(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., 1.1.1. (1) 1.1.1 (iii), V 0. V 0. (2) 1.1.1 (iv), y x, x. (3) x, y V, x y x y = x + ( y). x y x y. 2

(4) 0 x = 0. 0 C 0, 0 V.. (5) ( 1)x = x. 1.1.2 H., H, H x, y, (x, y), (i) (iv) : x, y, z H, α C, (i) (x, y) 0,, (x, y) = 0 z = 0. (ii) (x, y) = (y, x). (iii) (x + y, z) = (x, z) + (y, z). (iv) (αx, x) = α(x, y). (x, y) x y. 1.1.1 H, (1), (2) : x, y, z H, α C, (1) (x, y + z) = (x, y) + (x, z). (2) (x, αy) = α(x, y) H, x H, x = (x, x), x H.,. 3

1.1.1 H, x, (1) (3) : x, y H, α C, (1) x 0., x = 0, x = 0. (2) x + y x + y. (3) αx = α x., H x. H x, x = 1. H 0 x, e = x x x. 1.1.2 H., (1) (3) : x, y H, (1) (x, y) x y. ( ). (2) (x, y) = 1 4 ( x + y 2 x y 2 + i( x + iy 2 x iy 2 )). (3) x + y 2 + x y 2 = 2( x 2 + y 2 ). ( ). 1.1.3 H, H x y d(x, y) = y x., (1) (3) : (1) d(x, ; y) 0., x = y, d(x, y) = 0. 4

(2) d(x, y) = d(y, x). (3) d(x, z) d(x, y) + d(y, z). ( ) 1.1.3, H. H, H x H x. H {x n } H x, lim x n x = 0 n. lim x n = x, x n x, (n ) n., x n x., {x n }. {x n },, x n x m x n x + x x m lim x n x m = 0 m, n.., {x n },., H,. 1.1.3 H, H, d(x, y) = y x. 5

1.1.3, H, H {x n }, x H x n x. 1.1.4 H, (x, y) x y., x n x, y n y, lim (x n, y n ) = (x, y) n., x n x,. lim x n = x n 1.1.5( ) H 1., H, H 1 H.. H H {xn }., H {x n } {y n } x, y, x + y α αx x + y = {x n + y n }, αx = {αx n }, H., x y (x, y) = lim n (x n, y n )., H., x, y H, x y = lim n x n y n = 0 6

. x y., H., x, y, z, (1) (3) : (1) x x. (2) x y, y x. (3) x y, y z, x z. H H = H/, H, H 1 H. H H 1.,,..,. 1.1.4 H. H K, x, y K 0 α 1 α, αx + (1 α)y K. 7

1.1.6 H, K H, α = inf x K x., K {x n } lim x n = α n., {x n }. 1.1.6 {x n } K. 1.1.5 H, H.,. 1.1.1 L 2 (a, b) R (a, b), 2, L 2 (a, b)., f, g L 2 (a, b), (f, g) = b a f(x)g(x)dx., L 2 (R) = L 2 (, ).. 1.1.2 L 2 (a, b), {f n (x)} f(x), {f nk (x)}, (a, b) x f nk (x) f(x). 1.1.2 d 1, Ω R d., L 2 (Ω) Ω, 8

2. L 2 (Ω)., f, g L 2 (Ω), (f, g), (f, g) = f(x)g(x)dx Ω., L 2 (R d ). 1.1.3 l 2 x = {ξ n }, ξ n 2 <., l 2 x = {ξ n } y = (η n ) (x, y) = ξ n η n, l 2. 1.2,,.,. H., H x y, (x, y) = 0., x y. 0 H x. 9

H 0 {f 1, f 2,, f n, }, (f m, f n ) = 0, (m, n = 1, 2, )., {f n }, f n = 1, (n = 1, 2, 3, )., (f m, f n ) = δ mn, (mn = 1, 2, 3, )., δ mn, 1, (m = n), δ mn = 0, (m n). H {f 1, f 2,, f n } 1, 1, n α j f j = 0, (α j C, (1 j n)) j=1, α 1 = α 2 = = α n = 0. {f 1, f 2,, f n } 1, 1., H 0 {f 1, f 2,, f n } 1. 10

1.2.1( ) H 1 {f 1, f 2, },., e 1 = f 1 f 1, e 2 = f 2 (f 2, e 1 )e 1 f 2 (f 2, e 1 )e 1, e n = n 1 f n (f n, φ j )e j j=1, (n 2) n 1 f n (f n, e j )e j j=1, {e 1, e 2, }., e n f 1, f 2,, f n 1,, f n e 1, e 2, ;, e n 1. 1.2.1 L 2 ( π, ; π), (1) (3) : (1) {cos nx; n = 0, 1, 2, }. (2) {sin nx; n = 1, 2, }. (3) {cos nx; n = 0, 1, 2, } {sin nx; n = 1, 2, }. 1.2.2 L 2 ( π, ; π), (1) (3) : (1) { 1 2π, 1 π cos nx; n = 1, 2, }. (2) { 1 π sin nx; n = 1, 2, }. 11

(3) { 1 2π, 1 π cos nx, 1 π sin nx; n = 1, 2, }. H, {e n }., H x, x = α 1 e 1 + α 2 e 2 + + α n e n +., {α n }, α n = (x, e n ), (n = 1, 2, ). H x, {α n } x {e n }, x.,. 1.2.1( ) H, {e n } H., {α n } α n 2 <, x = α n e n, (x, e n ) = α n, (n 1) x H. 1.2.2( ) H {e n } 1.2.1., x H, (x, e n ) 2 x 2 12

., x H {e n }., x H {e n }.,. H {e n }, x H, (x, e n ) = 0, (n 1), x = 0. {e n }, e n,., H {x α }, α, (x, x α ) = 0, x = 0., {x α }. 1.2.3 H,.,.,. 1.2.4 H, {e n }, (1) (5) : (1) {e n }. (2) x, y H, (x, e n ) = (y, e n ), (n 1), x = y. 13

(3) x H, x = α n e n, α n = (x, e n ), (n 1). (4) x, y H, α n = (x, e n ), β n = (y, e n ), (n 1),. (x, y) = α n β n (5) x H,. (x, e n ) 2 = x 2 H., H {e n } H, 1.2.4., H {f n } H. H {e n }, {e n } H. H. 1.2.5 H, {e n } H., {e n }, x H ε > 0, {e n } 1 y = N α n e n 14

, x y < ε. 1.2.6 H, {e n } H., x H x = α n e n, α n = (x, e n ), (n 1), α = (α n ) l 2., x H α = (α n ) l 2.,. H = l 2 1.2.3 l 2, e n = (δ ni, i 1), (n 1), {e 1, e 2, }. 1.2.4 { 1 2π, 1 π cos nx, 1 sin nx; n = 1, 2, } π L 2 ( π, π). 1.2.5 L 2 ( π, π) f, : f(x) = 1 a 0 + 1 (a n cos nx + b n sin nx), 2π π 15

a n = 1 π π π b n = 1 π π π f(x) cos nxdx, (n = 0, 1, 2, ), f(x) sin nxdx, (n = 1, 2, ). L 2. 1.2.6 { 1 2π e inx ; n = 0, ±1, ±2, } L 2 ( π, π). 1.2.7 L 2 ( π, π) f, : f(x) = 1 a n e inx, 2π a n = 1 2π π n= π f(x)e inx dx, (n = 0, ±1, ±2, ). L 2. 1.2.8 { 1 π, 2 } cos nx; n = 1, 2, π L 2 (0, π). 1.2.9 { 2 } sin nx; n = 1, 2, π L 2 (0, π). 1.2.10 < a < b <., (a, b) { 1 b a e 2πinx/(b a) ; n = 0, ±1, ±2, } 16

L 2 (a, b). 1.2.11 d n P n (x) = 1 2 n n! dx n (x2 1) n, (n = 0, 1, 2, )., { 2n + 1 P n (x); n = 0, 1, 2, } 2 L 2 ( 1, 1). 1.2.12 dn H n (x) = ( 1) n e x2 dx n e x2, (n = 0, 1, 2, )., { 1 2 n n! π H n(x)e x2 /2 ; n = 0, 1, 2, } L 2 (, ). 1.2.13 L n (x) = e x dn dx n (xn e x ), (n = 0, 1, 2, )., { 1 n! L n(x)e x/2 ; n = 0, 1, 2, } L 2 (0, ). 17

1.3,. H M, H., M x, y α C, (i), (ii) : (i) x + y M. (ii) αx M. H, H M x, y H (x, y) M M.,. 1.3.1 H M H M H. 1.3.1, H, H, M H. H M N, (x, y) = 0, (x M, y N)., M N., M N = {0}., H x M, (x, y) = 0, (y M) 18

., x M. H A, A A = {u H; v A, (u, v) = 0}., : 1.3.2, A H. H M, M M., M M., M M = {0}. 1.3.3 H. M H, M H, M {0}., H x 0, x M. 1.3.4 H M, x H,. x = y + z, y M, z M 1.3.1 1.3.4,. (M ) = M H S, S 1 H L(S) L(S) = M S 19

., M = L(S) H.,. H., H x. x x., x, y H α C, (1) (4) : (1) x = x. (2) x + y = x + y. (3) αx = α x. (4) (x, y) = (x, y)., x H, x = x., x H,,, x 1 = x + x, x 2 = x x 2 2i x 1 = x 1, x 2 = x 2 x = x 1 + ix 2, x = x 1 ix 2. x 1 = Re (x), x 2 = Im (x), x 1 x, x 2 x., H 1 = {x H; x = x} 20

, H 1,. H = H 1 ih 1 21