Lý thuyết KgHilbert

Ch ’u ’ ong 5 KH ˆ ONG GIAN HILBERT 1. Kh´ai niˆe . m v ` ˆe khˆong gian Hilbert 1.1 T´ıch vˆo 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 . ., . : X ×X → R th ’ oa m˜an c´ac ¯di ` ˆeu kiˆe . n i) x, y = y, x, ∀x, y ∈ X, ii) x + y, z = x, z + y, z, ∀x, y, z ∈ X, iii) αx, y = αx, y, iv) x, x ≥ 0 x, x = 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 . ., . : 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 x, y = y, x. 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 ´ ˆat ¯d ’ ˘ ang th ´ ’ uc Schwarzt ∆ D ¯ i . nh l´y 1 V ´ ’ oi mo . i x, y ∈ X ta c´o |x, y| 2 ≤ x, x.y, y. Ch ´ ’ ung minh V ´ ’ oi mo . i x, y ∈ X v`a v ´ ’ oi mo . i α ∈ R ta c´o 0 ≤ x −αy, x − αy = x, x −2x, y + y, yα 2 = f(α). T ` ’ u ¯d´o 0 ≥ ∆  = x, y 2 − x, x.y, y. Vˆa . y |x, y| 2 ≤ x, x.y, y. ✷ 45 46 Ch ’u ’ ong 5. Khˆong gian Hilb ert 1.3 Nhˆa . n x´et Gi ’ a s ’ ’ u X l`a khˆong gian ti ` ˆen Hilbert. V ´ ’ oi mo . i x ∈ X, ¯d ˘ a . t x =  x, x. Ta c´o i) x ≥ 0. x = 0 ⇔ x = 0. ii) αx =  αx, αx =  αx, αx =  ααx, x = |α|  x, x = |α|x. iii) V ´ ’ oi mo . i x, y ∈ X, theo b ´ ˆat ¯d ’ ˘ ang th ´ ’ uc schwartz ta c´o x + y 2 = x + y, x + y = x, x + 2x, y + y, y ≤ x 2 + 2xy + y 2 = (x 2 + y 2 ) 2 . T ` ’ u ¯d´o x + y ≤ x + y. Suy ra x 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 x + y 2 + x −y 2 = 2(x 2 + y 2 ). Ch ´ ’ ung minh V ´ ’ oi mo . i x, y ∈ X ta c´o x + y 2 + x −y 2 = x + y, x + y = 2[x, x + y, y] = 2(x 2 + y 2 ).✷ 1.5 C´ac t´ınh ch ´ ˆat 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 αx + βy, z = αx, y + βy, z = αx, y + βy, z. T ’ u ’ ong t ’ u . , ta c´o x, αy + β z = αx, y + βx, z. 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 x n , y n  → x, y. Thˆa . t vˆa . y, ta c´o |x n , y n  −x, y| = |x n , y n  −x, y n  + x, y n  + x, y| = |x n − x, y n  + x, y n − y| ≤ x n − xy n  + xy n − y ≤ x n − xM + xy n − y → 0. 1.6 Khˆong gian Hilbert ✷ 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 R n l`a khˆong gian Hilbert v ´ ’ oi t´ıch vˆo h ’ u ´ ’ ong x, y = n  i=1 x i y i , x = (x 1 , x 2 , . . . , x n ), y = (y 1 , y 2 , . . . , y n ). • V´ı du . 2 Khˆong gian tuy ´ ˆen t´ınh L 2 (X, µ) l`a khˆong gian Hilbert v ´ ’ oi t´ıch vˆo h ’ u ´ ’ ong x, y =  X x(t)y(t)dµ. T´ıch phˆan t ` ˆon ta . i h ˜ ’ uu ha . n v`ı  X |x(t)y(t)|dµ ≤    X |x(t)| 2 dµ   1/2    X |y(t)| 2 dµ   1/2 (B ´ ˆat ¯d ’ ˘ ang th ´ ’ uc H¨older). • V´ı du . 3 Khˆong gian tuy ´ ˆen t´ınh l 2 l`a khˆong gian Hilbert v ´ ’ oi t´ıch vˆo h ’ u ´ ’ ong x, y = ∞  n=1 x n y n , x = (x n ), y = (y n ) ∈ l 2 . ∆ 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 x, y = 1 4 (x + y 2 + x −y 2 ), x, y ∈ X. ✷ 48 Ch ’u ’ ong 5. Khˆong gian Hilb ert 1.7 Liˆen hˆe . gi ˜ ’ ua c´ac khˆong gian . Tˆa . p X ❄ 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 ,  KG ti ` ˆen Hilbert D ¯ ’ u KG Hilb ert ❄ ✲ D ¯ K h`ınh b`ınh h`anh 2. T´ınh tr ’ u . c giao, h`ınh chi ´ ˆeu 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 x, y = 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 = 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 ⊥ α 1 y 1 + α 2 y 2 + . . . + α n y n . V`ı x, α 1 y 1 + α 2 y 2 + . . . + α n y n  = α 1 x, y 1  + α 2 x, y 2  + . . . + α n x, y n . iii) N ´ ˆeu x ⊥ y n , ∀n v`a y n → y th`ı x ⊥ y. V`ı x, y = lim n→∞ x, y n  = 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 x 1 , x 2 , . . . , x n ∈ S ta c´o x 1 + x 2 + . . . + x n  2 = x 1  2 + x 2  2 + . . . + x n  2 (D ¯ ’ ˘ ang th ´ ’ uc Pithagore). Ch ´ ’ ung minh L ´ ˆay n vector x 1 , x 2 , . . . , x n ∈ S. Gi ’ a s ’ ’ u α 1 x 1 + α 2 x 2 + . . . + α n x n = 0. Khi ¯d´o v ´ ’ oi mo . i j = 1, n ta c´o 0 = 0, x j  = α 1 x 1 + α 2 x 2 + . . . + α n x n , x j  = n  i=1 α i x i , x j  = α j x j , x j . V`ı x j = 0 nˆen x j , x i  = x j  2 = 0. Do ¯d´o α j = 0, ∀j = 1, n. T ` ’ u ¯d´o ta suy ra {x 1 , x 2 , . . . , 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 x 1 +x 2 +. . .+x n  2 =  n  i=1 x i , n  j=1 x j  = n  i=1 n  j=1 x i , x j  = n  i=1 x i , x i  = n  i=1 x i  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 ∞  n=1 x n hˆo . i tu . khi v`a ch ’ i khi chu ˜ ˆoi s ´ ˆo ∞  n=1 x n  2 hˆo . i tu . . Ch ´ ’ ung minh Go . i s n = n  i=1 x i , σ n = n  i=1 x i  2 . Theo ¯di . nh l´ı Pithagore, ∀n > m ta c´o s n − s m  2 = x m+1 + x m+2 + . . . + x n  2 = x m+1  2 + x m+2  2 + . . . + x n  2 = σ n − σ m . Do ¯d´o s n − s m  → 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 Ph ` ˆan b`u tr ’ u . c giao, h`ınh chi ´ ˆeu lˆen khˆong gian con ✷ 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 ⊥ . 50 Ch ’u ’ ong 5. Khˆong gian Hilb ert ∆ 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 x, y n  = 0, ∀n. Suy ra x, y = lim n→∞ x, y n  = 0, ngh ˜ ia l`a x ⊥ y. Vˆa . y x ⊥ [M]. ✷ ∆ 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 x −u > 0. T ` ’ u ¯d´o t ` ˆon ta . i d˜ay u n } n ⊂ M sao cho lim n→∞ x −u n  = 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 2x −(u n + u m ) 2 + u n − u m  2 = 2x − u n  2 + 2x −u m  2 . (5.1) V`ı u n +u m 2 ∈ M nˆen x − u n +u m 2  ≥ d. Khi ¯d´o 2x −(u n + u m ) 2 = 4x − u n + u m 2  2 ≥ 4d 2 . T ’ u (5.1) ta c´o 2(x −u n  2 + x −u m  2 ) ≥ 4d 2 + u n − u m  2 ≥ 0 (5.2) Cho qua gi ´ ’ oi ha . n (5.2) khi n, m → ∞ ta ¯d ’ u ’ o . c lim n,m→∞ u n −u m  = 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 x − y = lim n→∞ x −u n  = d. D ¯ ˘ a . t z = x −y th`ı x = y + z v`a z = d. Ta ch ´ ’ ung minh z ⊥ M. L ´ ˆay u ∈ M. V ´ ’ oi mo . i α ∈ R ta c´o z − αu, z − αu = z, z −2z, uα + u, uα 2 = z 2 − 2z, u + u 2 α 2 2. T´ınh tr . ’ uc giao, h`ınh chi ´ ˆeu 51 = d 2 + 2z, u + u 2 α 2 . M ˘ a . t kh´ac, z − αu, z − αu = z −αu 2 = x − (y + αu) 2 ≥ d 2 , (y + αu ∈ M) nˆen ta c´o d 2 − 2z, uα + u 2 α 2 ≥ d 2 hay u 2 α 2 − 2z, uα ≥ 0, ∀α ∈ R. T ` ’ u ¯d´o ∆  = z, u 2 ≤ 0. D ¯ i ` ˆeu n`ay x ’ ay ra khi v`a ch ’ i khi z, u = 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  + z  . Khi ¯d´o y − y  = z  − z. V`ı M v`a M ⊥ l`a c´ac khˆong gian con nˆen y − y  ∈ M, z  − z ∈ M ⊥ . Khi ¯d´o 0 = y − y  , z  − z = y −y  , y − y  . T ` ’ u ¯d´o y − y  = 0. Ta suy ra y = y  v`a z = z  . ✷  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 P x = y ≤  y 2 + z 2 = x (do ¯di . nh l´y Pithagore).  Hˆe . 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 = x, z = y + z, z = y, z + z, z = z, z. T ` ’ u ¯d´o z = 0. D ˜ ˆan ¯d ´ ˆen x = y ∈ M. Ta suy ra ¯d ’ u ’ o . c (M ⊥ ) ⊥ = M. Vˆa . y (M ⊥ ) ⊥ = M. ✷  Hˆe . 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 ⊥ ) ⊥ . 52 Ch ’u ’ ong 5. Khˆong gian Hilb ert 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]. Vˆa . y (M ⊥ ) ⊥ = M. ✷  Hˆe . 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}. ✷  Hˆe . 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. ✷ 3. Hˆe . tr ’ u . c chu ’ ˆan 3.1 Hˆe . tr ’ u . c chu ’ ˆan a) C´ac ¯di . nh ngh ˜ ia ✷ 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 e i , e j  = δ ij =  0 n ´ ˆeu i = 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). ✷ 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 = x, e 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 ∞  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 ∞  i=1 ξ 2 i ≤ x 2 , v ´ ’ oi ξ i = x, e 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  i=1 ξ i e i , (n = 1, 2, . . .) th`ı x = y n + n  i=1 ξ i e i . V ´ ’ oi j = 1, . . . , n, ta c´o y n , e j  = x − n  i=1 ξ i e i , e j  = x, e i  − n  i=1 ξ i e i , e j  = x, e j  −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 x 2 = y n + n  i=1 ξ i e i  2 = y n  2 + n  i= ξ i e i  2 = y n  2 + n  i=1 ξ 2 i ≥ n  i=1 ξ 2 i . Cho n → ∞ th`ı ∞  i=1 ξ 2 i ≤ x 2 . ✷  Hˆe . 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 ∞  i=1 ξ i e i luˆon hˆo . i tu . v`a (x − ∞  i=1 ξ i e i ) ⊥ e j , ∀j. Ch ´ ’ ung minh. V`ı ∞  i=1 ξ i e i  2 = ∞  i=1 ξ 2 i ≤ x 2 < ∞. nˆen theo ¯di . nh l´y (6) ta suy ra chu ˜ ˆoi ∞  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 x − ∞  i=1 ξ i e i , e j  = lim n→∞ x − n  i=1 ξ i e i , e j  = 0. Vˆa . y (x − ∞  i=1 ξ i e i ) ⊥ e j , ∀j. ✷ 3.2 Hˆe . tr ’ u . c chu ’ ˆan ¯d ` ˆay ¯d ’ u ✷ 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 = x, e 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 = ∞  i=1 ξ i e i . 54 Ch ’u ’ ong 5. Khˆong gian Hilb ert iii) V ´ ’ oi mo . i x ∈ X th`ı x 2 = ∞  i=1 ξ 2 i , (¯d ’ ˘ ang th ´ ’ uc Passerval). iv) V ´ ’ oi mo . i x ∈ X, y ∈ X th`ı x, y = ∞  i=1 ξ i η i v ´ ’ oi ξ i = x, e i , η i = y, e 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 − ∞  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 − ∞  i=1 ξ i e i = 0. Do ¯d´o x = ∞  i=1 ξ i e i . (ii) ⇒ (iv): V ´ ’ oi ξ i = x, e i , η j = y, e j , i, j = 1, 2, . . . ta c´o x, y =  ∞  i=1 ξ i e i , ∞  j=1 η j e j  =  lim n→∞ n  i=1 ξ i e i , lim n→∞ n  j=1 η j e j  = lim n→∞  n  i=1 ξ i e i , n  j=1 η j e j  = lim n→∞ n  i=1 ξ i η i e i , e i  = lim n→∞ n  i=1 ξ i η i = ∞  i=1 ξ i η i . (iv) ⇒ (iii): T ` ’ u (iv), cho y = x th`ı ta ¯d ’ u ’ o . c x 2 = x, x = ∞  i=1 ξ 2 i . (iii) ⇒ (i): Gi ’ a s ’ ’ u c´o (iii) v`a x ⊥ e i , ∀i. T ` ’ u ¯d´o ξ i = x, e i  = 0, ∀i. Suy ra x 2 = ∞  i=1 ξ 2 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 = ∞  i=1 ξ i e i = lim n→∞ n  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. ✷ ∆ 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 ∞  i=1 ξ 2 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 = ∞  i=1 ξ i e i , x 2 = ∞  i=1 ξ 2 i .