳
❚ø
φ(yn , zn ) → 0
{yn }
❣✐↔ t❤✐➳t s✉② r❛ ❝→❝ ❞➣②
✈➔
s✉② r❛
{zn }
{φ(yn , zn )}
❝ò♥❣ ❜à ❝❤➦♥✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✵✱ tç♥ t↕✐
g : [0, ∞) −→ [0, ∞)✱ g(0) = 0
❤➔♠ ❧ç✐✱ ❧✐➯♥ tö❝ ✈➔ t➠♥❣ ♥❣➦t
g( yn − zn ) ≤ zn + (yn − zn )
= yn
❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ tø ✭✶✳✼✮ ✈➔
2
2
− zn
2
2
− zn
s❛♦ ❝❤♦
− 2 yn − zn , j(zn )
− 2 yn , j(zn ) + 2 zn
2
= φ(yn , zn ).
❉♦ ✤â✱ tø
φ(yn , zn ) → 0
s✉② r❛
g✱
t❛ ♥❤➟♥ ✤÷ñ❝
yn − zn → 0✳
tö❝ ❝õ❛
g( yn − zn ) → 0✳
❚ø t➼♥❤ t➠♥❣ ♥❣➦t ✈➔ t➼♥❤ ❧✐➯♥
▼➺♥❤ ✤➲ ✶✳✶✺✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ❝❤➦t ✈➔ trì♥✳ ❈❤♦ C
❧➔ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ E ✈➔ ❝❤♦ x ∈ E ✳ ❑❤✐ ✤â✱ tç♥ t↕✐ ❞✉② ♥❤➜t
♣❤➛♥ tû x0 ∈ C s❛♦ ❝❤♦
φ(x0 , x) = inf{φ(z, x) : z ∈ C}.
❈❤ù♥❣ ♠✐♥❤✳
❱➻
E
zn → ∞
❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ ✈➔ ♥➳✉
✭◆❤➟♥ ①➨t ✶✳✺✮✱ ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸✱ tç♥ t↕✐
✭✶✳✽✮
x0 ∈ C
t❤➻
φ(zn , x) → ∞
s❛♦ ❝❤♦
φ(x0 , x) = inf{φ(z, x) : z ∈ C}.
❱➻
E
❧➔ ❧ç✐ ❝❤➦t ♥➯♥
t ∈ (0, 1)✱
✈➔ ♠å✐
.
2
❧➔ ❤➔♠ ❧ç✐ ❝❤➦t✳ ❉♦ ✤â ✈î✐ ♠å✐
φ(., x)
✈î✐
x1 = x2
t❛ ❝â
tx1 + (1 − t)x2
❙✉② r❛
x1 , x2 ∈ E
2
< t x1
2
❧➔ ❤➔♠ ❧ç✐ ❝❤➦t✳ ❉♦ ✈➟② ♣❤➛♥ tû
❚ø ▼➺♥❤ ✤➲ ✶✳✶✺✱ ♥➳✉
C
+ (1 − t) x2 2 .
x0
❧➔ ❞✉② ♥❤➜t✳
❧➔ ♠ët t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ❝❤➦t
E✱
t❤➻ t❛ ❝â →♥❤ ①↕
ΠC : E −→ C
①→❝ ✤à♥❤ ❜ð✐
φ(ΠC x, x) = inf φ(y, x),
y∈C
✈î✐ ♠å✐
x ∈ E✳
◆❤➟♥ ①➨t ✶✳✻✳
⑩♥❤ ①↕
ΠC
♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t tø
E
❧➯♥
C✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✈➔ ♣❤➨♣ ❝❤✐➳✉ tê♥❣
q✉→t trò♥❣ ♥❤❛✉✳
✶✽
▼➺♥❤ ✤➲ ✶✳✶✻✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ❝❤➦t ✈➔ trì♥✳ ❈❤♦ C
❧➔ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ E ✈➔ ❝❤♦ x ∈ E ✱ x0 ∈ C ✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣
✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮
x0 = ΠC x❀
z − x0 , j(x0 ) − j(x) ≥ 0 ✈î✐ ♠å✐ z ∈ C ✳
❜✮
❈❤ù♥❣ ♠✐♥❤✳
●✐↔ sû
z∈C
t ∈ (0, 1)✳
✈➔ ♠å✐
x0 = ΠC x✳
❱➻
C
❧➔ t➟♣ ❧ç✐ ♥➯♥
tz + (1 − t)x0 ∈ C
✈î✐ ♠å✐
❉♦ ✤â✱ t❛ ❝â
ϕ(x0 , x) ≤ φ(tz + (1 − t)x0 , x),
✈î✐ ♠å✐
z∈C
✈➔ ♠å✐
t ∈ (0, 1)✳
0 ≤ tz + (1 − t)x0
− x0
2
❚ø ✭✶✳✸✮✱ s✉② r❛
2
− 2 tz + (1 − t)x0 , j(x) + x
+ 2 x0 , j(x) − x
= tz + (1 − t)x0
2
− x0
2
2
2
− 2t z − x0 , j(x)
≤ 2t z − x0 , j(tz + (1 − t)x0 ) − 2t z − x0 , j(x)
= 2t z − x0 , j(tz + (1 − t)x0 ) − j(x) .
❈❤♦
t → 0+ ✱
t❛ ♥❤➟♥ ✤÷ñ❝
z − x0 , j(x0 ) − j(x) ≥ 0
✈î✐ ♠å✐
z ∈ C✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû ❜✮ ✤ó♥❣✳ ❑❤✐ ✤â✱ ✈î✐ ♠å✐
φ(z, x) − φ(x0 , x) = z
2
− 2 z, j(x) + x
= z
2
− x0
2
z ∈ C✱
2
tø ✭✶✳✸✮✱ t❛ ❝â
− x0
2
− 2 z − x0 , j(x)
≥ 2 z − x0 , j(x0 ) − 2 z − x0 , j(x)
= 2 z − x0 , j(x0 ) − j(x)
≥ 0.
❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷ñ❝
x0 = ΠC x✳
+ 2 x0 , j(x) − x
2
tỷ ỡ tr ổ
E
A : D(A) E 2E
ởt
ổ
ữủ ồ ỡ ợ ồ
tỷ
x, y D(A)
t
ổ õ
x y, u v 0, u A(x), v A(y).
ởt t tỷ ỡ
ỡ ỹ ỗ t
A : D(A) E 2E
ữủ ồ
G(A) = {(u, x) : x D(A), u A(x)}
ừ õ
ổ tỹ sỹ ự tr ỗ t ừ ởt t tỷ ỡ tr
ử
tỷ
A(x) = x3
t
ừ
R
A
A
ợ
xR
ởt t tỷ ỡ tr
R
R
s r ỗ t
ổ t tỹ sỹ ừ t ý ởt t tỷ ỡ tr
sỷ tỗ t ởt t tỷ ỡ
tỹ sỹ ỗ t ừ
ữ
ỡ ỹ tr
E
A
B
tr
õ tỗ t tỷ
(x0 , m)
/ G(A)
R
s ỗ t ừ
x0 R
s
ữ s r trữớ ủ
B
ự
(x0 , m) G(B)
A(x0 ) > m
A(x1 ) = m
A(x0 ) < m
rữớ ủ
sỷ
õ
x1
A(x0 ) > m
ừ ữỡ tr
x1 < x0
A(x) = m
tự
ỵ tr tr tỗ t
n = A(x2 ) (m, A(x0 ))
ứ
(x0 , m) G(B)
x2 (x1 , x0 )
s
(x2 , A(x2 )) G(A) G(B)
s
r
(x0 x2 )(m A(x2 )) 0.
x0 > x2
A(x2 ) m
t ợ
ổ t r trữớ ủ
rữớ ủ
sỷ
õ
x1
x1 > x0
A(x2 ) (m, A(x0 ))
ữ
A(x0 ) > m
A(x0 ) < m
ừ ữỡ tr
A(x) = m
tự
ỵ tr tr tỗ t
n = A(x2 ) (A(x0 ), m)
ứ
(x0 , m) G(B)
A(x1 ) = m
x2 (x0 , x1 )
s
(x2 , A(x2 )) G(A) G(B)
r
(x0 x2 )(m A(x2 )) 0.
s
x0 < x2
A(x2 ) m
t ợ
ổ t r trữớ ủ
ử
A
õ
A
ữ
A(x0 ) < m
ổ tỗ t t tỷ ỡ
sỹ ỗ t ừ
A(x2 ) (A(x0 ), m)
B
tr
R s ỗ t ừ B
ởt t tỷ ỡ ỹ tr
f : E R
ự tỹ
R
ởt ỗ tữớ ỷ tử
ữợ õ t tỷ ữợ
f (x) = {x E : f (y) f (x) y x, x , y E}
ởt t tỷ ỡ ỹ
ỵ ữợ t ởt ừ t tỷ ỡ
A
ỡ ỹ
ỵ
E ởt ổ ỗ trỡ A :
E 2E ởt t tỷ ỡ õ ừ A t tỷ
ỡ ỹ R(j + rA) = E ợ ồ r > 0
ứ ỵ s r
ợ ộ
xE
r>0
E
ởt ổ ỗ trỡ t
tỗ t t tỷ
xr
yr
s
0
j(xr x) + rAxr ;
0
j(yr ) j(x) + rAyr .
t sỹ tỗ t ừ tỷ
xr
yr
ữủ s r tứ ỵ
r t t
sỷ tỗ t
uE
s
0
j(u x) + rAu.
õ t õ
õ tứ t ỡ
1
1
j(x u) Au, j(x xr ) Axr .
r
r
ừ A s r
u xr , j(x u) j(x xr ) 0.
tữỡ ữỡ ợ
(x u) (x xr ), j(x u) j(x xr ) 0.