Giáo trình giải tích hàm

ˆ. GIAO
´ DU
` ¯DAO
` TA
BO
. C VA
.O
`’
ˆ` THO’
TRU’ONG
¯DA
. I HO
. C CAN

Gi´
ao tr`ınh

’ T´ICH HAM
`
GIAI
(Functional Analysis)

˜’ Kh´
anh
Biˆ
en soa.n: Ts. Nguyˆ
e˜n Huu
Ts. Lˆ
e Thanh T`
ung
Bˆo. mˆon To´an

Khoa Khoa ho.c Tu.’ nhiˆen
`’ ¯Da.i ho.c Cˆa`n Tho’
Tru’ ong

2013


`’ NOI
`
´ ¯DAU
ˆ
LOI
’ t´ıch h`am d¯u’o.’c viˆe´t cho ho.c viˆen cao ho.c ng`anh To´an. Gi´ao tr`ınh
Gi´ao tr`ınh Giai
´’ co’ ban
’ vˆe` Giai
’ t´ıch h`am nhu’ khˆong gian d¯.inh chuˆa’n, khˆong
cung cˆa´p c´ac kiˆe´n thuc
´’ nhˆa.n d¯u’o.’c, ngu’oi
’’ Tu
`’ c´ac kiˆe´n thuc
`’ d¯o.c c´o thˆe’ tu.’
gian Hilbert v`a l´
y thuyˆe´t to´an tu.
´’ c´ac chuyˆen d¯ˆe`kh´ac cua
’ Giai
’ t´ıch h`am nhu’ khˆong gian vecto’ tˆopˆo, khˆong
nghiˆen cuu
’ t´ıch phi tuyˆe´n... Gi´ao tr`ınh c´o thˆe’ d`
gian Sobolev, d¯a.i sˆo´ Banach, Giai

ung nhu’ t`ai
’ cho sinh viˆen d¯a.i ho.c.
liˆe.u tham khao
´’ vˆe` khˆong gian d¯.inh
Gi´ao tr`ınh gˆo`m 5 chu’ong.
Chu’ong
’
’ 1 tr`ınh b`ay c´ac kiˆe´n thuc
´’ thiˆe.u c´ac khˆong gian Lp . Chu’ong
chuˆa’n. Chu’ong
y co’
’ 2 gioi
’ 3 d¯ˆe`cˆa.p vˆe`c´ac nguyˆen l´
´’ vˆe` khˆong gian Hilbert. Chu’ong
’ cua
’ Giai
’ t´ıch h`am. Chu’ong
ban
’ 5 x´et
’ 4 nghiˆen cuu
’’
l´
y thuyˆe´t vˆe`c´ac to´an tu.
`’ dˆe˜ d¯ˆe´n kh´o v`a d¯u’o.’c
C´ac vˆa´n d¯ˆe` trong gi´ao tr`ınh d¯u’o.’c tr`ınh b`ay c´o hˆe. thˆo´ng tu
´’ minh d¯u’o.’c tr`ınh b`ay chi tiˆe´t, dˆe˜ hiˆe’u.
minh ho.a ba˘`ng c´ac v´ı du. cu. thˆe’. C´ac chung
´’ vˆa.n du.ng v`ao
`’ ho.c cung
’

Sau mˆo˜i chu’ong
up ngu’oi
cˆo´ kiˆe´n thuc,
’ c´o hˆe. thˆo´ng b`ai tˆa.p gi´
´’ Mˆo.t sˆo´ hu’ong
´’ dˆa˜n th´ıch ho.’p cho b`ai tˆa.p gi´
`’ d¯o.c d¯.inh
c´ac t`ınh huˆo´ng moi.
up ngu’oi
´’ d¯u’o.’c c´ach giai.
’
hu’ong
Trong qu´a tr`ınh biˆen soa.n khˆo’ng thˆe’ tr´anh d¯u’o.’c thiˆe´u s´ot. T´ac gia’ rˆa´t mong nhˆa.n
`’ ngu’oi
`’ d¯o.c vˆe`gi´ao tr`ınh n`ay.
d¯u’o.’c y
´ kiˆe´n d¯o´ng g´op qu´
y b´au tu


99

MU
. C LU
.C
`’ n´oi d¯ˆa`u
Loi
Chu’ong
ong gian d
¯.inh chuˆ

a’n
’ 1. Khˆ
§1. Khˆong gian tuyˆe´n t´ınh
§2. ¯Da.i cu’ong
’ vˆe`khˆong gian d¯.inh chuˆa’n
§3. Chuˆo˜i trong khˆong gian d¯.inh chuˆa’n

12

§4. Khˆong gian con

14

§5. Khˆong gian thu’ong
’

16

§6 T´ıch c´ac khˆong gian d¯.inh chuˆa’n
§7. To´an tu’’ tuyˆe´n t´ınh liˆen tu.c

17

˜’ ha.n chiˆe`u
§8 Khˆong gian d¯i.nh chuˆa’n huu

26

B`ai tˆa.p

28

1
7

19

Chu’ong
ac khˆ
ong gian Lp
’ 2. C´
´’
§1. C´ac bˆa´t d¯a˘’ ng thuc

33

§2. Khˆong gian Lp (X)

35

§3. Khˆong gian lp

38

§4. To´an tu’’ t´ıch phˆan trong khˆong gian LP [a, b]

38

§5. T´ıch chˆa.p

40

B`ai tˆa.p

41
’ giai
’ t´ıch h`
˘’ n cua
Chu’ong
ac nguyˆ
en l´ı co’ ba
am
’ 3. C´

§1. ¯Di.nh l´ı Hanh-Banach
§2. Nguyˆen l´ı bi. ch˘
a.n d¯ˆe`u

43
48

§3. Nguyˆen l´ı ´anh xa. mo’’
§4. ¯Di.nh l´ı d¯ˆo` thi. d¯o´ng

50

B`ai tˆa.p

53

52

Chu’ong
ong gian Hilbert
’ 4. Khˆ
§1. Kh´ai niˆe.m vˆe`khˆong gian Hilbert
§2. T´ınh tru.’c giao v`a h`ınh chiˆe´u

55
60


100

§3. Hˆe. tru.’c chuˆa’n
§4. Phiˆe´m h`am tuyˆe´n t´ınh v`a song tuyˆe´n t´ınh trong khˆong gian Hilbert

65

B`ai tˆa.p

75

72

Chu’ong
y thuyˆ
e´t to´
an tu’’
’ 5. L´

§1. To´an tu’’ liˆen ho.’p
§2. To´an tu’’ tu.’ liˆen ho.’p
§3. To´an tu’’ compact

79

’ to´an tu’’ liˆen tu.c
§4. Phˆo’ cua

85

B`ai tˆa.p

95

’
T`ai liˆe.u tham khao
Mu.c lu.c

98

81
83

99


Chu’ong
1
’

ˆ
ˆ’
KHONG
GIAN ¯DI.NH CHUAN
§1.

Khˆ
ong gian tuyˆ
e´n t´ınh

´’ co’ ban
’ vˆe`khˆong gian tuyˆe´n t´ınh.
Phˆa`n n`ay nha˘´c la.i mˆo.t c´ac kiˆe´n thuc

1.1
C.

Kh´
ai niˆ
e.m vˆ
e` khˆ
ong gian tuyˆ
e´n t´ınh

´’
`’ c´ac sˆo´ thu.’c R ho˘
`’ c´ac sˆo´ phuc
Trong gi´ao tr`ınh n`ay ta k´ı hiˆe.u K l`a tru’ong
a.c tru’ong

1.1.1 Khˆ
ong gian tuyˆ
e´n t´ınh
´’ ph´ep cˆo.ng + : X ×X →
`’ K l`a tˆa.p X c`
Khˆong gian tuyˆe´n t´ınh X trˆen tru’ong
ung voi
´
`
’
X v`a ph´ep nhˆan vˆo hu’ong
’ · : K × X → X thoa c´ac d¯iˆeu kiˆe.n
i) (x + y) + z = x + (y + z), ∀x, y, z ∈ X
ii) x + y = y + x, ∀x, y ∈ X
iii) ∃θ ∈ X : x + θ = x, ∀x ∈ X
iv) ∀x ∈ X, ∃ − x ∈ X : x + (−x) = θ;
v) α(x + y) = αx + αy, ∀α ∈ K, ∀x, y ∈ X
vi) (α + β)x = αx + βx, ∀αβ ∈ K, ∀x ∈ X
vii) (αβ)x = α(βx) ∀αβ ∈ K, ∀x ∈ X
viii) 1.x = x, ∀x ∈ X .
’ khˆong gian tuyˆe´n t´ınh d¯u’o.’c go.i l`a c´ac vecto.’
C´ac phˆa`n tu’’ cua
Nˆe´u K = R th`ı X l`a khˆong gian tuyˆe´n t´ınh thu.’c, nˆe´u K = C th`ı X l`a khˆong gian
´’
tuyˆe´n t´ınh phuc.
1


Chuong
’ ’ 1. Khˆ

ong gian d
¯.inh chuˆ
a’n

2

´’ c´ac ph´ep to´an cˆo.ng v`a nhˆan vˆo hu’ong
´’
• V´ı du. 1 Tˆa.p ho.’p Rn voi
x + y = (x1 + y1 , . . . , xn + yn )
αx = (αx1 , . . . , αxn )
trong d¯o´ α ∈ R, x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn l`a mˆo.t khˆong gian tuyˆe´n
t´ınh.
´’ hai
• V´ı du. 2 Tˆa.p ho.’p C([a, b]) = { f : [a, b] → R : f liˆen tu.c trˆen [a, b] } c`
ung voi
ph´ep to´an
(f + g)(x) = f (x) + g(x); ∀x ∈ [a, b]
(αf )(x)

= αf (x), ∀α ∈ R, ∀x ∈ [a, b]

trong d¯´o f, g ∈ C([a, b]), l`a mˆo.t khˆong gian tuyˆe´n t´ınh.
• V´ı du. 3 Tˆa.p ho.’p l2 = {x = (x1 , x2 , . . . , xn , . . .) : xn ∈ R,
´’ hai ph´ep to´an
voi

∑∞
n=1

|xn |2 < ∞} c`
ung

(xn ) + (yn ) = (xn + yn ),
α(xn ) = (αxn ),
´’ (xn ), (yn ) ∈ l2 , α ∈ K, l`a mˆo.t khˆong gian tuyˆe´n t´ınh.
voi
´’ ph´ep cˆo.ng d¯a thuc,
´’
´’ mˆo.t biˆe´n thu.’c trˆen R, voi
• V´ı du. 4 Tˆa.p ho.’p P c´ac d¯a thuc
´
´
´
`’ l`a mˆo.t khˆong gian
ph´ep nhˆan mˆo.t sˆo voi’ d¯a thuc
’ x´ac d¯.inh theo c´ach thˆong thu’ong
´
tuyˆen t´ınh.
1.1.2 ¯Dˆ
o.c lˆ
a.p tuyˆ
e´n t´ınh
Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh v`a x1 , x2 , . . . , xn ∈ X.
Tˆo’ng

n
∑

’ c´ac vecto’

αi xi , trong d¯o´ αi ∈ K, d¯u’o.’c go.i l`a mˆo.t tˆo’ ho.’p tuyˆe´n t´ınh cua

i=1

x1 , x2 , . . . , xn .
Hˆe. {x1 , x2 , . . . , xn } d¯u’o.’c go.i l`a d¯ˆo.c lˆa.p tuyˆe´n t´ınh nˆe´u
0, ∀i = 1, n.

n
∑

αi xi = 0 th`ı αi =

i=1

Hˆe. {x1 , x2 , . . . , xn } d¯∑
u’o.’c go.i l`a phu. thuˆ
o.c tuyˆe´n t´ınh nˆe´u hˆe. khˆong d¯oˆ. c lˆa.p tuyˆe´n
∑
n
n
2
´’ l`a ∃α1 , . . . , αn ,
t´ınh (tuc
i=1 αi xi = 0).
i=1 |αi | ̸= 0:
˜’ ha.n c´ac phˆa`n tu’’ cua
’
Tˆa.p M ⊂ X d¯u’o.’c go.i l`a d¯oˆ. c lˆa.p tuyˆe´n t´ınh nˆe´u mo.i hˆe. huu
´

M d¯ˆe`u d¯ˆo.c lˆa.p tuyˆen t´ınh.


1. Khˆ
ong gian tuyˆ
e´n t´ınh

3

1.1.3 Co’ so’’
Cho X l`a mˆo.t khˆong gian tuyˆe´n t´ınh. Mˆo.t tˆa.p con B ⊂ X d¯u’o.’c go.i l`a mˆo.t co’ so’’
’ X nˆe´u:
cua
B l`a tˆa.p d¯oˆ. c lˆa.p tuyˆe´n t´ınh.
˜’ ha.n c´ac phˆa`n tu’’ cua
’ mˆo.t sˆo´ huu
’ B,
Mo.i x ∈ X d¯ˆe`u l`a tˆo’ ho.’p tuyˆe´n t´ınh cua
nghi˜a l`a
n
∑
∀x ∈ X, ∃α1 , . . . , αn ∈ K, ∃x1 , . . . , xn ∈ B : x =
αi xi .
i=1

’ B d¯u’o.’c go.i l`a sˆo´ chiˆe`u cua
’ khˆong gian tuyˆe´n t´ınh X. K´ı hiˆe.u
Sˆo´ phˆa`n tu’’ cua
´
’

`
`
˜
˜’
dimX. Nˆeu B gˆom huu
’ ha.n phˆan tu’ th`ı X d¯u’o.’c go.i l`a khˆong gian tuyˆe´n t´ınh huu
ha.n chiˆe`u.
’ khˆong gian tuyˆe´n t´ınh X khi v`a chi’ khi
∆ ¯Di.nh l´
y 1.1 Tˆa.p ∅ ̸= B ⊂ X l`a co’ so’’ cua
´
´
´
B l`
a tˆa.p ho.’p d¯ˆo.c lˆa.p tuyˆen t´ınh tˆoi d¯a.i (tuc
o.c lˆa.p tuyˆe´n t´ınh v`a nˆe´u M ⊃ B,
’ l`a B d¯ˆ
M ̸= B th`ı M phu. thuˆo.c tuyˆe´n t´ınh).
∆ ¯Di.nh l´
y 1.2 Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh n chiˆe`u v`a x1 , x2 , . . . , xn l`a mˆo.t
’ X. Khi d¯´o mo.i x ∈ X d¯ˆe`u d¯u’o.’c biˆe’u diˆe˜n duy nhˆa´t da.ng
co’ so’’ cua
n
∑
x=
αi xi .
i=1

1.2

C´
ac khˆ
ong gian tuyˆ
e´n t´ınh

1.2.1 Khˆ
ong gian con
✷ ¯Di.nh nghi˜a 1 Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh v`a Y ⊂ X. Y d¯u’o.’c go.i l`a khˆong
’ X nˆe´u ∀α, β ∈ K, ∀x, y ∈ Y ta c´o αx + βy ∈ Y .
gian con cua
• V´ı du. 5
i) Ta.p ho.’p l1 gˆo`m tˆa´t ca’ c´ac d˜
ay sˆo´ x = (xn ) sao cho
’ khˆong gian c´ac d˜
khˆong gian con cua
ay sˆo´.

∑∞
n=1

|xn | < ∞ l`a mˆo.t

ii) Tˆa.p ho.’p c´ac h`am sˆo´ liˆen tu.c trˆen d¯oa.n [a, b], k´ı hiˆe.u C[a,b] , l`a mˆo.t khˆong gian
’ khˆong gian c´ac h`am sˆo´ F([a, b]).
con cua
’ X th`ı ∩α Yα c˜
∆ ¯Di.nh l´
y 1.3 Nˆe´u {Yα } l`a ho. c´ac khˆong gian con cua
ung l`a khˆong
’ X.

gian con cua
’ ho. tˆa´t ca’
✷ ¯Di.nh nghi˜a 2 Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh v`a M ⊂ X. Giao cua
´
´
´’ M , d¯u’o.’c
’ X chua
’ X chua
c´ac khˆong gian con cua
’ M l`a khˆong gian con nho’ nhˆat cua
’’ M , k´ı hiˆe.u l`a L(M ) hay span(M ).
go.i l`a khˆong gian con sinh boi
Khi L(M ) = X ta n´oi M sinh ra X v`a M l`a tˆa.p c´ac phˆa`n tu’’ sinh.


4

Chuong
’ ’ 1. Khˆ
ong gian d
¯.inh chuˆ
a’n

˜’ ha.n c´ac phˆa`n tu’’ cua
’
∆ ¯Di.nh l´
y 1.4 L(M ) l`a tˆa.p tˆa´t ca’ c´ac tˆ
o’ ho.’p tuyˆe´n t´ınh huu
´
M , tuc

’ l`a
n
∑
L(M ) = {
αi xi : αi ∈ K, xi ∈ M, n ∈ N}.
i=1

1.2.2 Khˆ
ong gian thu’ ong
’
’ X. ¯Di.nh nghi˜a
Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh v`a Y l`a khˆong gian con cua
quan hˆe. ∼ trong X nhu’ sau:
∀x1 , x2 ∈ X, x1 ∼ x2 ⇔ x1 − x2 ∈ Y.
´’ tu’ong
Khi d¯o´ ∼ l`a quan hˆe. tu’ong
Tˆa.p thu’ong
’ d¯u’ong
’ x + Y l`a
’ d¯u’ong.
’
’ X/Y c´ac lop
´’ hai ph´ep to´an
mˆo.t khˆong gian tuyˆe´n t´ınh voi
(x + Y ) + (y + Y ) = (x + y) + Y
α(x + Y ) = αx + Y
1.2.3 T´ıch Descartes c´
ac khˆ
ong gian tuyˆ
e´n t´ınh

Cho ho. {Xα } c´ac khˆong gian tuyˆe´n t´ınh. ¯Da˘. t X =

∏

Xα . Trong X ta d¯.inh

α

´’ mo.i (xα )α , (yα )α ∈ X
nghi˜a hai ph´ep to´an nhu’ sau: voi
(xα ) + (yα ) = (xα + yα )
λ(xα ) = (λxα )
´’ hai ph´ep to´an l`a khˆong gian tuyˆe´n t´ınh, d¯u’o.’c go.i l`a t´ıch Descartes
X c`
ung voi
’ ho. c´ac khˆong gian Xα .
cua
1.2.4 Tˆ
o’ng c´
ac khˆ
ong gian tuyˆ
e´n t´ınh
✷ ¯Di.nh nghi˜a 3 Gia’ su’’ X l`a mˆo.t khˆong gian tuyˆe´n t´ınh v`a Y1 , Y2 , . . . , Yn l`a c´ac
’ X. Khi d¯o´
khˆong gian con cua
Y ={

n
∑

yi : yi ∈ Yi , i = 1, n}

i=1

´’ c´ac Yi , i = 1, n, d¯u’o.’c go.i l`a tˆo’ng cua
’ c´ac khˆong gian
l`a khˆong gian con nho’ nhˆa´t chua
n
∑
con Yi , i = 1, n. K´ı hiˆe.u Y =
Yi .
i=1

Nˆe´u Yi ∩ Yj = {0}, (i ̸= j) th`ı Y =

n
∑
i=1

’ c´ac khˆong
Yi d¯u’o.’c go.i l`a tˆo’ng tru.’c tiˆe´p cua

gian con Yi , i = 1, n. K´ı hiˆe.u Y = Y1 ⊕ Y2 ⊕ . . . ⊕ Yn .


1. Khˆ
ong gian tuyˆ
e´n t´ınh

5

´’ mo.i y ∈ Y d¯ˆe`u c´o su.’ biˆe’u
∆ ¯Di.nh l´
y 1.5 Y = Y1 ⊕ Y2 ⊕ . . . ⊕ Yn khi v`a chi’ khi voi
´’ yi ∈ Yi , i = 1, n.
diˆe˜n duy nhˆa´t da.ng y = y1 + y2 + . . . + yn voi
’ Z. Nˆe´u dim X < ∞
⊙ Ch´
uy
´ Nˆe´u X = Y ⊕ Z th`ı Y d¯u’o.’c go.i l`a phˆa`n b`
u d¯a.i sˆo´ cua
th`ı dim X = dim Y + dim Z.
˜’ ta c´o X/Y ≃ Z. Sˆo´ chiˆe`u cua
’ X/Y d¯u’o.’c go.i l`a sˆo´ d¯ˆo´i chiˆe`u cua
’ Y , k´ı
Hon
’ nua,
hiˆe.u l`a codim Y . Khi d¯´o ∞ > dim X = dim Y + codim Y .

1.3

’’ tuyˆ
To´
an t u
e´n t´ınh

1.3.1 To´
an tu’’ tuyˆ
e´n t´ınh
´

`’ K. Anh
✷ ¯Di.nh nghi˜a 4 Gia’ su’’ X, Y l`a khˆong gian tuyˆe´n t´ınh trˆen tru’ong
xa.
A : X → Y d¯u’o.’c go.i l`a to´an tu’’ tuyˆe´n t´ınh nˆe´u ∀α1 , α2 ∈ K, ∀x1 , x2 ∈ X ta c´o
A(α1 x1 + α2 x2 ) = α1 Ax1 + α2 Ax2 .
’ A,
KerA = A−1 (0) = {x ∈ X : Ax = 0} l`a ha.t nhˆan (ha.ch) cua
’ cua
’ A.
ImA = {Ax : x ∈ X} l`a anh
Nˆe´u A l`a song ´anh th`ı ta n´oi A l`a ph´ep d¯a˘’ ng cˆa´u tuyˆe´n t´ınh v`a X v`a Y l`a hai
´’ nhau.
khˆong gian tuyˆe´n t´ınh d¯a˘’ ng cˆa´u voi
’’
• V´ı du. 6 X´et A : Rk → Rm x´ac d¯.inh boi
´’ yi =
A(x1 , x2 , . . . , xk ) = (y1 , y2 , . . . , yn ) voi

k
∑

aij xj

(i = 1, . . . , m),

j=1

’ ma trˆa.n
trong d¯´o aij l`a hˆe. sˆo´ cua


a11 a12 . . . a1k
 a21 a22 . . . a2k


am1 am2 . . . amk






`’ Rk v`ao Rm .
Ta thˆa´y A l`a to´an tu’’ tuyˆe´n t´ınh tu
´
• V´ı du. 7 Cho C[a,b] = {x(t) : x(t) liˆen tu.c trˆen [a, b]}. Anh
xa. A : C[a,b] → C[a,b] cho
’’
boi
∫
b

Ax(t) =

K(t, s)x(s)ds,
a

´’ K(t, s) l`am h`am liˆen tu.c trˆen h`ınh vuˆong a ≤ t, s ≤ b, l`a to´an tu’’ tuyˆe´n t´ınh, d¯u’o.’c
voi
´’ ha.ch K(t, s).
go.i l`a to´an tu’’ t´ıch phˆan voi

Chuong
’ ’ 1. Khˆ
ong gian d
¯.inh chuˆ
a’n

6

• V´ı du. 8 Cho X l`a tˆa.p c´ac h`am liˆen tu.c v`a bi. ch˘
a.n trˆen R+ = {t ∈ R : t ≥ 0}.
’
To´an tu’
∫ ∞
Lx(s) =
e−st x(t)dt
0

l`a to´an tu’’ tuyˆe´n t´ınh, d¯u’o.’c go.i l`a ph´ep biˆe´n d¯ˆ
o’i Laplace.
∆ ¯Di.nh l´
y 1.6 Gia’ su’’ X, Y l`a c´ac khˆong gian tuyˆe´n t´ınh v`a A : X → Y l`a to´an tu’’
tuyˆe´n t´ınh. Nˆe´u A c´o to´an tu’’ ngu’o.’c A−1 th`ı A−1 c˜
ung l`a to´an tu’’ tuyˆe´n t´ınh.
´’ minh
Chung
´’ mo.i y1 , y2 ∈ ImA v`a voi
´’ mo.i α1 , α2 ∈ K. Khi d¯´o tˆo`n ta.i x1 , x2 ∈ X sao cho
Voi

y1 = Ax1 , y2 = Ax2 . Ta c´o
A(α1 x1 + α2 x2 ) = α1 Ax1 + α2 Ax2 = α1 y1 + α2 y2 .
Suy ra
A−1 (α1 y1 + α2 y2 ) = α1 x1 + α2 x2 = α1 A−1 y1 + α2 A−1 y2 .
Vˆa.y A−1 l`a to´an tu’’ tuyˆe´n t´ınh.
∆ ¯Di.nh l´
y 1.7 Gia’ su’’ X, Y l`a hai khˆong gian tuyˆe´n t´ınh v`a A : X → Y l`a to´an tu’’
´
tuyˆen t´ınh. Khi d¯´o
’ X,
i) KerA l`a khˆong gian con cua
’ Y,
ii) ImA l`a khˆong gian con cua
iii) dim X = dim KerA + dim ImA,
iv) X/KerA ≃ ImA.
1.3.2 Khˆ
ong gian c´
ac to´
an tu’’ tuyˆ
e´n t´ınh
Cho hai khˆong gian tuyˆe´n t´ınh X, Y . Go.i L(X, Y ) l`a tˆa.p ho.’p tˆa´t ca’ c´ac to´an
`’ X v`ao Y . X´ac d¯.inh hai ph´ep to´an trˆen L(X, Y ) nhu’ sau: ∀A, B ∈
tu’’ tuyˆe´n t´ınh tu
L(X, Y )
(A + B)x = Ax + Bx,
(αA)x = αAx
´’ hai ph´ep to´an l`a khˆong gian tuyˆe´n t´ınh.
Khi d¯o´ L(X, Y ) c`
ung voi

1.4

Phiˆ
e´m h`
am tuyˆ
e´n t´ınh, khˆ
ong gian liˆ
en ho.’p

To´an tu’’ tuyˆe´n t´ınh A : X → K d¯u’o.’c go.i l`a phiˆe´m h`am tuyˆe´n t´ınh x´ac d¯.inh trˆen
X.
Khˆong gian L(X, K) c´ac phiˆe´m h`am tuyˆe´n t´ınh x´ac d¯.inh trˆen X d¯u’o.’c go.i l`a khˆong
’ X. K´ı hiˆe.u X ∗ = L(X, K).
gian liˆen ho.’p cua
’ X.
Khˆong gian X ∗∗ = L(X ∗ , K) d¯u’o.’c go.i l`a khˆong gian liˆen ho.’p thu´’ hai cua


2. ¯Da.i cu’ong
vˆ
e` khˆ
ong gian d
¯.inh chuˆ
a’n
’

§2.
2.1

7

vˆ
e` khˆ
ong gian d
¯.inh chuˆ
a’ n
’
¯Da.i cu’ ong
Chuˆ
a’n v`
a khˆ
ong gian d
¯.inh chuˆ
a’n

´’ K = R
`’ vˆo hu’ong
✷ ¯Di.nh nghi˜a 5 Gia’ su’’ X l`a khˆong gian tuyˆe´n t´ınh trˆen tru’ong
’
’
(ho˘
a.c K = C). Chuˆan trˆen X l`a mˆo.t h`am (phiˆe´m h`am) ∥. ∥ x´ac d¯.inh trˆen X thoa
`
m˜
an c´ac d¯iˆeu kiˆe.n sau:
i) ∥x∥ ≥ 0, ∀x ∈ X,
∥x∥ = 0 ⇔ x = 0,
ii) ∥αx∥ = |α|∥x∥, ∀x ∈ X, ∀α ∈ K,
iii) ∥x + y∥ ≤ ∥x∥ + ∥y∥, ∀x, y ∈ X.
´’ chuˆa’n ∥ . ∥ x´ac d¯.inh trˆen X d¯u’o.’c go.i l`a khˆong

Khˆong gian tuyˆe´n t´ınh X c`
ung voi
gian d¯i.nh chuˆa’n. K´ı hiˆe.u (X, ∥.∥).
`’ K = R (hay K = C) th`ı ta go.i (X, ∥.∥) l`a khˆong gian d¯.inh chuˆa’n thu.’c
Nˆe´u tru’ong
´’
(hay phuc).
´’ hai ph´ep to´an cˆo.ng v`a
• V´ı du. 9 Tˆa.p Rn c´ac bˆo. n sˆo´ thu.’c x = (x1 , x2 , . . . , xn ) voi
’
´’ l`a khˆong gian tuyˆe´n t´ınh. X´et chuˆan
nhˆan vˆo hu’ong
∥x∥ =

n
∑

|xi |2 ,

(x1 , x2 , . . . , xn ) ∈ Rn .

i=1

Chuˆa’n n`ay d¯u’o.’c go.i l`a chuˆa’n Euclide trong Rn . Khi d¯o´ (Rn , ∥.∥) l`a khˆong gian
d¯.inh chuˆa’n, d¯u’o.’c go.i l`a khˆong gian Euclide n chiˆe`u.
´’ c´ac ph´ep to´an cˆo.ng v`a
• V´ı du. 10 Tˆa.p ho.’p C[a,b] c´ac h`am sˆo´ liˆen tu.c trˆen [a, b] voi
´’ mˆo.t sˆo´ l`a mˆo.t khˆong gian tuyˆe´n t´ınh. Hon
´’ chuˆa’n
˜’ voi

nhˆan voi
’ nua,
∥x∥ = max |x(t)|
t∈[a,b]

th`ı C[a,b] l`a khˆong gian d¯.inh chuˆa’n.
´’ hai ph´ep to´an cˆo.ng v`a
ay sˆo´ thu.’c bi. ch˘
a.n c`
ung voi
• V´ı du. 11 Tˆa.p ho.’p tˆa´t ca’ c´ac d˜
’
’
´
´
nhˆan vˆo hu’ong
’ l`a mˆo.t khˆong gian d¯.inh chuˆan voi’ chuˆan
∥x∥ = sup |xn |.
n

∞

Khˆong gian n`ay d¯u’o.’c k´ı hiˆe.u l`a l .
ay sˆo´ thu.’c x = (xn )n sao cho
• V´ı du. 12 K´ı hiˆe.u l2 l`a tˆa.p ho.’p c´ac d˜

∞
∑
n=1

´’ trˆen c´ac d˜
X´et hai ph´ep cˆo.ng v`a nhˆan vˆo hu’ong
ay. ¯Da˘. t

x2n hˆo.i tu..


Chuong
’ ’ 1. Khˆ
ong gian d
¯.inh chuˆ
a’n

8

(
∥x∥ =

∞
∑

) 12
x2n

,

n=1

th`ı l2 l`a mˆo.t khˆong gian d¯.inh chuˆa’n.
• V´ı du. 13 Gia’ su’’ 1 ≤ p < ∞. Go.i Lp ([a, b]) l`a tˆa.p ho.’p c´ac h`am f : [a, b] → R (hay

∫b
C) sao cho f d¯o d¯u’o.’c v`a a |f (x)|p dx < ∞. Ta d¯.inh nghi˜a
f = g trˆen [a, b] ⇔ f (x) = g(x) h.k.n trˆen [a, b]
´’ trˆen c´ac h`am.
v`a x´et hai ph´ep cˆo.ng v`a nhˆan vˆo hu’ong
p
’
¯Di.nh nghi˜a chuˆan trˆen L ([a, b]) nhu’ sau:
 b
1/p
∫
∥f ∥p =  |f (x)|p dx .
a

´’ chuˆa’n trˆen l`a khˆong gian d¯.inh chuˆa’n.
Khi d¯o´ Lp ([a, b]) c`
ung voi
´’ mo.i x, y ∈ X, d¯a˘. t
⊕ Nhˆ
a.n x´
et Gia’ su’’ X l`a khˆong gian d¯.inh chuˆa’n. Voi
d(x, y) = ∥x − y∥
th`ı d l`a mˆo.t metric trˆen X. Khi d¯´o (X, d) l`a mˆo.t khˆong gian metric.
Ta c´o
´’ ph´ep ti.nh tiˆe´n),
i) d(x + z, y + z) = d(x, y) (d bˆa´t biˆe´n d¯ˆo´i voi
ii) d(αx, αy) = |α|d(x, y),
iii) d(x, y) = ∥x − y∥ = ∥(x − y) − 0∥ = d(x − y, 0).
´’ metric
Theo trˆen ta thˆa´y mo.i khˆong gian d¯.inh chuˆa’n d¯ˆe`u l`a khˆong gian metric voi

`
d¯u’o.’c x´ac d¯.inh nhu’ trˆen. Do d¯´o mo.i kh´ai niˆe.m, mˆe.nh d¯ˆe d¯u
´ng cho khˆong gian metric
’
`
d¯ˆeu d¯u
´ng cho khˆong gian d¯.inh chuˆan.

2.2

Chuˆ
a’n tu’ ong
d
¯u’ ong
’
’

✷ ¯Di.nh nghi˜a 6 Gia’ su’’ ∥.∥1 , ∥.∥2 l`a hai chuˆa’n x´ac d¯.inh trˆen X. Go.i τ1 , τ2 l`a hai
’’ ∥.∥1 , ∥.∥2 .
tˆopˆo gˆay nˆen boi
Chuˆa’n ∥.∥1 d¯u’o.’c go.i l`a ma.nh hon
’ ∥.∥2 nˆe´u τ1 ⊃ τ2 .
Chuˆa’n ∥.∥1 v`a ∥.∥2 tu’ong
’ d¯u’ong
’ nˆe´u τ1 = τ2 .
∆ ¯Di.nh l´
y 1.8 Chuˆa’n ∥.∥1 ma.nh hon
’ chuˆa’n ∥.∥2 khi v`a chi’ khi ∃c > 0 sao cho
´’ minh.
Chung

∥x∥2 ≤ c.∥x∥1 ,

∀x ∈ X.


2. ¯Da.i cu’ong
vˆ
e` khˆ
ong gian d
¯.inh chuˆ
a’n
’

9

Go.i
S1 (x0 , r) = {x ∈ X : ∥x − x0 ∥1 < r},
S2 (x0 , r) = {x ∈ X : ∥x − x0 ∥2 < r}
´’ ∥.∥1 , ∥.∥2 .
lˆa`n lu’o.’t l`a h`ınh cˆa`u mo’’ tˆam x0 , b´an k´ınh r d¯ˆo´i voi
’’ Ta thˆa´y 0 ∈ S2 (0, 1)
(⇒) Gia’ su’’ τ1 ⊃ τ2 . V`ı S2 (0, 1) l`a τ2 mo’’ nˆen S2 (0, 1) l`a τ1 mo.
’ S2 (0, 1). Do d¯o´ tˆo`n ta.i r > 0 sao cho S1 (0, r) ⊂ S2 (0, 1).
nˆen 0 l`a τ1 d¯iˆe’m trong cua
rx
Khi d¯o´ ∀x ∈ X, x ̸= 0 d¯a˘. t u = 2∥x∥
th`ı ∥u∥1 =
1
S2 (0, 1). Do d¯o´

rx
= ∥u∥2 < 1.
2∥x∥1 2

r
2

< r. Suy ra u ∈ S1 (0, r) ⊂

Dˆa˜n d¯ˆe´n ∥x∥2 < 2r ∥x∥ = c∥x∥1 .
´’ xay
’ ra.
Khi x = 0 th`ı d¯a˘’ ng thuc
Vˆa.y ∥x∥2 ≤ c∥x∥1 (c = 2r ), ∀x ∈ X.
´’ minh τ2 ⊂ τ1 .
(⇐) Ta chung
´’ mo.i G ∈ τ2 th`ı G l`a τ2 -mo.
’’ Lˆa´y bˆa´t k`
’
Voi
y x0 ∈ G. V`ı x0 l`a τ2 -¯
diˆe’m trong cua
´
`
G nˆen tˆon ta.i r > 0 sao cho S2 (x0 , r) ⊂ G. Ta s˜
e chung
’ minh
r
S1 (x0 , ) ⊂ S2 (x0 , r) ⊂ G.
c

´’ mo.i x ∈ S1 (x0 , r ) ta c´o ∥x − x0 ∥1 < r . Tu
`’ d¯o´
Thˆa.t vˆa.y, voi
c
c
r
∥x − x0 ∥2 ≤ c∥x − x0 ∥1 < c. = r.
c
’ G. Do x0 t`
Do d¯o´ x0 l`a τ1 -¯
diˆe’m trong cua
uy y
´ nˆen G l`a τ1 -mo’’ hay G ⊂ τ1 .
△ Hˆ
e. qua’ 1 Chuˆa’n ∥.∥1 v`a ∥.∥2 tu’ong
’ d¯u’ong
’ khi v`a chi’ khi ∃c1 , c2 , 0 < c1 < c2 sao
cho c1 .∥x∥1 ≤ ∥x∥2 ≤ c2 .∥x∥1 , ∀x ∈ X.

2.3

Su.’ hˆ
o.i tu. trong khˆ
ong gian d
¯.inh chuˆ
a’n

✷ ¯Di.nh nghi˜a 7 D˜
ay {xn } hˆo.i tu. d¯ˆe´n x trong khˆong gian d¯.inh chuˆa’n X nˆe´u
lim ∥xn − x∥ = 0. K´ı hiˆe.u lim xn = x hay xn → x (n → ∞).

n→∞
n→∞
´’ chuˆa’n ∥x∥ = max |x(t)|, x´et d˜
• V´ı du. 14 Trong khˆong gian d¯.inh chuˆa’n C[0,1] voi
ay
t∈[0,1]

xn (t) =

sin(2πnt)
.
n2


Chu’ong
ong gian d
¯.inh chuˆ
a’n
’ 1. Khˆ

10

x
1
x1

0.75
0.5

x2

0.25
0.2

0.4

0.6

x3
0.8

1

t

-0.25
-0.5
-0.75
-1

’ d˜
H1. Ba sˆo´ ha.ng d¯ˆa`u tiˆen cua
ay {xn }.

´’ h`am ha˘`ng 0. Thˆa.t vˆa.y, ta c´o
`’ h`ınh trˆen ta thˆa´y h`ınh nhu’ d˜
Tu
ay hˆo.i tu. toi
∥xn − 0∥ =

1
1
∥ sin(2πnt)∥ ≤ 2 → 0.
2
n
n

✸ T´ınh chˆ
a´t
i) Nˆe´u xn → x th`ı ∥xn ∥ → ∥x∥ (Chuˆa’n l`a h`am liˆen tu.c trˆen X).
Thˆa.t vˆa.y, ta c´o
∥xn ∥ = ∥(xn − x) + x∥ ≤ ∥xn − x∥ + ∥x∥
hay
Tu’ong
’ tu.’, ta c´o

∥xn ∥ − ∥x∥ ≤ ∥xn − x∥.

(1.1)

∥x∥ − ∥xn ∥ ≤ ∥xn − x∥.

(1.2)

`’ 1.1 v`a 1.2, ta suy ra
Tu
| ∥xn ∥ − ∥x∥ | ≤ ∥xn − x∥ → 0.
ii) Nˆe´u d˜
ay {xn } hˆo.i tu. th`ı ∃K sao cho ∥xn ∥ ≤ K, ∀n.
Thˆa.t vˆa.y, nˆe´u d˜

ay (xn ) hˆo.i tu. th`ı d˜
ay sˆo´ {∥xn ∥} hˆo.i tu.. Nˆen d˜
ay sˆo´ {∥xn ∥} bi.
ch˘
a.n. Do d¯o´ ∃K: ∥xn ∥ ≤ K, ∀n.
iii) Nˆe´u xn → x, yn → y th`ı xn + yn → x + y. Nˆe´u xn → x, αn → α th`ı
´’ l`a c´ac ph´ep to´an x + y v`
αn xn → αx ( tuc
a αx liˆen tu.c).
Thˆa.t vˆa.y, ta c´o
∥(xn + yn ) − (x + y)∥ ≤ ∥xn − x∥ + ∥yn − y∥ → 0,
∥αn xn − αx∥ = ∥αn (xn − x) + x(αn − α)∥ ≤ |αn |∥xn − x∥ + |αn − α|∥x∥ → 0.
`’
• Hˆ
o.i tu. d
¯ˆ
e`u v`
a hˆ
o.i tu. tung
d
¯iˆ
e’m
´’ chuˆa’n
X´et khˆong gian C[a,b] tˆa´t ca’ c´ac h`am liˆen tu.c trˆen d¯oa.n [a, b] voi
∥f ∥ = max |f (t)|, f ∈ C[a,b]
t∈[a,b]


2. ¯Da.i cu’ong
vˆ

e` khˆ
ong gian d
¯i.nh chuˆ
a’n
’

11

v`a d˜
ay {fn } ∈ C[a,b] .
`’ d¯iˆe’m d¯ˆe´n f nˆe´u lim |fn (t)−f (t)| = 0, ∀t ∈ [a, b].
D˜
ay {fn } d¯u’o.’c go.i l`a hˆo.i tu. tung
n→∞

D˜
ay {fn } d¯u’o.’c go.i l`a hˆo.i tu. d¯ˆe`u d¯ˆe´n f nˆe´u lim ∥fn − f ∥ = 0.
n→∞
`’ d¯iˆe’m. Tuy nhiˆen, d¯iˆe`u
’ d˜
Ta thˆa´y su.’ hˆo.i tu. d¯ˆe`u cua
ay h`am k´eo theo hˆo.i tu. tung
ngu’o.’c la.i khˆong d¯u
´ng. Cha˘’ ng ha.n trong khˆong gian C[0,1] , x´et d˜
ay h`am {gn } x´ac d¯.inh
nhu’ sau:
 n
nˆe´u 0 ≤ t ≤ 2−n
 2 t
gn (t) =

2 − 2n t nˆe´u 2−n ≤ t ≤ 21−n

´’ t kh´ac
0
voi
´’ mˆo˜i t ∈ [0, 1].
V`ı gn ̸= 0, ta c´o ∥gn ∥ ̸= 0. ¯Da˘. t fn = ∥ggnn ∥ . Ta dˆe˜ thˆa´y fn (t) → 0 voi
´’ mo.i n ∈ N nˆen d˜
Tuy nhiˆen, v`ı ∥fn ∥ = 1 voi
ay {fn } khˆong hˆo.i tu. vˆe`0 theo chuˆa’n.
1

g (t)
n

0

2.4

1−n

2

1

t

Khˆ
ong gian Banach

ay trong khˆong gian d¯.inh chuˆa’n. D˜
ay {xn }n
✷ ¯Di.nh nghi˜a 8 Cho {xn }n l`a mˆo.t d˜
’ (hay d˜
d¯u’o.’c go.i l`a d˜
ay co’ ban
ay Cauchy) nˆe´u ∥xm − xn ∥ → 0 khi m, n → ∞.
⊕ Nhˆ
a.n x´
et
’
Nˆe´u d˜
ay {xn }n hˆo.i tu. th`ı {xn }n l`a d˜
ay co’ ban.
’ mˆe.nh d¯ˆe`trˆen n´oi chung khˆong d¯u
´ng. Cha˘’ ng ha.n, x´et P([0, 1])
¯Diˆe`u ngu’o.’c la.i cua
´’ x´ac d¯.inh trˆen [0, 1] voi
´’ chuˆa’n ∥P ∥ = maxx∈[0,1] |P (x)|,
l`a khˆong gian c´ac d¯a thuc
P (x) ∈ P([0, 1]). ¯Da˘. t
xn
x2
+ . . . + , n = 1, 2, . . .
Pn (x) = 1 + x +
2!
n!
´’ ha.n cua
’ nhung
’

Ta thˆa´y {Pn } l`a d˜
ay co’ ban
’ n´o khˆong hˆo.i tu. trong P([0, 1]) v`ı gioi
´’
n´o khˆong l`a d¯a thuc.
✷ ¯Di.nh nghi˜a 9 Khˆong gian d¯.inh chuˆa’n X d¯u’o.’c go.i l`a khˆong gian Banach
1

1

nˆe´u X

`’ d¯ˆa`u tiˆen d¯ua
Stephan Banach (1892-1945), nh`a to´an ho.c Balan, ngu’oi
y thuyˆe´t tˆo’ng qu´at vˆe`
’ ra l´


Chu’ong
ong gian d
¯.inh chuˆ
a’n
’ 1. Khˆ

12

´’ metric d(x, y) = ∥x − y∥, tuc
´’ l`a mo.i d˜
’ trong
ay co’ ban

l`a khˆong gian metric d¯ˆa`y voi
`
X d¯ˆeu hˆo.i tu..

Stefan Banach
• V´ı du. 15 C´ac khˆong gian d¯.inh chuˆa’n Rn , Cn , C([a, b]), lp , l∞ l`a c´ac khˆong gian
Banach.
am d¯^
a`y kh^
ong gian) Gia’ su’’ X l`a khˆong gian d¯.inh chuˆa’n khˆong
∆ ¯Di.nh l´
y 1.9 (L`
´’ X sao cho X tr`
d¯ˆ
a`y. Khi d¯´o tˆo`n ta.i mˆo.t khˆong gian Banach X ∗ chua
u mˆa.t trong
∗
X .
´’ minh
Chung
´’ d(x, y) = ∥x − y∥. Ba˘`ng c´ach l`am d¯ˆa`y khˆong
X´et khˆong gian metric (X, d), voi
gian n`ay ta d¯u’o.’c khˆong gian metric d¯ˆa`y X ∗ . Ta cˆa`n xˆay du.’ng c´ac ph´ep to´an d¯ˆe’ n´o
tro’’ th`anh khˆong gian d¯.inh chuˆan nhˆa.n X l`am khˆong gian con.
Lˆa´y x, y ∈ X ∗ . V`ı X = X ∗ nˆen tˆo`n ta.i c´ac d˜
ay (xn ), (yn ) trong X hˆo.i tu. lˆa`n lu’o.’t
’ trong X ⊂ X ∗ nˆen ta d¯.inh nghi˜a
d¯ˆe´n x, y. Dˆe˜ thˆa´y (xn + yn ), (λxn ) l`a c´ac d˜
ay co’ ban
λx = lim λxn , x + y = lim (xn + yn ).

n→∞

n→∞

Ta kiˆe’m tra thˆa´y c´ac d¯.inh nghi˜a n`ay x´ac d¯.inh c´ac ph´ep to´an d¯ˆe’ X ∗ l`a mˆo.t
´’ chuˆa’n trˆen X ∗ d¯u’o.’c cho boi
’’
khˆong gian d¯.inh chuˆa’n nhˆa.n X l`am khˆong gian con voi
∗
∗
∗
∥x∥ = d (x, 0), trong d¯´o d l`a metric trˆen X .

§3.

Chuˆ
o˜i trong khˆ
ong gian d
¯.inh chuˆ
a’ n

✷ ¯Di.nh nghi˜a 10 Gia’ su’’ {xn }n l`a mˆo.t d˜
ay trong khˆong gian d¯.inh chuˆa’n.
Tˆo’ng vˆo ha.n x1 + x2 + . . . + xn + . . . d¯u’o.’c go.i l`a mˆo.t chuˆo˜i trong khˆong gian d¯.inh
∞
∑
’
chuˆan X. K´ı hiˆe.u
xn .
n=1

’ ˆong n˘
khˆong gian d¯.inh chuˆa’n d¯ˆ
a`y trong luˆa.n ´an tiˆe´n si˜ cua
am 1920


3. Chuˆ
o˜i trong khˆ
ong gian d
¯.inh chuˆ
a’n

13

’ chuˆo˜i.
Phˆa`n tu’’ sn = x1 + x2 + . . . + xn d¯u’o.’c go.i l`a tˆo’ng riˆeng thu´’ n cua
Nˆe´u d˜
ay {sn }n hˆo.i tu. vˆe´ phˆa`n tu’’ s th`ı ta n´oi chuˆo˜i hˆo.i tu. v`a c´o tˆo’ng l`a s. K´ı hiˆe.u
∞
∑
s=
xn .
n=1

Chuˆo˜i

∞
∑

xn d¯u’o.’c go.i l`a hˆo.i tu. tuyˆe.t d¯ˆo´i nˆe´u chuˆo˜i sˆo´

∞
∑

∥xn ∥ hˆo.i tu..

n=1

n=1

´’ minh d¯u’o.’c d¯.inh l´ı sau d¯ˆay:
Ta dˆe˜ d`ang chung
∞
∞
∑
∑
∆ ¯Di.nh l´
y 1.10 Nˆe´u c´ac chuˆo˜i
xn ,
yn hˆo.i tu. v`a c´o tˆ
o’ng l`a x, y th`ı c´ac chuˆo˜i
∞
∑
n=1

(xn + yn ),

∞
∑

n=1

n=1

αxn c˜
ung hˆo.i ty. v`a c´o tˆo’ng l`a x + y, αx.

n=1

∆ ¯Di.nh l´
y 1.11 Gia’ su’’ X l`a khˆong gian Banach. Khi d¯´
o chuˆ
o˜i

∞
∑

xn hˆ
o.i tu. khi

n=1

v`a chi’ khi ∀ε > 0, ∃N > 0 sao cho ∀n > N , ∀p ta c´o
∥xn+1 + xn+2 + . . . + xn+p ∥ < ε.
´’ mimh
Chung
∑∞
o.i tu. ⇔
n=1 xn hˆ

{sn =

∑n
k=1

xk } hˆo.i tu.

⇔

’ (do X Banach)
{sn } l`a d˜
ay co’ ban

⇔

∀ε > 0, ∃N > 0: ∀n > N, ∀p ta c´o
∥sn+p − sn ∥ = ∥xn+1 + xn+2 + . . . + xn+p ∥ < ε.

∑
∆ ¯Di.nh l´
y 1.12 Nˆe´u X l`a khˆong gian Banach v`a ∞
a chuˆo˜i hˆo.i tu. tuyˆe.t d¯ˆo´i
n=1 xn l`
∞
∞
∞
∑
∑
∑

trong X th`ı chuˆo˜i
xn hˆo.i tu. v`
a c´o ∥
xn ∥ ≤
∥xn ∥.
n=1

´’ minh
Chung
V`ı chuˆo˜i sˆo´

∑∞
n=1

n=1

n=1

∥xn ∥ hˆo.i tu. nˆen ∀ε > 0, ∃N sao cho ∀n > N , ∀p ta c´o

∥xn+1 + xn+2 + . . . + xn+p ∥ ≤ ∥xn+1 ∥ + ∥xn+2 ∥ + . . . + ∥xn+p ∥ < ε.
Theo d¯.inh l´
y (1.11) ta suy ra chuˆo˜i

∞
∑

xn hˆo.i tu.. M˘
a.t kh´ac, ∀n ta c´o

n=1

∥x1 + x2 + . . . + xn ∥ ≤ ∥x1 ∥ + ∥x2 ∥ + . . . + ∥xn ∥ ≤

∞
∑
n=1

∥xn ∥.


Chu’ong
ong gian d
¯.inh chuˆ
a’n
’ 1. Khˆ

14

Cho n → ∞ ta d¯u’o.’c
∥

∞
∑
n=1

xn ∥ ≤

∞
∑

∥xn ∥.

n=1

∆ ¯Di.nh l´
y 1.13 Khˆong gian d¯.inh chuˆa’n X l`a khˆong gian Banach khi v`a chi’ khi mo.i
chuˆo˜i hˆo.i tu. tuyˆe.t d¯ˆo´i trong X d¯ˆe`u hˆo.i tu..
´’ minh
Chung
´’ minh nˆe´u mo.i chuˆo˜i hˆo.i tu. tuyˆe.t d¯ˆo´i l`a
’ chung
Theo d¯.inh l´
y trˆen, ta chi c`on phai
`
’
hˆo.i tu. th`ı X l`a d¯ˆay d¯u.
´’ mo.i sˆo´ tu.’ nhiˆen k tˆo`n ta.i sˆo´ tu.’
’ trong X. Khi d¯´o voi
Gia’ su’’ {xn }n l`a d˜
ay co’ ban
´
nhiˆen nk (xnk > xk−1 ) sao cho voi’ mo.i n, m ≥ nk ta c´o ∥xm − xn ∥ < 21k .
1
¯Da˘. c biˆe.t ∥xnk+1 − xnk ∥ < 2k .
X´et chuˆo˜i (*): xn1 + (xn2 − xn1 ) + (xn3 − xn2 ) + . . .
∑
V`ı chuˆo˜i ∞
o.i tu. nˆen chuˆo˜i (*) hˆo.i tu. vˆe` phˆa`n tu’’ x ∈ X. Khi
k=1 ∥xnk+1 − xnk ∥ hˆ

d¯´o
x = lim sk = lim [xn1 + (xn2 − xn1 ) + . . . + (xnk − xnk−1 )] = lim xnk ∈ X.
k→∞

k→∞

k→∞

Ta c´o
∥xn − x∥ ≤ ∥xn − xnk ∥ + ∥xnk − x∥.
Cho n → ∞ th`ı nk → ∞, ta c´o limn→∞ xn = x ∈ X.
Vˆa.y X l`a khˆong gian Banach.

§4.

Khˆ
ong gian con

✷ ¯Di.nh nghi˜a 11 Gia’ su’’ X l`a khˆong gian d¯i.nh chuˆa’n v`a Y l`a khˆong gian con tuyˆe´n
´’ chuˆa’n d¯o´
’ X. Khi d¯o´ chuˆa’n trˆen X ha.n chˆe´ trˆen Y c˜
t´ınh cua
ung l`a chuˆa’n trˆen Y , voi
’
’ khˆong gian d¯.inh chuˆa’n
Y l`a khˆong gian d¯.inh chuˆan d¯u’o.’c go.i l`a khˆong gian con cua
X.
’ X.
Nˆe´u Y d¯o´ng th`ı ta n´oi Y l`a khˆong gian con d¯´ong cua

’’ Y .
span(Y ) = L(Y ) d¯u’o.’c go.i l`a khˆong gian con d¯´ong sinh boi
’ khˆong gian d¯.inh chuˆ
∆ ¯Di.nh l´
y 1.14 Nˆe´u Y l`a khˆong gian con cua
a’n X th`ı Y l`a
’ X.
khˆ
ong gian con d¯´ong cua
´’ minh
Chung
´’ minh Y l`a khˆong gian con cua
’ X.
Hiˆe’n nhiˆen Y d¯´ong. Ta chi’ cˆa`n chung


15

4. Khˆ
ong gian con

∀x, y ∈ Y , tˆo`n ta.i c´ac d˜
ay {xn }, {yn } trong Y sao cho xn → x, yn → y.
’ X nˆen αxn +βyn ∈ Y , ∀α, β ∈ K. Khi d¯´o αxn +βyn →
V`ı Y l`a khˆong gian con cua
αx + βy. Do d¯´o αx + βy ∈ Y .
’ khˆong gian d¯.inh chuˆa’n
∆ ¯Di.nh l´
y 1.15 (Riesz) Nˆe´u Y l`a khˆong gian con d¯´
ong cua

’’ Y v`a z sao
X th`ı ∀z ∈
/ Y , ∀ε > 0, ∃x0 thuˆ
o.c khˆong gian con tuyˆe´n t´ınh gˆay nˆen boi
cho ∥x0 ∥ = 1 v`a ∥x0 − y∥ > 1 − ε, ∀y ∈ Y .
´’ minh
Chung
V`ı z ∈
/ Y v`a Y d¯´ong nˆen
d = d(z, Y ) = inf ∥z − y∥ > 0.
y∈Y

´’ δ =
∀ε > 0 d¯u’ b´e (c´o thˆe’ chon 0 < ε < 1), voi
∃y0 ∈ Y : d ≤ ∥z − y0 ∥ < d + δ.
¯Da˘. t x0 =
∥x0 ∥ = 1.

z−y0
∥z−y0 ∥

εd
1−ε

> 0, theo d¯.inh nghi˜a inf

’’ Y v`a z v`a
th`ı x0 thuˆo.c khˆong gian con tuyˆe´n t´ınh gˆay nˆen boi

Khi d¯o´ ∀y ∈ Y , ta c´o

∥x0 − y∥ =

z − y0
1
−y =
∥z − (y0 + ∥z − y0 ∥y)∥
∥z − y0 ∥
∥z − y0 ∥

V`ı Y l`a khˆong gian con v`a y0 , y ∈ Y nˆen y0 + ∥z − y0 ∥y ∈ Y . Do d¯´o ∥z − (y0 +
∥z − y0 ∥y)∥ > d.
M˘
a.t kh´ac,
∥z − y0 ∥ < d + δ ⇒

1
1
>
∥z − y0 ∥
d+δ

Do d¯´o
∥x0 − y∥ >

d
δ
=1−
=1−ε
d+δ
d+δ

’ khˆong gian d¯.inh chuˆ
△ Hˆ
e. qua’ 2 Nˆe´u Y l`a khˆong gian con d¯´
ong cua
a’n X, Y ̸= X
th`ı ∀ε > 0, ∃x0 ∈
/ Y , ∥x0 ∥ = 1 sao cho ∥x0 − y∥ > 1 − ε, ∀y ∈ Y .
´’ minh
Chung
´’ minh cua
’ d¯.inh l´ı
V`ı Y ̸= X nˆen tˆo`n ta.i z ∈ X sao cho z ∈
/ Y . Theo c´ach chung
z−y0
´
`
’
˘
1.15, d¯a.t x0 = ∥z−y0 ∥ ta d¯u’o.’c d¯iˆeu phai chung
’ minh.


Chuong
ong gian d
.inh chu
an
1. Kh

16

Đ5.

Kh
ong gian thu ong


ã B
o d
¯ˆ
e`
Gia’ su’’ X l`a khˆong gian d¯.inh chuˆa’n, x0 ∈ X v`
a Y ⊂ X. Nˆe´u Y d¯´
ong th`ı x0 + Y
d¯´
ong.
ong gian thu’ ong
• Xˆ
ay du.’ ng khˆ
’
’ X.
Gia’ su’’ X l`a khˆong gian d¯.inh chuˆa’n v`a Y l`a khˆong gian con d¯´ong cua
Ta c´o khˆong gian thu’ong
’ tuyˆe´n t´ınh X/Y . Trang bi. cho X/Y mˆo.t chuˆa’n nhu’
sau:
∥˜
x∥ = inf ∥z∥ = inf ∥x + y∥, (˜
x ∈ X/Y ).
z∈˜
x

y∈Y

Ta c´o
i) ∥˜
x∥ = inf z∈˜x ∥z∥ ≥ 0, ∀˜
x ∈ X/Y .
* x˜ = ˜0 ⇒ ∥˜
x∥ = 0 (v`ı 0 ∈ x˜),
* Nˆe´u ∥˜
x∥ = 0 th`ı ∀n, ∃xn ∈ x˜ : 0 ≤ ∥xn ∥ < n1 .
Ta d¯u’o.’c d˜
ay {xn } ⊂ x˜ v`a xn → 0. V`ı x˜ d¯o´ng (do Y d¯´ong) nˆen 0 ∈ x˜. Do d¯´o
˜
x˜ = 0 ∈ X/Y .
ii) ∥α˜
x∥ = inf ∥αz∥ = |α| inf ∥z∥ = |α|∥˜
x∥.
z∈˜
x

z∈˜
x

iii) Cho x˜, y˜ ∈ X/Y , ∀n, ∃xn ∈ x˜, ∃yn ∈ y˜ sao cho
∥xn ∥ < ∥˜
x∥ +

1
,

2n

∥yn ∥ < ∥˜
y∥ +

1
2n

⇒ ∥˜
x + y˜∥ ≤ ∥xn + yn ∥ ≤ ∥xn ∥ + ∥yn ∥ < ∥˜
x∥ + ∥˜
y∥ +

1
n

Cho n → ∞ ta d¯u’o.’c ∥˜
x + y˜∥ ≤ ∥˜
x∥ + ∥˜
y ∥.
Vˆa.y X/Y l`a khˆong gian d¯.inh chuˆa’n. Khˆong gian n`ay d¯u’o.’c go.i l`a khˆong gian d¯.inh
’ khˆong gian d¯i.nh chuˆa’n X theo khˆong gian con d¯o´ng Y .
chuˆa’n thu’ong
’ cua
’ X
∆ ¯Di.nh l´
y 1.16 Nˆe´u X l`
a khˆong gian Banach v`a Y l`a khˆong gian con d¯o´ng cua
th`ı X/Y l`a mˆo.t khˆong gian Banach.
´’ minh

Chung
´’ minh mo.i chuˆo˜i hˆo.i tu. tuyˆe.t d¯ˆo´i trong X/Y d¯ˆe`u hˆo.i tu..
Ta chung
∞
∞
∑
∑
´
’
˜
˜
˜
’
Gia su’ chuˆoi
x˜n hˆo.i tu. tuyˆe.t d¯ˆoi trong X/Y , nghia l`a chuˆoi
∥˜
xn ∥ hˆo.i tu..
n=1

n=1

´’ mˆo˜i sˆo´ tu.’ nhiˆen n, tˆo`n ta.i
’ chuˆa’n trong khˆong gian thu’ong,
Theo d¯.inh nghi˜a cua
voi
’
un ∈ x˜n sao cho


6. T´ıch c´

ac khˆ
ong gian d
¯i.nh chuˆ
a’n

17

∞
∞
∑
∑
1
⇒
∥u
∥
<
∥x˜n ∥ + 1.
n
2n
n=1
n=1
∞
∞
∑
∑
Do d¯o´
un hˆo.i tu.. Go.i x =
un . Ta c´o x˜ ∈ X/Y . V`ı x−(u1 +u2 +. . .+un ) ∈

∥un ∥ < ∥x˜n ∥ +

n=1

n=1

x˜ − (x˜1 + x˜2 + . . . + x˜n ) nˆen

∥˜
x − (x˜1 + x˜2 + . . . + x˜n )∥ ≤ ∥x − (u1 + u2 + . . . + un )∥ → 0.
Vˆa.y chuˆo˜i

∞
∑

x˜n hˆo.i tu..

n=1

§6.

T´ıch c´
ac khˆ
ong gian d
¯i.nh chuˆ
a’ n

• Xˆ
ay du.’ ng khˆ
ong gian t´ıch
Gia’ su’’ (X1 , ∥.∥1 ), (X2 , ∥.∥2 ), . . . , (Xn , ∥.∥n ) l`a c´ac khˆong gian d¯.inh chuˆa’n. Ta c´o

n
∏
´
khˆong tuyˆen t´ınh t´ıch X =
Xi . ¯Da˘. t
1=1

∥x∥ =

n
∑

∥xi ∥i ,

x = (x1 , x2 , . . . , xn ) ∈ X.

i=1

´’ minh ∥ . ∥ l`a chuˆa’n trˆen X.
Ta chung
i) ∥x∥ > 0 (Hiˆe’n nhiˆen).
∑
∥x∥ = 0 ⇔ ni=1 ∥xi ∥i = 0 ⇔ xi = 0 ∀i = 1, n ⇔ x = (0, . . . , 0).
ii) ∀x ∈ X, ∀α ∈ K ta c´o
∥αx∥ =

n
∑

∥αxi ∥i = |α|

i=1

n
∑

∥xi ∥i = |α|∥x∥.

i=1

iii) ∀x = (x1 , x2 , . . . , xn ), (y1 , y2 , . . . , yn ) ∈ X
∥x + y∥ =

n
∑

∥xi + yi ∥i ≤

i=1

=

n
∑
i=1

∥xi ∥i +

n
∑

(∥xi ∥i + ∥yi ∥i )

i=1
n
∑

∥yi ∥i = ∥x∥ + ∥y∥.

i=1

’ c´ac
Khi d¯o´ (X, ∥.∥) l`a khˆong gian d¯.inh chuˆa’n, d¯u’o.’c go.i l`a t´ıch Descartes cua
khˆong gian d¯.inh chuˆa’n (Xi , ∥.∥i ).
a’n, X =
∆ ¯Di.nh l´
y 1.17 Gia’ su’’ (Xi , ∥.∥i ) (i = 1, . . . , n) l`a c´ac khˆong gian d¯.inh chuˆ


Chu’ong
ong gian d
¯.inh chuˆ
a’n
’ 1. Khˆ

18

n
∏

Xi , xk = (xk1 , . . . , xkn )), x0 = (x01 , . . . , x0n ) ∈ X. Khi d¯´
o xk → x0 (k → ∞) khi v`a

i=1

chi’ khi xki → x0i (k → ∞), (i = 1, . . . , n).
∆ ¯Di.nh l´
y 1.18 Gia’ su’’ {(Xi , ∥.∥i )}i (i = 1, n) l`a c´ac khˆong gian d¯.inh chuˆ
a’n. Khi
n
∏
d¯´
o X =
Xi l`a khˆong gian Banach khi v`a chi’ khi Xi (i=1,. . . , n) l`a khˆong gian
i=1

Banach.
´’ minh
Chung
’ trong Xi (i = 1, . . . , n). X´et d˜
(⇒) Gia’ su’’ {xki }k l`a d˜
ay co’ ban
ay
{xk = (x1 , . . . , xi−1 , xki , xi+1 , . . . , xn )}k
´’ xj cˆo´ d¯.inh trong Xj (j ̸= i). Ta c´o
voi
k
∥xm − xk ∥ = ∥xm
i − xi ∥i → 0 (k, m → ∞).

´’ to’ {xk }k l`a d˜
’ trong X. V`ı X l`a khˆong gian Banach nˆen xk → x0 ∈ X.
ay co’ ban
Chung
’
’
Gia su’ x0 = (x1 , . . . , xi−1 , x0i , xi+1 , . . . , xn ). Khi d¯o´
∥xki − x0i ∥i ≤

n
∑

∥xki − x0i ∥i = ∥xk − x0 ∥ → 0.

i=1

Do d¯´o d˜
ay {xki } hˆo.i tu. trong Xi . Vˆa.y Xi (i = 1, . . . , n) l`a khˆong gian Banach.
’ trong X. Khi d¯o´ ∀i = 1, n ta c´o
(⇐) Gia’ su’’ {xk = (x1 ,k , x2 , . . . , xkn )}k l`a d˜
ay co’ ban
k
∥xm
i − xi ∥i ≤

n
∑

k
∥xm

i − xi ∥i = ∥xm − xk ∥ → 0,

(m, k → 0).

i=1

’ trong Xi . V`ı Xi l`a khˆong gian Banach nˆen xki → x0i ,
ay co’ ban
Suy ra {xki }k l`a d˜
0
0
i = 1, n. ¯Da˘. t x0 = (x1 , x2 , . . . , x0n ) th`ı x0 ∈ X v`a
∥xk − x0 ∥ =

n
∑

∥xki − x0i ∥i → 0,

(k → ∞).

i=1

Do d¯´o xk → x0 ∈ X. Vˆa.y X l`a khˆong gian Banach.
⊕ Nhˆ
a.n x´
et V`ı R l`a khˆong gian Banach nˆen theo d¯.inh l´ı 1.18 ta c´o Rn l`a khˆong
gian Banach.

7. To´
an tu’’ tuyˆ
e´n t´ınh liˆ
en tu.c

§7.
7.1

19

’ tuyˆ
To´
an tu
e´n t´ınh liˆ
en tu.c
’’ tuyˆ
To´
an t u
e´n t´ınh liˆ
en tu.c

✷ ¯Di.nh nghi˜a 12 Cho hai khˆong gian d¯.inh chuˆa’n X, Y v`a to´an tu’’ tuyˆe´n t´ınh
A : X → Y . Ta n´oi
A liˆen tu.c ta.i x0 ∈ X nˆe´u xn → x0 th`ı Axn → Ax0 .
A liˆen tu.c trˆen X nˆe´u A liˆen tu.c ta.i mo.i x ∈ X.
A bi. ch˘
a.n nˆe´u tˆo`n ta.i M > 0 sao cho ∥Ax∥Y ≤ M ∥x∥X ∀x ∈ X.
∆ ¯Di.nh l´
y 1.19 Gia’ su’’ X, Y l`
a hai khˆong gian d¯.inh chuˆ

a’n v`a A : X → Y l`
a to´an
´
’
’
tu’ tuyˆen t´ınh. Khi d¯´o A liˆen tu.c khi v`a chi khi A bi. ch˘
a.n.
´’ minh
Chung
´’ mo.i x ∈ X,
´’ hˆe´t ta chung
´’ minh tˆo`n ta.i sˆo´ M ≥ 0 sao cho ∥Ax∥ ≤ M , voi
(⇒) Tru’oc
∥x∥ = 1.
´’ mo.i n ∈ N, tˆo`n ta.i xn ∈ X, ∥xn ∥ = 1 sao cho ∥Axn ∥ > n.
Gia’ su’’ ngu’o.’c la.i, voi
∥Axn ∥
xn
1
xn
¯Da˘. t xn′ = n th`ı ∥xn ∥ = n v`a ∥Axn ∥ = ∥A( n )∥ = n > 1. Ta c´o xn → 0
nhung
0 = A(0). Do d¯o´ A khˆong liˆen tu.c ta.i x = 0 (Vˆo l´ı).
’ Axn
´’ minh tˆo`n ta.i M ≥ 0 sao cho ∥Ax∥ ≤ M ∥x∥, ∀x ∈ X.
Tiˆe´p d¯ˆe´n ta chung
′

′

′

′

´’ mo.i x ∈ X, x ̸= 0, theo trˆen ta c´o
Voi
x
∥Ax∥
M ≥ ∥A(
)∥ =
,
∥x∥
∥x∥
hay
∥Ax∥ ≤ M ∥x∥, x ̸= 0.
´’ xay
’ ra.
Ta thˆa´y khi x = 0 th`ı d¯a˘’ ng thuc
Vˆa.y A bi. ch˘
a.n.
(⇐) Gia’ su’’ A bi. ch˘
a.n. Khi d¯o´ tˆo`n ta.i sˆo´ M ≥ 0 sao cho ∥Ax∥ ≤ M ∥x∥, ∀x ∈ X.
Lˆa´y x ∈ X t`
uy y
´ v`a xn → x. Ta c´o
∥Axn − Ax∥ = ∥A(xn − x)∥ ≤ M.∥xn − x∥ → 0.
Suy ra Axn → Ax. Do d¯´o A liˆen tu.c ta.i x.
Vˆa.y A liˆen tu.c trˆen X.
∆ ¯Di.nh l´
y 1.20 Gia’ su’’ X, Y l`

a hai khˆong gian d¯.inh chuˆ
a’n v`a A : X → Y l`
a to´an
´
’
’
tu’ tuyˆen t´ınh. Khi d¯´o A liˆen tu.c trˆen X khi v`a chi khi A liˆen tu.c ta.i 0.
´’ minh
Chung


Chu’ong
ong gian d
¯.inh chuˆ
a’n
’ 1. Khˆ

20

(⇒) Hiˆe’n nhiˆen.
(⇐) Gia’ su’’ A liˆen tu.c ta.i 0. Lˆa´y t`
uy y
´ x ∈ X v`a gia’ su’’ xn → x. Ta c´o xn − x → 0
nˆen
lim Axn − Ax = lim A(xn − x) = A(0) = 0
n→∞

n→∞

Suy ra lim Axn = Ax.

n→∞

Do d¯o´ A liˆen tu.c ta.i x. Vˆa.y A liˆen tu.c trˆen X.
• V´ı du. 16 Cho
T :

R2 → R3
.
(x, y) → (3x + y, x − 3y, 4y)

´’ mo.i (x, y) ∈ R2 ta c´o
Dˆe˜ d`ang ta thˆa´y T l`a to´an tu’’ tuyˆe´n t´ınh. Ngo`ai ra, voi
∥T (x, y)∥ = ∥(3x + y, x − 3y, 4y)∥
= [(3x + y)2 + (x − 3y)2 + (4y)2 ]1/2
= (10x2 + 26y 2 )1/2
√
26(x2 + y 2 )1/2
≤
√
=
26∥(x, y)∥.
Do d¯o´ ph´ep biˆe´n d¯ˆo’i tuyˆe´n t´ınh T bi. ch˘
a.n.
• V´ı du. 17 X´et X = Rk v`a Y = Rm . Go.i {e1 , e2 , . . . , ek } v`a {u1 , u2 , . . . , um } l`a c´ac
´’ mo.i x = (ξ1 , . . . , ξk ) ∈ Rk v`a y = (η1 , . . . , ηm ) ∈ Rm ta c´o
’ Rk v`a Rm th`ı voi
co’ so’’ cua
x=

k

∑

ξ j ej ,

y=

j=1

m
∑

ηi ui .

i=1

X´et to´an tu’’ A : Rk → Rm x´ac d¯.inh nhu’ sau:
A(ξ1 , . . . , ξk ) = (η1 , . . . , ηm )
´’
voi
ηi =

k
∑

aij ξj

(i = 1, 2, . . . , m)

j=1

trong d¯´o Aej = (a1j , . . . , amj ), j = 1, . . . , k. Ma

a11 a12 . . .
 a21 a22 . . .

 ... ... ...
am1 am2 . . .
’ to´an tu’’ A.
go.i l`a ma trˆa.n cua

trˆa.n

a1k
a2k 

... 
amk


7. To´
an tu’’ tuyˆ
e´n t´ınh liˆ
en tu.c

21

Ta thˆa´y A l`a to´an tu’’ tuyˆe´n t´ınh.
(n)

(n)

(0)

(0)

Nˆe´u xn = (ξ1 , . . . , ξk ) → x0 = (ξ1 , . . . , ξk ) th`ı do su.’ hˆo.i tu. trong Rk l`a hˆo.i
(n)
(0)
tu. theo to.a d¯oˆ. nˆen ξj → ξj (j = 1, . . . , k). Do d¯´o
∑
∑
(n)
(0)
(0)
ηi (n) = kj=1 aij ξj → kj=1 aij ξj = ηi
´’ l`a Axn → Ax0 .
tuc
Vˆa.y A liˆen tu.c.
`’ khˆong gian d¯.inh chuˆa’n
✷ ¯Di.nh nghi˜a 13 Gia’ su’’ A l`a to´an tu’’ tuyˆe´n t´ınh liˆen tu.c tu
X v`ao khˆong gian d¯.inh chuˆa’n Y . Khi d¯o´ tˆo`n ta.i sˆo´ M ≥ 0 sao cho
∥Ax∥ ≤ M ∥x∥ , ∀x ∈ X.
’ m˜
’ to´an tu’’ tuyˆe´n
Sˆo´ M ≥ 0 nho’ nhˆa´t thoa
an d¯iˆe`u kiˆe.n trˆen d¯u’o.’c go.i l`a chuˆa’n cua
t´ınh liˆen tu.c A. K´ı hiˆe.u ∥A∥. Ta c´o
i) ∥Ax∥ ≤ ∥A∥.∥x∥, ∀x ∈ X
ii) Nˆe´u ∥Ax∥ ≤ M.∥x∥, ∀x ∈ X th`ı ∥A∥ ≤ M .
∆ ¯Di.nh l´

y 1.21
∥A∥ =

∥Ax∥
= sup ∥Ax∥.
x∈X,x̸=0 ∥x∥
x∈X,∥x∥=1
sup

´’ minh
Chung
Ta c´o
∥Ax∥ ≤ M ∥x∥, ∀x ∈ X

⇔

Do d¯o´
∥Ax∥
∥A∥ = sup
= sup A
x∈X,x̸=0 ∥x∥
x∈X,x̸=0

∥Ax∥
≤ M, ∀x ∈ X, x ̸= 0.
∥x∥
(

x
∥x∥

)
=

sup

∥Az∥.

z∈X,∥z∥=1

´’ chuˆa’n ∥x∥ =
• V´ı du. 18 Cho khˆong gian C[a,b] c´ac h`am liˆen tu.c trˆen [a, b] voi
max |x(t)|.
a≤t≤b

X´et to´an tu’’ t´ıch phˆan Ax(t) =

∫b

K(t, s).x(s)ds.

a

´’ mˆo˜i x ∈ C[a,b] ta c´o
Voi
∥Ax(t)∥ = maxa≤t≤b

∫b

K(t, s)s(s)ds

a

≤ max |x(s)| max
a≤s≤b

∫b

a≤t≤b a

|K(t, s)|ds = ∥x∥ max

∫b

a≤t≤b a

|K(t, s)|ds.