Lý thuyết KgHilbert

tr ’u.c giao trong khˆong gian Hilbert X... tr ’u.c chu ’ˆan trong khˆong gian Hilbert X.. tr ’u.c chu ’ˆan trong khˆong gian Hilbert X... tr ’u.c chu ’ˆan trong khˆong gian Hilbert X.

Ch ’ u ’ ong 5

1 Kh´ ai niˆ e.m v ` ˆ e khˆ ong gian Hilbert

1.1 T´ıch vˆ o h ’u´’ ong

a) D¯ i.nh ngh˜ia

* Cho khˆong gian tuy ´ˆen t´ınh X trˆen tr ’u`’ong R T´ıch vˆo h ’u´’ ong trˆen X l`a ´anh xa.

h., i : X × X → R th ’oa m˜an c´ac ¯di `ˆeu kiˆe.n

i) hx, yi = hy, xi, ∀x, y ∈ X,

ii) hx + y, zi = hx, zi + hy, zi, ∀x, y, z ∈ X,

iii) hαx, yi = αhx, yi,

iv) hx, xi ≥ 0

hx, xi = 0 ⇔ x = 0.

* N ´ˆeu X l`a khˆong gian tuy ´ˆen t´ınh trˆen tr ’u`’ong C th`ı t´ıch vˆo h ’u´’ong trˆen X l`a ´anh

xa h., i : X × X → C th ’oa m˜an c´ac ¯di `ˆeu kiˆe.n ii), iii) v`a iv) ’’o trˆen, c`on ¯di `ˆeu kiˆe.n i)

¯

d ’u ’o.c thay b’’oi hx, yi = hy, xi.

Khˆong gian tuy ´ˆen t´ınh X c`ung v´’oi t´ıch vˆo h ’u´’ong x´ac ¯di.nh trˆen X ¯d ’u ’o.c go.i l`a

khˆong gian ti `ˆen Hilbert

1.2 B ´ ˆ at ¯ d ’˘ ang th´’ uc Schwarzt

∆ D¯ i.nh l´y 1 V´’ oi mo.i x, y ∈ X ta c´o

|hx, yi|2 ≤ hx, xi.hy, yi.

Ch´’ung minh

V´’oi mo.i x, y ∈ X v`a v´’oi mo.i α ∈ R ta c´o

0 ≤ hx − αy, x − αyi = hx, xi − 2hx, yi + hy, yiα2 = f (α).

T`’u ¯d´o

0 ≥ ∆ 0 = hx, yi2− hx, xi.hy, yi.

45

1.3 Nhˆ a.n x´et

Gi ’a s ’’u X l`a khˆong gian ti `ˆen Hilbert V´’oi mo.i x ∈ X, ¯d˘a.t

kxk =qhx, xi.

Ta c´o

i) kxk ≥ 0.

kxk = 0 ⇔ x = 0.

ii) kαxk =qhαx, αxi =qαhx, αxi =qαhαx, xi = |α|qhx, xi = |α|kxk.

iii) V´’oi mo.i x, y ∈ X, theo b ´ˆat ¯d ’˘ang th´’uc schwartz ta c´o

kx + yk2 = hx + y, x + yi = hx, xi + 2hx, yi + hy, yi

≤ kxk2+ 2kxkkyk + kyk2 = (kxk2+ kyk2)2.

T`’u ¯d´o kx + yk ≤ kxk + kyk.

Suy ra kxk l`a chu ’ ˆan trˆen X.

V`ı vˆa.y ta th ´ˆay mo.i khˆong gian ti `ˆen Hilbert ¯d `ˆeu l`a khˆong gian ¯di.nh chu ’ˆan v´’oi chu ’ˆan trˆen Do ¯d´o mo.i kh´ai niˆe.m, mˆe.nh ¯d `ˆe trong khˆong gian ¯di.nh chu ’ˆan ¯d `ˆeu ¯d´ung cho khˆong gian Hilbert

1.4 ¯ ’˘ D ang th´’ uc h`ınh b`ınh h` anh

∆ D¯ i.nh l´y 2 V´’ oi mo.i x, y ∈ X ta c´o

kx + yk2+ kx − yk2 = 2(kxk2+ kyk2).

Ch´’ung minh

V´’oi mo.i x, y ∈ X ta c´o

kx + yk2+ kx − yk2 = hx + y, x + yi = 2[hx, xi + hy, yi] = 2(kxk2+ kyk2).2

1.5 C´ ac t´ınh ch ´ ˆ at

i) T´ıch vˆo h ’u´’ong l`a phi ´ˆem h`am song tuy ´ˆen t´ınh

Thˆa.t vˆa.y,

∀x, y ∈ X v`a ∀α, β ∈ K ta c´o

hαx + βy, zi = hαx, yi + hβy, zi = αhx, yi + βhy, zi.

T ’u ’ong t ’u., ta c´o hx, αy + βzi = αhx, yi + βhx, zi.

1 Kh´ai ni.ˆem v `ˆe khˆong gian Hilbert 47

ii) Gi ’a s ’’u X l`a khˆong gian ti `ˆen Hilbert N ´ˆeu {x n } n , {y n } n ⊂ X, x n → x, y n → y

th`ı hx n , y n i → hx, yi.

Thˆa.t vˆa.y, ta c´o

|hx n , y n i − hx, yi| = |hx n , y n i − hx, y n i + hx, y n i + hx, yi|

= |hx n − x, y n i + hx, y n − yi|

≤ kx n − xkky n k + kxkky n − yk ≤ kx n − xkM + kxkky n − yk → 0.

1.6 Khˆ ong gian Hilbert

2 D¯ i.nh ngh˜ia 1 Khˆong gian ti `ˆen Hilbert ¯d ’u ¯d ’u ’o.c go.i l`a khˆong gian Hilbert

• V´ı du 1 Khˆong gian tuy ´ˆen t´ınh Rn l`a khˆong gian Hilbert v´’oi t´ıch vˆo h ’u´’ong

hx, yi =

n

X

i=1

x i y i , x = (x1, x2, , x n ), y = (y1, y2, , y n ).

• V´ı du 2 Khˆong gian tuy ´ ˆen t´ınh L2(X, µ) l`a khˆong gian Hilbert v´’oi t´ıch vˆo h ’u´’ong

hx, yi =

Z

X

x(t)y(t)dµ.

T´ıch phˆan t `ˆon ta.i h˜’uu ha.n v`ı

Z

X

|x(t)y(t)|dµ ≤



 Z

X

|x(t)|2dµ





1/2

 Z

X

|y(t)|2dµ





1/2

(B ´ˆat ¯d ’˘ang th´’uc H¨older).

• V´ı du 3 Khˆong gian tuy ´ ˆen t´ınh l2 l`a khˆong gian Hilbert v´’oi t´ıch vˆo h ’u´’ong

hx, yi =

∞

X

n=1

x n y n , x = (x n ), y = (y n ) ∈ l2.

∆ D¯ i.nh l´y 3 V´’ oi mo.i khˆong gian ti ` ˆen Hilbert X ¯ d ` ˆeu t ` ˆon ta.i mˆo.t khˆong gian Hilbert

X ∗ ch´’ ua X sao cho X l`a khˆong gian con tr`u mˆa.t trong X ∗

∆ D¯ i.nh l´y 4 Mo.i khˆong gian Banach th ’oa m˜an ¯d’˘ang th´’uc h`ınh b`ınh h`anh ¯d ` ˆeu l`a khˆong gian Hilbert.

Ch´’ung minh

T´ıch vˆo h ’u´’ong cho b ’’oi

hx, yi = 1

4(kx + yk

2 + kx − yk2), x, y ∈ X 2

1.7 Liˆ en hˆ e gi˜’ua c´ac khˆong gian

?

Metric

d

KG metric

Ph´ep to´an

+,

?

Ph´ep to´an

+,

KG tuy ´ˆen t´ınh Metric

d

Metric b ´ˆat bi ´ˆen v`a thu `ˆan nh ´ˆat

?

KG ¯di.nh chu ’ˆan

?

D

¯ ’u

KG Banach

?

T´ıch vˆo h ’u´’ong

h, i

KG ti `ˆen Hilbert D

¯ ’u

KG Hilbert

?

-D

¯ K h`ınh b`ınh h`anh

2 T´ınh tr ’u.c giao, h`ınh chi´ ˆ eu

2.1 Vector tr ’u.c giao

a) D¯ i.nh ngh˜ia

Gi ’a s ’’u X l`a khˆong gian ti `ˆen Hilbert

* Hai vector x, y ∈ X ¯d ’u ’o.c go.i l`a tr ’u.c giao n ´ˆeu hx, yi = 0 K´ı hiˆe.u x ⊥ y.

* Hˆe S ⊂ X ¯d ’u ’o.c go.i l`a hˆe tr ’u.c giao n ´ˆeu n ´ˆeu c´ac vector c ’ua S tr ’u.c giao v´’oi nhau

t`’ung ¯dˆoi mˆo.t (i.e ∀x, y ∈ S, x 6= y th`ı x ⊥ y).

b) T´ınh ch ´ˆat

i) N ´ˆeu x ⊥ y th`ı y ⊥ x,

x ⊥ x ⇔ x = 0,

0 ⊥ x, ∀x ∈ X

ii) N ´ˆeu x ⊥ y i , ∀i = 1, n th`ı x ⊥ α1y1+ α2y2 + + α n y n

V`ı

hx, α1y1+ α2y2+ + α n y n i = α1hx, y1i + α2hx, y2i + + α n hx, y n i.

iii) N ´ˆeu x ⊥ y n , ∀n v`a y n → y th`ı x ⊥ y.

V`ı hx, yi = lim n→∞ hx, y n i = 0.

2 T´ınh tr.’uc giao, h`ınh chi ´ˆeu 49

c) D¯ i.nh l´y Pithagore

∆ D¯ i.nh l´y 5 Gi ’a s ’’ u S l`a mˆo.t hˆe tr ’u.c giao g ` ˆom c´ac vector kh´ac 0 Khi ¯ d´o S l`a hˆe.

¯

dˆo.c lˆa.p tuy ´ ˆen t´ınh H ’ on n˜’ ua, v´’ oi n vector x1, x2, , x n ∈ S ta c´o

kx1+ x2+ + x n k2 = kx1k2+ kx2k2+ + kx n k2 (D ¯ ’˘ang th´’uc Pithagore).

Ch´’ung minh

L ´ˆay n vector x1, x2, , x n ∈ S Gi ’a s ’’u α1x1+ α2x2+ + α n x n = 0 Khi ¯d´o v´’oi

mo.i j = 1, n ta c´o

0 = h0, x j i = hα1x1+ α2x2+ + α n x n , x j i =

n

X

i=1

α i hx i , x j i = α j hx j , x j i.

V`ı x j 6= 0 nˆen hx j , x i i = kx j k2 6= 0 Do ¯ d´o α j = 0, ∀j = 1, n T`’u ¯d´o ta suy ra

{x1, x2, , x n } l`a hˆe ¯dˆo.c lˆa.p tuy ´ ˆen t´ınh Vˆa.y S l`a hˆe ¯dˆo.c lˆa.p tuy ´ˆen t´ınh

Ngo`ai ra ta c´o

kx1+x2+ .+x n k2 = h

n

X

i=1

x i ,

n

X

j=1

x j i =

n

X

i=1

n

X

j=1

hx i , x j i =

n

X

i=1

hx i , x i i =

n

X

i=1

kx i k2 2

∆ D¯ i.nh l´y 6 Gi ’a s ’’ u {x n } n l`a hˆe tr ’u.c giao trong khˆong gian Hilbert X Khi ¯d´o chu ˜ˆoi

∞

X

n=1

x n hˆo.i tu khi v`a ch ’i khi chu ˜ ˆoi s ´ ˆo

∞

X

n=1

kx n k2 hˆo.i tu

Ch´’ung minh

Go.i s n=

n

X

i=1

x i , σ n =

n

X

i=1

kx i k2 Theo ¯di.nh l´ı Pithagore, ∀n > m ta c´o

ks n − s m k2 = kx m+1 + x m+2 + + x n k2

= kx m+1 k2+ kx m+2 k2+ + kx n k2 = σ n − σ m

Do ¯d´o ks n − s m k → 0 (n, m → ∞) khi v`a ch ’i khi σ n − σ m → 0 (n, m → ∞) Do

X l`a khˆong gian ¯d `ˆay nˆen {s n } n hˆo.i tu khi v`a ch’i khi {σ n } n hˆo.i tu 2

2.2 Ph ` ˆ an b` u tr ’u.c giao, h`ınh chi´ ˆ eu lˆ en khˆ ong gian con

2 D¯ i.nh ngh˜ia 2 Gi ’a s ’’u X l`a khˆong gian Hilbert v`a M, N ⊂ X.

* Vector x ¯ d ’u ’o.c go.i l`a tr ’u.c giao v´’oi tˆa.p M n ´ ˆeu x ⊥ y, ∀y ∈ M K´ı hiˆe.u x ⊥ M.

* Tˆa.p M ¯d ’u ’o.c go.i l`a tr ’u.c giao v´’oi tˆa.p N n ´ ˆeu x ⊥ y, ∀x ∈ M, ∀y ∈ N K´ı hiˆe.u

M ⊥ N.

* Ta th ´ˆay tˆa.p t ´ˆat c ’a c´ac vector tr ’u.c giao v´’oi tˆa.p M l`a mˆo.t khˆong gian con ¯d´ong

c ’ua X, khˆong gian con n`ay go.i l`a ph ` ˆan b`u tr ’u.c giao c’ua M, k´ı hiˆe.u l`a M ⊥

∆ D¯ i.nh l´y 7 Gi ’a s ’’ u X l`a khˆong gian ti ` ˆen Hilbert, M ⊂ X v`a [M] l`a khˆong gian con

¯

d´ong c ’ua X gˆay nˆen b ’’ oi M N ´ ˆeu x ⊥ M th`ı x ⊥ [M].

Ch´’ung minh V´’oi mo.i y ∈ [M] th`ı y = lim n→∞ y n, v´’oi y n l`a mˆo.t t ’ˆo h ’o.p tuy ´ˆen t´ınh (h˜’uu ha.n) c´ac ph `ˆan t ’’u c’ua M V`ı x ⊥ M nˆen x ⊥ y n , ∀n T`’u ¯ d´o hx, y n i = 0, ∀n Suy ra

hx, yi = lim

n→∞ hx, y n i = 0, ngh˜ ia l`a x ⊥ y Vˆa.y x ⊥ [M] 2

∆ D¯ i.nh l´y 8 Gi ’a s ’’ u M l`a khˆong gian con ¯ d´ong c ’ua khˆong gian Hilbert X Khi ¯ d´o mo.i x ∈ X ¯d ` ˆeu bi ’ ˆeu di ˜ ˆen duy nh ´ ˆat da.ng x = y + z, v´’oi y ∈ M, z ∈ M ⊥ , trong ¯ d´o y l`a ph ` ˆan t ’’ u c ’ua M g ` ˆan x nh ´ ˆat.

Ch´’ung minh

* Khi x ∈ M th`ı ta c´o th ’ˆe vi ´ˆet x = x + 0, v´’ oi 0 ⊥ M.

* Khi x / ∈ M.

V`ı M ¯d´ong nˆen

d = d(x, M ) = inf

u∈M kx − uk > 0.

T`’u ¯d´o t `ˆon ta.i d˜ay ku n } n ⊂ M sao cho lim

n→∞ kx − u n k = d.

+ Ta ch´’ung minh {u n } n l`a d˜ay Cauchy

Thˆa.t vˆa.y, ´ap du.ng ¯d ’˘ang th´’uc h`ınh b`ınh h`anh cho x − u n v`a x − u m ta c´o

k2x − (u n + u m )k2+ ku n − u m k2 = 2kx − u n k2+ 2kx − u m k2. (5.1) V`ı u n +u m

2 ∈ M nˆen kx − u n +u m

2 k ≥ d Khi ¯d´o

k2x − (u n + u m )k2 = 4kx − u n + u m

2 k

2 ≥ 4d2.

T ’u (5.1) ta c´o

2(kx − u n k2+ kx − u m k2) ≥ 4d2+ ku n − u m k2 ≥ 0 (5.2) Cho qua gi´’oi ha.n (5.2) khi n, m → ∞ ta ¯d ’u ’o.c lim n,m→∞ ku n − u m k = 0 Do ¯ d´o {u n } n

l`a d˜ay Cauchy trong M.

V`ı M ¯ d´ong trong X ¯d `ˆay nˆen M ¯d `ˆay Do ¯d´o d˜ay {u n } n hˆo.i tu v `ˆe ph `ˆan t ’’u y thuˆo.c

M Khi ¯ d´o ta c´o kx − yk = lim

n→∞ kx − u n k = d.

D

¯ ˘a.t z = x − y th`ı x = y + z v`a kzk = d Ta ch´’ ung minh z ⊥ M.

L ´ˆay u ∈ M V´’ oi mo.i α ∈ R ta c´o

hz − αu, z − αui = hz, zi − 2hz, uiα + hu, uiα2 = kzk2− 2hz, ui + kuk2α2

2 T´ınh tr.’uc giao, h`ınh chi ´ˆeu 51

= d2+ 2hz, ui + kuk2α2.

M˘a.t kh´ac,

hz − αu, z − αui = kz − αuk2 = kx − (y + αu)k2 ≥ d2, (y + αu ∈ M)

nˆen ta c´o

d2 − 2hz, uiα + kuk2α2 ≥ d2

hay

kuk2α2− 2hz, ukα ≥ 0, ∀α ∈ R.

T`’u ¯d´o

∆0 = hz, ui2 ≤ 0.

D

¯ i `ˆeu n`ay x ’ay ra khi v`a ch ’i khi hz, ui = 0 hay z ⊥ M Do ¯ d´o z ∈ M ⊥.

T´om la.i, ta c´o x = y + z v´’oi y ∈ M v`a z ∈ M ⊥

+ S ’u bi ’ˆeu di ˜ˆen l`a duy nh ´ˆat

Gi ’a s ’’u x = y + z = y 0 + z 0 Khi ¯d´o y − y 0 = z 0 − z V`ı M v`a M ⊥ l`a c´ac khˆong

gian con nˆen y − y 0 ∈ M, z 0 − z ∈ M ⊥ Khi ¯d´o 0 = hy − y 0 , z 0 − zi = hy − y 0 , y − y 0 i.

T`’u ¯d´o y − y 0 = 0 Ta suy ra y = y 0 v`a z = z 0 2

¯ Ch´u ´y Theo ¯di.nh l´y (8), mo.i x ∈ X ¯d `ˆeu c´o bi ’ˆeu di ˜ˆen x = y + z, trong ¯ d´o y l`a

ph `ˆan t ’’u c’ua M g ` ˆan x nh ´ˆat, ¯d ’u ’o.c go.i l`a h`ınh chi ´ˆeu c ’ua x lˆen khˆong gian con M D¯ ˘a.t

P (x) = y th`ı P l`a to´an t ’’u ¯d ’u ’o.c go.i l`a to´an t ’’u chi ´ˆeu lˆen khˆong gian con M R˜o r`ang

P l`a to´an t ’’u tuy ´ˆen t´ınh H ’on n˜’ua, P liˆen tu.c v`ı

kP xk = kyk ≤

q

kyk2+ kzk2 = kxk (do ¯ di.nh l´y Pithagore).

4 Hˆ e qu ’a 1 N ´ ˆeu M l`a khˆong gian con ¯ d´ong c ’ua khˆong gian Hilbert X th`ı

(M ⊥)⊥ = M.

Ch´’ung minh

* V`ı M ⊥ M ⊥ nˆen M ⊂ (M ⊥)⊥

* M˘a.t kh´ac, l ´ˆay x ∈ (M ⊥)⊥ th`ı x ⊥ M ⊥ Theo ¯di.nh l´y (8) ta c´o x = y + z v´’oi

y ∈ M v`a z ∈ M ⊥ Khi ¯d´o

0 = hx, zi = hy + z, zi = hy, zi + hz, zi = hz, zi.

T`’u ¯d´o z = 0 D ˜ˆan ¯d ´ˆen x = y ∈ M Ta suy ra ¯ d ’u ’o.c (M ⊥)⊥ = M.

4 Hˆ e qu ’a 2 Gi ’a s ’’u X l`a khˆong gian Hilbert, M ⊂ X v`a [M] l`a khˆong gian con ¯d´ong

c ’ua X gˆay nˆen b ’’ oi M Khi ¯ d´o

[M] = (M ⊥)⊥

Ch´’ung minh.

* V´’oi mo.i x ∈ M ⊥ th`ı x ⊥ M nˆen x ⊥ [M] T`’u ¯ d´o M ⊥ ⊥ M Do ¯ d´o [M] ⊂ (M ⊥)⊥

* M˘a.t kh´ac, v`ı M ⊂ [M] nˆen M ⊥ ⊃ [M] ⊥ T`’u ¯d´o (M ⊥)⊥ ⊂ ([M] ⊥)⊥ = [M].

4 Hˆ e qu ’a 3 Gi ’a s ’’u X l`a khˆong gian Hilbert, M ⊂ X v`a [M] l`a khˆong gian con ¯d´ong

gˆay nˆen b ’’ oi M Khi ¯ d´o X = [M] khi v`a ch ’i khi n ´ ˆeu x ⊥ M th`ı x = 0.

Ch´’ung minh D¯ ’ˆe ´y r`˘ang X ⊥ = {0} Theo hˆe qu ’a 2, ta th ´ ˆay X = [M] t ’u ’ong ¯d ’u ’ong v´’oi X = (M ⊥)⊥ V`ı M ⊥ d´ong nˆen ta c´o¯

M ⊥ = ((M ⊥)⊥)⊥ = X ⊥ = {0}.

2

4 Hˆ e qu ’a 4 Gi ’a s ’’u M l`a khˆong gian con c’ua khˆong gian Hilbert X Khi ¯d´o M tr`u

mˆa.t trong X khi v`a ch ’i khi x ⊥ M th`ı x = 0.

Ch´’ung minh V`ı M = [M] nˆen ta c´o ¯di `ˆeu ph ’ai ch´’ung minh 2

3 Hˆ e tr ’u.c chu ’ˆan

3.1 Hˆ e tr ’u.c chu ’ˆan

a) C´ac ¯di.nh ngh˜ia

2 D¯ i.nh ngh˜ia 3 Gi ’a s ’’u X l`a khˆong gian Hilbert Hˆe {e i } i ⊂ X ¯d ’u ’o.c go.i l`a hˆe tr ’u.c chu ’ˆan n ´ˆeu

he i , e j i = δ ij =

(

0 n ´ˆeu i 6= j

1 n ´ˆeu i = j

(i.e Hˆe tr ’u.c chu ’ˆan l`a hˆe tr ’u.c giao v`a chu ’ˆan h´oa)

2 D¯ i.nh ngh˜ia 4 Gi ’a s ’’u {e i } i l`a hˆe tr ’u.c chu ’ˆan trong khˆong gian Hilbert X Khi ¯d´o

v´’oi mo.i x ∈ X, s ´ ˆo ξ i = hx, e i i ¯d ’u ’o.c go.i l`a hˆe s ´ˆo Fourier c ’ua x ¯d ´ˆoi v´’oi e i v`a chu ˜ˆoi

∞

X

i=1

ξ i e i go.i l`a chu ˜ˆoi Fourier (hay khai tri ’ˆen Fourier) c ’ua x theo hˆe {e i } i

b) B ´ˆat ¯d ’˘ang th´’uc Bessel

∆ D¯ i.nh l´y 9 (B ´^at ¯d ’˘ang th´’uc Bessel) Gi ’a s ’’ u {e i } i l`a hˆe tr ’u.c chu ’ˆan trong khˆong gian Hilbert X Khi ¯ d´o v´’ oi mo.i x ∈ X ta c´o

∞

X

i=1

ξ2

i ≤ kxk2, v´’ oi ξ i = hx, e i i, ∀i.

3 H.ˆe tr.’uc chu ’ˆan 53

Ch´’ung minh V´’oi mo.i x ∈ X, ¯d˘a.t y n = x −

n

X

i=1

ξ i e i , (n = 1, 2, ) th`ı x = y n+

n

X

i=1

ξ i e i V´’oi j = 1, , n, ta c´o

hy n , e j i = hx −

n

X

i=1

ξ i e i , e j i = hx, e i i −

n

X

i=1

ξ i he i , e j i = hx, e j i − ξ j = 0.

T`’u ¯d´o y n ⊥ ξ i e i , ∀i = 1, n Theo ¯di.nh l´y Pithagore ta c´o

kxk2 = ky n+

n

X

i=1

ξ i e i k2 = ky n k2+

n

X

i=

kξ i e i k2 = ky n k2+

n

X

i=1

ξ2i ≥

n

X

i=1

ξ i2.

Cho n → ∞ th`ı

∞

X

i=1

4 Hˆ e qu ’a 5 Gi ’a s ’’u {e i } i l`a hˆe tr ’u.c chu ’ˆan trong khˆong gian Hilbert X Khi ¯d´o v´’oi mo.i x ∈ X chu ˜ ˆoi

∞

X

i=1

ξ i e i luˆon hˆo.i tu v`a (x −

∞

X

i=1

ξ i e i ) ⊥ e j , ∀j.

Ch´’ung minh V`ı

∞

X

i=1

kξ i e i k2 =

∞

X

i=1

ξ2

i ≤ kxk2 < ∞.

nˆen theo ¯di.nh l´y (6) ta suy ra chu ˜ˆoi

∞

X

i=1

ξ i e i hˆo.i tu

M˘a.t kh´ac v ’oi mo.i j v`a n > j ta c´o

hx −

∞

X

i=1

ξ i e i , e j i = lim

n

X

i=1

ξ i e i , e j i = 0.

Vˆa.y (x −

∞

X

i=1

3.2 Hˆ e tr ’u.c chu ’ˆan ¯d ` ˆ ay ¯ d’u

2 D¯ i.nh ngh˜ia 5 Hˆe tr ’u.c chu ’ˆan {e i } i d ’u ’o.c go.i l`a ¯d `¯ ˆay ¯d ’u n ´ˆeu x ⊥ e i , ∀i th`ı x = 0.

Hˆe tr ’u.c chu ’ˆan ¯d `ˆay ¯d ’u ¯d ’u ’o.c go.i l`a c ’o s’’o c’ua khˆong gian Hilbert

∆ D¯ i.nh l´y 10 Gi ’a s ’’ u {e i } i l`a hˆe tr ’u.c chu ’ˆan trong khˆong gian Hilbert X v`a ξ i =

hx, e i i (i = 1, 2, ) l`a hˆe s ´ ˆo Fourier c ’ua x ¯ d ´ ˆoi v´’ oi e i Khi ¯ d´o c´ac mˆe.nh ¯d ` ˆe sau l`a

t ’ u ’ ong ¯ d ’ u ’ ong

i) {e i } i l`a hˆe tr ’u.c chu ’ˆan ¯d ` ˆay ¯ d ’u.

ii) V´’ oi mo.i x ∈ X th`ı x =

∞

X

i=1

ξ i e i

iii) V´’ oi mo.i x ∈ X th`ı kxk2 =

∞

X

i=1

ξ i2, (¯ d ’˘ ang th´’ uc Passerval).

iv) V´’ oi mo.i x ∈ X, y ∈ X th`ı hx, yi =

∞

X

i=1

ξ i η i v´’ oi ξ i = hx, e i i, η i = hy, e i i.

v) Hˆe {e i } i tuy ´ ˆen t´ınh tr`u mˆa.t trong X (ngh˜ia l`a L({e i }) = X).

Ch´’ung minh

(i) ⇒ (ii): Ta c´o (x −

∞

X

i=1

ξ i e i ) ⊥ e j , ∀j V`ı {e i } i l`a hˆe tr ’u.c chu ’ˆan ¯d `ˆay ¯d ’u nˆen

x −

∞

X

i=1

ξ i e i = 0 Do ¯d´o x =

∞

X

i=1

ξ i e i

(ii) ⇒ (iv): V´’ oi ξ i = hx, e i i, η j = hy, e j i, i, j = 1, 2, ta c´o

hx, yi = h

∞

X

i=1

ξ i e i ,

∞

X

j=1

η j e j i = h lim

n→∞

n

X

i=1

ξ i e i , lim

n→∞

n

X

j=1

η j e j i

= lim

n

X

i=1

ξ i e i ,

n

X

j=1

η j e j i = lim

n→∞

n

X

i=1

ξ i η i he i , e i i = lim

n→∞

n

X

i=1

ξ i η i =

∞

X

i=1

ξ i η i

(iv) ⇒ (iii): T`’u (iv), cho y = x th`ı ta ¯d ’u ’o.c

kxk2 = hx, xi =

∞

X

i=1

ξ i2.

(iii) ⇒ (i): Gi ’a s ’’u c´o (iii) v`a x ⊥ e i , ∀i T`’u ¯ d´o ξ i = hx, e i i = 0, ∀i Suy ra kxk2 =

∞

X

i=1

ξ2

i = 0 Do ¯d´o x = 0.

(ii) ⇒ (v): Gi ’a s ’’u c´o (ii) Khi ¯d´o v´’oi mo.i x ∈ X ta c´o

x =

∞

X

i=1

ξ i e i = lim

n→∞

n

X

i=1

ξ i e i

Ta th ´ˆay x l`a gi´’oi ha.n c’ua mˆo.t d˜ay c´ac t ’ˆo h ’o.p tuy ´ˆen t´ınh c´ac ph `ˆan t ’’u e i nˆen

x ∈ L({e i }).

(v) ⇒ (i): Gi ’a s ’’u c´o (v) v`a x ⊥ e i , ∀i T`’u ¯ d´o x ⊥ L({e i }) Suy ra x ⊥ L({e i }) Theo

hˆe qu ’a (4) ta suy ra x = 0 Vˆa.y {e i } i l`a hˆe tr ’u.c chu ’ˆan ¯d `ˆay ¯d ’u 2

∆ D¯ i.nh l´y 11 ((Riesz-Fisher)) Gi ’a s ’’ u {e i } i l`a hˆe tr ’u.c chu ’ˆan ¯d ` ˆay ¯ d ’u trong khˆong gian Hilbert X N ´ ˆeu d˜ ay s ´ ˆo {ξ i } i th ’oa m˜ an

∞

X

i=1

ξ2

i < ∞ th`ı c´o mˆo.t vector duy nh ´ ˆat

x ∈ X nhˆa.n ξ i l`am hˆe s ´ ˆo Fourier v`a x =

∞

X

i=1

ξ i e i , kxk2 =

∞

X

i=1

ξ2

i