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1 I , : ~/math/functional-analysis/functional-analysis-1.tex

2 Fourier R n R n Schwarz R n Euclid (1) Gram-Schmidt QR (1) (2) (2) Riesz C n 21 4 Hilbert Bessel Parseval Lax-Milgram, Stampacchia Lax-Milgram Stampacchia A 29 B 30 B

3 C 31 C C C D 33 PDF 2

4 1 1.1 Fourier 1 CG [10] () 1.2 Hilbert 1. Schwarz 2. Bessel 3. () () 2, 3 Euclid 1 1 3

5 1.3 Gram-Schmidt Lanczos 1.4 Fourier Fourier () Fourier 4

6 2 R n 2.1 x, y (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3 (x, y) = (x ) (y ) cos θ (θ x y ) R R n n R n n n x x x T t x a v 1,, v m R n { m } Span(v 1,, v m ) := λ j v j ; (λ 1,, λ m ) R m a x A T 2.2 R n Schwarz n x = (x 1,, x n ) T, y = (y 1,, y n ) T R n (2.1) (x, y) def. = n x i y i i=1 (x, y) x y ( ) R n (1) x R n (x, x) 0. x = 0 (x, x) = 0. (2) x, y, z R n λ, µ R (λx + µy, z) = λ(x, z) + µ(y, z). (3) x, y R n (x, y) = (y, x). 5

7 (1), (2), (3) (, ) (2.1) (x, y) R n Euclid (x, y) = y T x (AB) T = B T A T (Ax, y) = y T (Ax) = (y T A)x = (A T y) T x = (x, A T y) A (Ax, y) = (x, Ay) ( ) A R n n (i), (ii) (i) A (ii) x, y R n (Ax, y) = (x, Ay). (i) = (ii) (ii) = (i) x, y R n (Ax, y) = (x, Ay) (Ax, y) = (x, A T y) (x, Ay) = (x, A T y). x Ay = A T y ( (x, Ay A T y) x = Ay A T y Ay A T y = 0) y A = A T. Schwarz (Schwarz ) x, y R n (x, y) 2 (x, x)(y, y). x y 1 ( ) x y 1 t tx + y 0. (tx + y, tx + y) > 0. (x, x)t 2 + 2(x, y)t + (y, t) > 0. t 2 4 = (x, y) 2 (x, x)(y, y) < 0. (x, y) 2 < (x, x)(y, y). x y (x, y) 2 = (x, x)(y, y) 6

8 2.3 R n Euclid Euclid Euclid (Euclid ) x R n x def. = (x, x) = n x i 2 i=1 x Euclid () R n Euclid (1), (2), (3) (1) x R n x 0. x = 0 (2) x R n, λ R λx = λ x. (3) x, y R n x + y x + y. ( x y ) 7

9 () (, ) R n Euclid (1) x, y R n x y def. (x, y) = 0. x y (2) x R n, M R n x M def. y M x y x M (3) M R n, N R n M N def. ( x M) ( y N) x y M N (4) M R n M def. = {x R n ; x M} M (, ) R n Euclid Euclid (1) (9) (1) x y y x, M N N M. (2) () x y x 2 + y 2 = x + y 2. (3) ( (parallelogram theorem)) x + y 2 + x y 2 = 2( x 2 + y 2 ). (4) ( (polarization identity)) (x, y) = 1 ( x + 4 y 2 x y 2 ). (5) {0} = R n, (R n ) = {0}. (6) M R n M R n (7) M 1 M 2 R n (M 1 ) (M 2 ). (8) M R n M (M ). (9) M R n M M = {0}. (1) (x, y) = (y, x) 8

10 (2) x + y 2 = (x + y, x + y) = (x, x + y) + (y, x + y) = (x, x) + (x, y) + (y, x) + (y, y) = x 2 + 2(x, y) + y 2 (x, y) = 0 x + y 2 = x 2 + y 2. (3) x + y 2 + x y 2 = (x + y, x + y) + (x y, x y) = [(x, x) + (x, y) + (y, x) + (y, y)] + [(x, x) (x, y) (y, x) + (y, y)] = 2 [(x, x) + (y, y)] = 2( x 2 + y 2 ). (4) 1 ( x + y 2 x y 2) = 1 {[(x, x) + (x, y) + (y, x) + (y, y)] [(x, x) (x, y) (y, x) + (y, y)]} 4 4 = 1 [(x, y) + (y, x)] = (x, y). 2 (5) x R n (x, 0) = 0, x 0 x {0}. R n {0}. {0} = R n. x (R n ) x : (x, x) = 0. x = 0. (R n ) = {0}. (6) x, y M z M λ, µ R (x, z) = (y, z) = 0. (λx + µy, z) = λ(x, z) + µ(y, z) = λ 0 + µ 0 = 0. λx + µy M. M (7) (8) x M y M (x, y) = 0. x (M ). M (M ). (9) x M M x : (x, x) = 0. x = 0. M M {0}. M M = {0}. M R n M R n V V V V V = R n R 3 V (1 ) V V (2 ) V V V () V R n dim V = n dim V. 9

11 2.4.1 () Pappus ( Pappus (B.C.260 ) ( ) Perga Apollonius (B.C ) ) ( ) ( ) () V R n (V ) = V. V (V ) V V dim ( (V ) ) = n dim(v ) = n (n dim V ) = dim V V 1, V 2 R n (1) (V 1 + V 2 ) = (V 1 ) (V 2 ). (2) (V 1 V 2 ) = (V1 ) + (V2 ). 2.5 (1) 1. v ( 0) u p p Span (v) (2.2) p = λv, λ R (p u) v (p u, v) = 0. (2.2) (λv u, v) = 0. 10

12 v 0 λ = (u, v) (v, v) p = λv = p = (u, v) v 2 (u, v) (v, v) v. p Span(v), p u v p = ( ) v v u, v v n = v ( v ) v v p = (u, n)n (u, n) u v u v θ (u, n) = u cos θ u Span(v) ( ) () p u p u = min x u x Span (v) x u 2 = x p 2 + p u 2 x u 2 x p v 1, v 2 u Span (v 1, v 2 ) 11

13 Span(v) x x = λv 1 + µv 2, λ, µ R (x u, v i ) = 0 (i = 1, 2) i.e. (x, v i ) = (u, v i ) (i = 1, 2) 1 λ(v 1, v 1 ) + µ(v 2, v 1 ) = (u, v 1 ) λ(v 1, v 2 ) + µ(v 2, v 2 ) = (u, v 2 ) ( (v 1, v 1 ) (v 2, v 1 ) (v 1, v 2 ) (v 2, v 2 ) ) ( λ µ ) = ( (u, v 1 ) (u, v 2 ) ). Schwarz 1 1 λ, µ (Gram-Schmidt ) ( ) R n V x R n y M, (x y) V y y x V (orthogonal projection) M ( ) R n V x, y R n (i) y x V (ii) x y = min x v. v V [(i)= (ii)] y x V z V 3 x, y, z y y z V (x y, y z) = 0. x z 2 = (x y) + (y z) = x y 2 + y z 2. x z 2 x y 2, y z = 0 z = y. x y = min x z. z V 1 Gram 12

14 [(ii) = (i)] h V f : R R f(t) = x (y th) 2 = x y + th 2 (t R) (y th) V f 0 f (0) = 0 f(t) = x y 2 + 2t(x y, h) + t 2 h 2 (x y, h) = 0. h V x y V (1 ) v 0 V = Span(v) = {λv; λ R} u R n V ( ) w = (u, v) (v, v) v 1 Schwarz Schwarz Schwarz y = 0 y 0 x V = Span(v) w = λy, (x, y) λ = 3 0, w, x (y, y) (x, x) = x 2 = x w 2 + w 2 w 2 = λ 2 y 2 = ( ) 2 (x, y) y 2 = (y, y) () (x, x)(y, y) (x, y) 2. (x, y)2 (y, y). (x w, x w) = 0 w = x x y Gram-Schmidt Schmidt () 13

15 2.6.1 (Schmidt ) u 1,, u m R n v 1,, v m v 1 = u 1, v i = u i i 1 (u i, v j ) (v j, v j ) v j (v i, v j ) = 0 (1 j < i m), Span(u 1,, u i ) = Span(v 1,, v i ) (CG ) (i = 1, 2,, m) (Gram-Shimidt ) u 1,, u m R n v 1 = u 1, w 1 = 1 (v1, v 1 ) v 1, i 1 v i = u i (u i, w j )w j, w i = w 1,, w m (w i, w j ) = δ ij (1 j, i m), 1 (vi, v i ) v i (i = 2, 3,, m) Span(u 1,, u i ) = Span(w 1,, w i ) (i = 1, 2,, m) ( ) R n V V v 1,, v m V s.t. V = Span(v 1,, v m ) and (v i, v j ) = δ ij (1 i j m). 2.7 QR (QR ) ( ) ( ) (QR ) QR [7] 14

16 2.8 (1) (2) (, (projection theorem)) V R n x R n V y V v 1,, v m m y = (x, v j )v j V v 1,, v m x R n V y v i : m (2.3) λ 1,, λ m R s.t. y = λ j v j. (x y) V {v i } (2.3) ( m ) (y, v i ) = λ j v j, v i = (x y, v i ) = 0 (i = 1, 2,, m) (x, v i ) = (y, v i ) (i = 1, 2,, m) m λ j (v j, v i ) = λ i = (x, v i ) (i = 1, 2,, m) y = m (x, v j )v j x V y z = x y x = y + z, y V, z V y + z = y + z x = y + z, y V, z V y = y and z = z. y y = z z V V = {0} 0 () R n = V V 15 m λ j δ ji = λ i

17 2.9 V 1, V 2 R n (V 1 V 2 ) V 1 + V 2 = {v 1 + v 2 ; v 1 V 1, v 2 V 2 } V 1 V 2 V 1 V 2 () V 1 V 2 x V 1 V 2 x = v 1 + v 2 (v 1 V 1, v 2 V 2 ) v 1, v 2 x = v 1 + v 2 (v 1 V 1, v 2 V 2 ) v 1 v 1 = v 2 v 2 V 1, V 2 V 1 V 2 V 1 V 2 0 : v 1 = v 1, v 2 = v 2. v 1 v 1 = v 2 v 2 = [5] V 1 +V 2 V 1 V 2 [6] V 1 V (2) () V R n dim(v ) = n dim V. R n u 1,, u n m u 1,, u m V Gram-Schmidt v 1,, v n Span(v 1,, v m ) = Span(u 1,, u m ) = V v 1,, v m V V = Span(v m+1,, v n ) x R n n x = (x, v j )v j x V (x, v j ) = 0 (j = 1, 2,, m) x V x Span(v m+1,, v n ). V Span(v m+1,, v n ). V Span(v m+1,, v n ) (2.4) V (V ) = V (2.4) 16

18 2.11 ( ) ( ) V R n P x = x V P : R n R n V () R n V P (1) P (2) P : P 2 = P. (3) P : P T = P. (1) x 1 R n, x 2 R n λ 1 R, λ 2 R P x 1 = y 1, P x 2 = y 2 y 1, y 2 V x 1 y 1 V, x 2 y 2 V. λ 1 y 1 + λ 2 y 2 V (λ 1 x 1 + λ 2 x 2 ) (λ 1 y 1 + λ 2 y 2 ) = λ 1 (x 1 y 1 ) + λ 2 (x 2 y 2 ) V. P (λ 1 x 1 + λ 2 x 2 ) = λ 1 y 1 + λ 2 y 2 = λ 1 P x 1 + λ 2 P x 2. (2) x R n P x = y y V, x y V y V, y y = 0 V P y = y. P 2 x = P x. P 2 = P. (3) V v 1,, v m ( m ) (P x, y) = (x, v j )v j, y = (x, P y) = ( x, ) m (y, v j )v j = m (x, v j )(v j, y), m (y, v j )(x, v j ). (P x, y) = (x, P y). P T = P P 2 = P, P T = P P V () P R n n P 2 = P, P T = P V = {x R n ; P x = x} = ker(i P ) R n V V P 17

19 V = ker(i P ) R n x R n P x = y P 2 = P P y = P (P x) = (P P )x = P 2 x = P x = y V y V. v V P v = v (x y, v) = (x, v) (y, v) = (x, v) (P x, v) = (x, v) (x, P v) = (x, v) (x, v) = 0. x y V. y x V P 2 = P, P T = P P R n n V ker(i P ) P R n n ker(i P ) = {x R n ; P x = x} = image P. ker(i P ) = {x R n ; (I P )x = 0} = {x R n ; P x = x}. {x R n ; P x = x} image P P 2 = P {x R n ; P x = x} image P y image P x R n s.t. P x = y P y = P (P x) = P 2 x = P x = y y {x R n ; P x = x} () P R n n Q def. = I P ( I ) (1), (2), (3) (1) Q (2) R n = image P + image Q. (3) image P image Q. (4) x R n x = y + z, y image P, z image Q y = P x, z = Qx. (1) Q Q 2 = (I P ) 2 = I 2 IP P I + P 2 = I 2P + P 2 = I 2P + P = I P = Q, Q T = (I P ) T = I T P T = I P = Q. (2) I = P + (I P ) = P + Q x R n x = P x + Qx image P + image Q. R n image P + image Q. (3) x, y R n (P x, Qy) = (x, P Qy) = (x, P (I P )y) = (x, P y P 2 y) = (x, P y P y) = (x, 0) = 0 image P image Q. 18

20 (4) x = y + z, y image P, z image Q y image P P y = y. z image Q = ker(i Q) = ker P P z = 0. P x = P y + P z = y + 0 = y. Qx = Qy + Qz = 0 + z = z Riesz (Riesz (Riesz representation theorem)) f R n f : R n R a R n f(x) = (x, a) (x R n ) Riesz ( R n ) R n a = (a 1,, a n ) T f(x) = (x, a) = a 1 x a n x n a j = f(e j ), e j = (δ ij ) (j = 1, 2,, n) a a V = {x R n ; f(x) = 0} V u a u a = λu λ λ f(x) = (x, a) = (x, λu) x = u f(u) = (u, λu) = λ(u, u) λ = f(u) (u, u). f(x) = (x, a), a def. = f(u) (u, u) u f = 0 (a = 0 ) f 0 V def. = ker f = {x R n ; f(x) = 0} R n V R n. y R n \ V y V z u = y z u V u 0 x R n y = x f(x) f(u) u 19

21 f(y) = f(x) f(x) f(u) = f(x) f(x) = 0 f(u) y ker f = V. (y, u) = 0 ( (y, u) = x f(x) ) f(u) u, u = (x, u) f(x) (u, u) f(u) f(x) = f(x) (u, u) = (x, u). f(u) (x, u)f(u) (u, u) = a = f(u) (u, u) u ( x, f(u) ) (u, u) u. 20

22 3 C n Hermite C z z z C n n C n n n x x x T t x 21

23 4 Hilbert SpanL () X C X X (x, y) x, y C (i), (ii), (iii) (i) x, x 0. x = 0 (ii) x, y = y, x. (iii) αx + βy, z = α x, z + β y, z (), (inner product space, pre-hilbert space) Banach Hilbert (Hilbert space) () R n Euclid Hilbert () C n Hermite Hilbert (L 2 ) (l 2 ) f, g = Ω l 2 def. = {x = (x 1, x 2, ); f(x)g(x) dµ(x) x j 2 < }. 22

24 (, ) X (1) X {u j } j N (orthonormal system) (u j, u k ) = δ jk (2) X {u j } j N (complete orthonormal system) (orthonaormal basis) {u j } : x X {α n } n N X N N v α j u j 0 (N ) {u j } j N X x n x = α n u n i.e. lim n x α j u j = 0 n=1 α n = (x, u n ) x 2 = (x, u n ) 2. n=1 4.5 Bessel (Bessel ) X {u n } n N x, u j 2 x 2 x X n N (x X). n x, u j u j x Span(u 1, u 2,, u n ) n 2 n 2 x, u j u j + x x, u j u j = x 2. n 2 x, u j u j x 2. 23

25 n 2 x, u j u j = n x, u j u j 2 = n x, u j 2 n x, u j 2 x 2. n x, u j 2 x 2. X {u j } x X x, u j u j j Cauchy X Hilbert X {u n } n N x X p = x, u j u j x M := Span{u j } p M, (x p) M. 4.6 Parseval () X (separable) X D D = X () Hilbert X {φ n } n N {x n } n N Schmidt {u n } n N (u i, u j ) = δ ij (i, j = 1, 2, ), () n N Span(x 1,, x n ) = Span(u 1,, u n ). 24

26 4.7.3 () Hilbert X {φ n } 5 (i) (v) (i) V := Span{φ n }, M := V M = X. (ii) u X u = n u, φ n φ n. (iii) u X u 2 = n u, φ n 2. (iv) u, v X u, v = n u, φ n v, φ n. (v) u X n u, φ n = 0 = u = () X Hilbert V x X V y δ = inf x z z V (y n ) n N V N x y n δ (n ) 2( x y n 2 + x y m 2 ) = (x y n )+(x y m ) 2 + (x y n ) (x y m ) 2 = 4 x y n + y m + y 2 m y n y m y n 2 = 2( x y n 2 + x y m 2 ) 4 x y 2 n + y m 2 2( x y n 2 + x y m 2 ) 4δ 2. n, m 2(δ 2 + δ 2 ) 4δ 2 = 0 {y n } n N Cauchy lim n y n = y V y V. x y = min x z. z V x y V h V f(θ) = x y θh 2 θ = 0 δ f (0) = 0. f(θ) = x y 2 2θRe (x y, h) + θ 2 h 2 25

27 f (0) = Re (x y, h) Re (x y, h) = 0. h ih 0 = Re (x y, ih) = Re ( i(x y), h) = Im (x y, h). (x y, h) = 0. h V x y V V Hilbert (V ) = V ( ) K Hilbert X x X x y = min x v v K y K y K Re (x y, v y) 0 (v K) 26

28 5 Lax-Milgram, Stampacchia 5.1 Lax-Milgram (Lax Milgram ) H Hilbert a: H H R H φ H (1), (2) (1) (W) (W) Find u H s.t. (5.1) a(u, v) = φ, v (v H). (2) a (W) u (V) (V) Find u H s.t. { } 1 1 (5.2) a(u, u) φ, u = min a(v, v) φ, v. 2 v H 2 u H (5.1) (5.2) (V) F (v) def. = 1 a(v, v) φ, v 2 F (u) = 0 (5.1) 27

29 5.2 Stampacchia (Stampacchia ) H Hilbert K H a H φ H (i), (ii) (i) (W S ) (W S ) Find u K s.t. (5.3) a(u, v u) φ, v u (v K) (ii) a (W S ) u (V S ) (V S ) Find u K s.t. { } 1 1 (5.4) a(u, u) φ, u = min a(v, v) φ, v. 2 v K 2 u K (5.3) (5.4) (V S ) F (v) def. = 1 a(v, v) φ, v 2 F (u) = 0 (5.2) K = u 0 + V = {u 0 + v; v V } (u 0 H, V : H ) a(u, v u) φ, v u (v H) a(u, v) = φ, v (v H). 28

30 A 29

31 B B.1 B.1.1 V 1, V 2 R n 3 (i) V 1 + V 2 V 1 V 2 (ii) V 1 V 2 = {0}. (iii) dim(v 1 + V 2 ) = dim V 1 + dim V 2. B.1.2 ( ) V 1, V 2 V 1 V 2 = {0} R n V 1 + V 2 V 1 V 2 V 1 V 2 V 1 V 2 V 1 V 2 = {0} V 1 + V 2 30

32 C C.1 C.1.1 (von Neumann) X (X X ) x + y 2 + x y 2 = 2( x 2 + y 2 ) (x, y X) Yosida [11] C.2 C.2.1 (, ) X V x X (C.1) x = y + z, y V, z V y x V y x V y V, x y V ( ) (C.1) y, z x = y + z = y + z, y, y V, z, z V y y = z z V V 0 : y y = z z = 0. y = y, z = z. 31

33 C.3 x 0 (x x, a) = 0 l x 0 x a (x, a) = c (x 0 x, a) a a 2 l = (x 0 x, a). a l = (x 0, a) c. a R 2 (x 0, y 0 ) ax + by = c l = ax 0 + by 0 c a2 + b 2. R 3 (x 0, y 0, z 0 ) ax + by + cz = d l = ax 0 + by 0 + cz 0 d a2 + b 2 + c 2. 32

34 D 33

35 [1], I, II, (1966, 1969). [2], 1, 2,, (1997, 199?). [3],, (1980, 1999). [4],, (1994). [5],, (1966). [6],, (1985). [7],,,, (1994). [8],,,, (1988). [9], Fourier, (2003). [10],,, ( ), (1978). [11] Kôsaku Yosida, Functional analysis, sixth edition, Springer (1980). 34

2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C

2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C 2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe