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

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1 2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA n 1 ɛ-δ K2-2S04-00 : C

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

3 2S III IV K201 : April 16, 2004 Version : 1.1 α R X X < 0.01 (1 + X) α =1+αX + O(X 2 ) O(X 2 ) ɛ(x) :=(1+X) α (1 αx) ɛ(x)/x 2 X 1. (1) 1 x (2) F (x) =Sin 1 x F (X) 0 2 (3) F (x) 3 0 (4) f(x) =Sin 1 x +2 1 x 2 lim x 0 f(x) 2 x x 2 2. D = {(x, y) R 2 : 2x y 1, x 3y 1} (x 3y) 100 dxdy D f : D f(d) =E x, x D f(x) =f(x ) E x = x 3. (1) z C D = {z :1/2 < z < 2} f(z) =z 2 D (2) C ( z = x ) + yi( (x, ) y) R 2 f : Re z Re z 2 Im z Im z 2 D 0 4. R 2 R 2 ( ) ( ) ( ) ( ) x Re z Re z 2 u f : = = y Im z Im z 2 v K2-2S04-01 : C

4 2S III IV K201 ( ) (1) u, v x, y f 1 1 (2) X := x 1, Y := y 1, U := u, V := v 2 U, V X, Y (3) X, Y ( ) ( ) 2 A U X A V Y A ( ) (4) f (5) 1 1 ( ) 1 f 1 3 u 1, u 2, u 3 R 3 R 3 p R 3 3 x 1,x 2,x 3 1 p = x 1 u 1 + x 2 u 2 + x 3 u 3 R 3 5. u 1, u 2, u 3 R 3 R 3 x 1 u 1 + x 2 u 2 + x 3 u 3 = 0 x 1 = x 2 = x 3 = e 1 = 0, e 2 = 1, e 3 = 0 R u 1 = 0, u 2 = 1, u 3 = 1 e 1, e 2, e 3 u 1, u 2, u u 1, u 2, u 3 R 3 A u 1 = Au 1, u 2 = Au 2, u 3 = Au 3 R 3 Hint. p R 3 3 x 1,x 2,x 3 p = x 1 u 1 + x 2 u 2 + x 3 u 3 p := A 1 p u i K2-2S04-01 : C

5 2S III IV K R 3 n R n 2. 7 u 1, u 2, u 3 R 3 A u 1 = Au 1, u 2 = Au 2, u 3 = Au 3 R 3 u i A 1 3. (1) (2) 4. 0 tan x sin x lim x 0 x 2 e x sin x x x 2 lim x 0 x 3 ( ) 1 4 f f 3 5. dxdy (D :1 x 2 + y 2 4) (x 2 + y 2 ) m D K2-2S04-01 : C

6 2S III IV K202 Jacobi : April 23, 2004 Version : 1.1 Jacobi R n R m x 1 u 1 u 1 (x 1,,x n ) f :.. =. u m (x 1,,x n ) x n u m n = 26, m = 10 x 1. x n = a 1. a n Jacobi Jacobian matrix J(a 1,...,a n )= ( ) ui (a 1,...,a n ) x j 1 i m,1 j n m n f a 1. a n = X j := x j a j,u i := u i b i Taylor X j U 1 X 1. = J(a 1,...,a n ). + O(X X 2 n). b 1. b m U m X n O(X X2 n ) n O(X X2 n ) n =3 X j X1 2 + X2 2 + X f Jacobi 1 f : R R Jacobi K2-2S04-02 : A

7 2S III IV K202 A n f : R n R n x f(x) =Ax E R n n E f(e) = det A E det A =0 f(e) =0 1. E Lebesgue E f(e) Jacobian 1. I = D dxdy (x 2 + y 2 ) m (D :1 x 2 + y 2 4) (1) ( ) ( ) ( ) r x r cos θ f : := θ y r sin θ f(e) =D rθ- E (2) I N E N 2 E 1... E N 2 E i (r i,θ i ) D i := f( E i ), (x i,y i ):=f(r i,θ i ) R := r r i, Θ:= θ θ i, X = x x i, Y = y y i f : E i D i (3) i D i E i (4) N D i I D i x i,y i, D i r i,θ i, E i (5) I drdθ E 2. I = (x 2 + y 2 + z 2 )dxdydz (D : x 2 + y 2 + z 2 1) D K2-2S04-02 : A

8 2S III IV K202 (1) 3 (x, y, z) =(x 0,y 0,z 0 ) x y2 0 + z2 0 (x, y, z) = (x 0,y 0,z 0 ) x, y, z I (2) r x r cos θ cos φ f : θ y := r cos θ sin φ φ z r sin θ f(e) =D rθφ- E (3) I N E N 3 E 1... E N 3 E i (r i,θ i,φ i ) D i := f( E i ), (x i,y i,z i ):=f(r i,θ i,φ i ) R := r r i, Θ:=θ θ i, Φ=φ φ i X = x x i, Y = y y i, Z = z z i f : E i D i (4) i D i E i (5) I D i (x 2 i + y 2 i + z 2 i ) x i, y i,z i, D i r i, θ i,φ i, E i (6) I drdθdφ E u = u(x, y) (x, y) =(a, b) Taylor u(x, y) =u(a, b)+u x (a, b)(x a)+u y (a, b)(y b) + 2!( 1 uxx (a, b)(x a) 2 +2u xy (a, b)(x a)(y b)+u yy (a, b)(y b) 2) + 1. u(x, y) = x 2 + y 2 (x, y) =(1, 2) 2 Taylor (1) (2) (1 + X) α =1+αX + α(α 1) 2 X 2 + K2-2S04-02 : A

9 2S III IV K (1) u(x, y) =sin(3x +2y) (x, y) =(0, 0) 2 Taylor sin X = X X 3 /3! + (2) v(x, y) =sin(x +4y) (x, y) =(0, 0) 2 Taylor (3) (u, v) =f(x, y) :=(sin(3x +2y), sin(x +4y)) (x, y) =(0, 0) f (4) (x n+1,y n+1 )=f(x n,y n ) (x 1,y 1 )=(1/1000, 1/1000) (x n,y n ) a, b, c > 0 I = ( x2 a + y2 2 b + z2 2 c )dxdydz x2 (D : 2 a + y2 2 b + z2 2 c 1) 2 D 4. 1 I I N := { D i N 2 } I N := {Jacobian E i I N } I N,I N I (N ) K2-2S04-02 : A

10 2S III IV K203 : April 30, 2004 Version : 1.2 (1) office(a439) (2)office hour Cafe David 2 V P Q PQ O OP P f (1) Q 1 f(q) (2) P, Q V f( OP + OQ) = f( OP) + f( OQ) (3) Q V α R f(α OQ) = αf( OQ) f V V 1. O f f(o) = O Q 100 Q A 35 1 B 42 K2-2S04-03 : B

11 2S III IV K (1) 2 u 1, u 2 O Q (x 1,x 2 ) 1 OQ = x 1 u 1 + x 2 u 2 Q =(x 1,x 2 ) A Q (2) Q (x 1,x 2 ) A 1 (3x 1 +2x 2,x 1 +4x 2 ) A Q 100 ( α 1 α 2 ) ( = ) 100 ( Q 100 α 1 u 1 + α 2 u 2 2 A =( )100 x 1 x 2 ) 42 (1) 2 v 1, v 2 Q (y 1,y 2 ) 1 OQ = y 1 v 1 + y 2 v 2 Q =(y 1,y 2 ) B Q (2) Q (y 1,y 2 ) B (2y 1, 5y 2 ) B Q 100 ( ) 100 ( y 1 y 2 ) ( = y y 2 Q y 1 v y 2 v B A ) K2-2S04-03 : B

12 2S III IV K203 2 B A A P =( ) P 1 AP =( )=:B OQ = x 1 u 1 + x 2 u 2 =(u 1, u 2 ) A f( OQ) = ( ) ( ) 3x 1 +2x 2 x 1 Of(Q) = (u 1, u 2 ) =(u 1, u 2 )A x 1 +4x 1 x 2 ( ) ( ) ( =(u 1, u 2 )P (P 1 AP )P 1 x =(u 1, u 2 )P P ( ) 2 1 (v 1, v 2 ):=(u 1, u 2 )P =(u 1, u 2 ) =(2u 1 u 2, u 1 + u 2 ) 1 1 ( y 1 y 2 f( OQ) = Of(Q) = (u 1, u 2 )P ) ( := P 1 ( x 1 x 2 x 2 ) ( = 1 3 ) ( 2 0 P x 1 x ( x 1 x 2 )( ). x 1 x 2 ) ) ( =(v 1, v 2 ) 2x 1 5x 2 x 1 x 2 ). ) =2x 1 v 1 +5x 2 v 2 2 (u 1, u 2 ) P (u 1, u 2 )P =(v 1, v 2 ) (u 1, u 2 ) ( x 1 x 2 ) (v 1, v 2 ) P 1 ( x 1 x 2 ) (u 1, u 2 ) A (v 1, v 2 ) P 1 AP K2-2S04-03 : B

13 2S III IV K203 V R (vector space) a, b V a + b V (unique ) α R, a V αa V a, b, c V, α, β R (1) (a + b)+c = a +(b + c). (2) a + b = b + a. (3) 0 V 0 + a = a + 0 (4) a V a V a + a = a + a = 0 (5) (α + β)a = αa + βa. (6) α(a + b) =αa + αb. (7) α(βa) =(αβ)a (8) 1 a = a V = R n 1. R C C V = C n C V = R n C i R n 2. R (1) Poly d := {f : f d } (2) R := {(x 1,x 2,...): x j R} 3. Poly d Poly d := {f : f d } 4. Poly := {f : f } R R 0 := {(x 1,x 2,...) R : n n 0} V =Poly 2 f V 2 a 1,a 2,a 3 unique f = f(x) =a 1 + a 2 x + a 3 x 2 u 1 = u 1 (x) = K2-2S04-03 : B

14 2S III IV K203 1, u 2 = u 2 (x) =x, u 3 = u 3 (x) =x 2 f = a 1 u 1 + a 2 u 2 + a 3 u 3 =(u 1, u 2, u 3 ) a 1 a 2 a 3 5. D V f(x) d dx f(x) V D : V V D(f) =(u 1, u 2, u 3 )A a 1 a 2 a 3 3 A f V b 1, b 2, b 3 unique f = f(x) =b 1 + b 2 x + b 3 (x 1) 2 v 1 = v 1 (x) =1, v 2 = v 2 (x) =x, v 3 = v 3 (x) =(x 1) 2 f = b 1 v 1 + b 2 v 2 + b 3 v 3 =(v 1, v 2, v 3 ) b 1 b 2 b 3 6. (u 1, u 2, u 3 )P =(v 1, v 2, v 3 ) 3 P P P 1 7. D : V V D(f) =D(b 1 v 1 + b 2 v 2 + b 3 v 3 )=(v 1, v 2, v 3 )B b 1 b 2 b 3 3 B 8. 2 (v 1, v 2, v 3 )=(1,x 1, (x 1) 2 ) K2-2S04-03 : B

15 2S III IV K R (1) V 1 = {a = {a n } n=1 : a n+2 = a n+1 + a n }. { } d 2 (2) V 2 = f = f(x) : f(x) =f(x), f: R R. dx2 (3) V 3 = {a = A : A m n }. 2. R ) (1) U 1 = {a = {a n } n=1 : a n+1 =2a n +1}. (2) U 2 = {f = f(x) :f d Z }. (3) U 3 = {a = A : A n } n =2 x x y y x n 10 n x 10 n+1 x >0, n Z, 10 n x 10 n+1 x <0 x R 3., (1) m n 3n +4m =1 (2) x 2x 1 > 0 (3) x y x 2 + y 2 2xy (4) x y x 2 3y +1=0 K2-2S04-03 : B

16 2S III IV K (1) x R, y R, x+ y<0 (2) x, y R, z Z, x+ y z<0 (3) x R, x 2 + x +1< 0 (4) x >0, y < 10, y > x 2 +1 K2-2S04-03 : B

17 2S III IV K204 : May 7, 2004 Version : 1.1 V R V W V a, b W a + b W α R a W αa W 1. W V 0 W 2. W = {a = {a n } n=1 : a n+2 = a n+1 + a n } R 3. W = {a = {a n } n=1 : a n+1 =2a n +1} R Span u 1,...,u n V {a 1 u a n u n : a 1,...,a n R} V span{u 1,...,u n } u 1,...,u n 4. span{u 1,...,u n } =: W V 5. Poly d =span { 1,x,x 2,...,x d} =span{1, (x 1), (x 1)(x 2),...,(x 1)(x 2) (x d)} 1. E V {a 1 u a n u n : n N, u 1,...,u n E, a 1,...,a n R} span{e} E Poly =span { 1,x,x 2,... }.span{e} V E K2-2S04-04 : B

18 2S III IV K204 u 1,...,u n V a V (a 1,...,a n ) a = a 1 u a n u n =(u 1,...,u n ). u 1,...,u n V V n a 1 a n V V R n V a a 1. a n R n 6. b = b 1 u b n u n V α R a + b αa R n R n V V R n 2. E V V x V n (a 1,...,a n ) x = a 1 u a n u n Poly Poly =span { 1,x,x 2,... } 7. u 1,...,u n V (1) span{u 1,...,u n } = V (2) u 1,...,u n K2-2S04-04 : B

19 2S III IV K204 span{u 1,...,u n } = V R u 1,...,u n V span{u 1,...,u n } 8. {f = f(x) :f : R R} R Ṽ f 1 = f 1 (x), f 2 = f 2 (x) f 1 + f = f 1 (x) +f 2 (x) Ṽ, α 1 f = αf 1 (x) Ṽ V 1 =span{1, cos x, cos 2 x, cos 3 x} V 2 =span{1, cos x, cos 2x, cos 3x} V 3 =span{sin x, sin x cos x, sin cos 2 x} (1) 1, cos x, cos 2 x, cos 3 x V 1 Hint. 7 (2) V 1 = V 2 (1, cos x, cos 2 x, cos 3 x)p =(1, cos x, cos 2x, cos 3x) 4 P (3) 1, cos x, cos 2x, cos 3x V 1 = V 2 (4) sin x, sin x cos x, sin cos 2 x V 3 (5) T : V 1 V 3 V 1 f = f(x) T f = df (x) dx V 3. T (6) f = f(x) =a 1 + a 2 cos x + a 3 cos 2 x + a 4 cos 3 x =(1, cos x, cos 2 x, cos 3 x) a 1 a 2 a 3 a 4 K2-2S04-04 : B

20 2S III IV K204 T f =(sinx, sin x cos x, sin cos 2 x)a a 1 a 2 a 3 A T 1, cos x, cos 2 x, cos 3 x 1, cos x, cos 2x, cos 3x A a 4 1. V =Poly 2 (1) 1, x,x 2 V. (2) 1, x+1, (x +1) 2 V (3) span{1, π,x,2x +1} (4) (1, x, x 2 )P =(1, x+1, (x +1) 2 ) (5) T : V V V f = f(x) T f =(x +1) d (f(x)) V. dx T (6) f = f(x) =a 1 + a 2 x + a 3 x 2 =(1,x,x 2 ) a 1 a 2 a 3 T f =(1,x,x 2 )A a 1 a 2 A T 1, x,x 2 T 1, x+1, (x +1) 2 B (7) B = P 1 AP a 3 K2-2S04-04 : B

21 2S III IV K204 A B A B A, B A B B A A B B A A B A B B A A B B A A B B A (1) x, y > 0 x 2 + y 2 > 0 (2) 2x 1 > 0 4x 1 > 0 (3) f : R R (4) u 1,...,u n V u 1,...,u n (5) u 1,...,u n V span{u 1,...,u n } = V K2-2S04-04 : B

22 2S III IV K205 : May 14, 2004 Version : V = R e 1 = 0, e 1 2 = 0, e 0 3 = 1, e 0 4 = 0, f : V V A = f : V a =(e 1, e 2, e 3, e 4 ) a 1 a 2 a 3 f(a) =(e 1, e 2, e 3, e 4 )A a 1 a 2 a 3 V a 4 a 4 e 1 + e 2, e 2 + e 3, e 2 e 4, e 2 + e 4 f B 2. a 1,...,a n C N (1) b C N span{a 1,...,a n } α 1,...,α n a 1 + α 1 b, a 2 + α 2 b,..., a n + α n b (2) a 1 a 2, a 2 a 3,..., a n 1 a n, a n 3 K2-2S04-05 : B

23 2S III IV K205 A, B A = B (1) A B (2) B A (3) A = B 3. A, B A = B (1) A Z 6 B Z 2 3 (2) A =span{x, 2x, x 2, 3x 2 }, B =span{2, x 2, (x 1) 2 }, A, B Poly 2. (3) A =span{x, x +2,x 2 }, B =span{2, x 2, (x 1) 2 }, A, B Poly 2. (4) A = {x R :0 x 1} B = {(x, y) R 2 : x + 1 x =1}. (5) A = {(x, y) R 2 : x 0 y 0} B = {(x, y) R 2 : x + y x y 0}. (6) A = {(x, y) R 2 : x 0 y 0} B = {(x, y) R 2 : x + y + x y 0}. (Hints for (4), (5) and (6): ) K2-2S04-05 : B

24 2S III IV K206-1 : May 21, 2004 Version : 1.2 Otis 26 U Otis A Aretha Aretha V Otis, U, A, V,Aretha Otis f U V U, V O U,O V f O U O V Otis f A Aretha Aretha U Otis A 2 U, V f : U V f(u) Im f U f (image) Im f V Im f f rank f rank f := dim(im f). V V dim V = {0} dim({0}) =0 V dim V =0 V = {0 V } D :Poly 2 Poly 10 D : f = f(x) d f(x) =D(f) dx Im D =Poly 1 rank f =dim(poly 1 )=2 f A rank f =ranka 1. D :Poly Poly D : f = f(x) d f(x) =D(f) dx Im D =Poly rank f 2. f : U V Im f V K2-2S04-06 : C

25 2S III IV K f : U V u 1,...,u n U U n n (1) span{f(u 1 ),...,f(u n )} =Imf. (2) f(u 1 ),...,f(u n ) Im f (3) f(u 1 ),...,f(u n ) V 4. Otis U Aretha V f A A f : U V U V f : U V f(a) =0 (= 0 V (kernel) Ker f V ) a U Ker f := {a U : f(a) =0} = f 1 ({0}). Ker f U D :Poly 2 Poly 10 D : f = f(x) d f(x) =D(f) dx Ker D =Poly 0 =span{1} Poly 2 Ker D D Poly f : U V Ker f U 6. D 2 :Poly 3 Poly 3 D 2 : f = f(x) d2 dx 2 f(x) =D2 (f) (1) D 2 (2) Poly 3 1, x, x 2, x 3 D 2 (1), D 2 (x), D 2 (x 2 ), D 2 (x 3 ) Im D 2 K2-2S04-06 : C

26 2S III IV K206-3 f : U V f, V dim(im f)+dim(kerf) =dimu. n + n = n + = 7. f : U V (1) dim U =3, dim(im f) =1, dim(ker f) =2. (2) dim U =3, dim(im f) =2, dim(ker f) =1. (3) dim U =5, dim(im f) =5, dim(ker f) =0. (4) dim U =5, dim(im f) =0, dim(ker f) =5. (5) dim U =, dim(im f) =, dim(ker f) =1. (6) dim U =, dim(im f) =, dim(ker f) = (7) dim U =, dim(im f) = 2004, dim(ker f) =. dim(ker f) =0 Ker f = {0 U } 8. f : U V (1) Ker f = {0 U } (2) f f(a) =f(a ) V = a = a U. (3) f u 1,...,u n U Ker f = {0 U } f(u 1 ),...,f(u n ) Im f =span{u 1,...,u n } K2-2S04-06 : C

27 2S III IV K206-4 Ker f = {0 U } Im f = V f(u 1 ),...,f(u n ) V U f V dim U = n =dimv =dim(imf) Im f = V f V (capacity) U V f Im f = V f : R R, f :(x 1,x 2,...) x 1 Im f = R 2 f : U V Im f = f(u) =V f (surjective) b V a U f(a) =b f : U V (bijective) (isomorphism) f : U V dim(im f) =dimv f : U V (1) dim U =3, dim(im f) =1, dim(ker f) =2 f. (2) dim U =3, dim(im f) =2, dim(ker f) =1 f. (3) dim U =5, dim(im f) =5, dim(ker f) =0. (4) dim U =5, dim(im f) =0, dim(ker f) =5. (5) dim U =, dim(im f) =, dim(ker f) =1. (6) dim U =, dim(im f) =, dim(ker f) = 2004 f. (7) dim U =, dim(im f) = 2004, dim(ker f) =. K2-2S04-06 : C

28 2S III IV K I : U =Poly 2 V =Poly 3 I : f = f(x) x 0 f(t)dt = I(f) (1) I (2) I (3) U =Poly 2 1, x,x 2 I(1), I(x), I(x 2 ) Im I V x 2 (4) V =Poly 3 1, x, 2, x 3 U 3 I A f :(1,x,x 2 ) a 1 a 2 a 3 (1, x, x 2 2, x 3 3 )A (5) A rank A f rank f a 1 a 2 a 3 (1) f : N N, f : n 2n. (2) f : N 2N, f : n 2n 2N ( ) ( ) (3) f : R 2 R 2 x x 3y, f : y 2x ( ) ( ) (4) f : R 2 R 2 x x + y, f :. y xy K2-2S04-06 : C

29 2S III IV K207-1 : May 28, 2004 Version : 1.1 Otis Aretha Otis X Otis A Aretha Aretha Y Otis, X, A, Y,Aretha Otis f X X Y Otis f A Aretha X A Y f f : X Y X Y X Y 1. X = Y = R,f : X Y y = f(x) := x X = Y = R {0},g : X Y y = g(x) :=x/ x X X Y X K2-2S04-07 : C

30 2S III IV K207-2 f : X Y X Y f (injective) x, x X x x f(x) f(x ) x, x X f(x) =f(x ) Y x = x f : X Y f(x) =Y f (surjective) y Y x X f(x) =y f : X Y (bijective) 2 n X = {1, 2,...,n} X X ( ) K2-2S04-07 : C

31 2S III IV K207-3 X i j (i, j) ( ) =(2, 3) (3, 4) (1, 2) (2, 3) (3, 4) X = {1, 2,...,n} f : X X n =4 f : X X, f(1) = 2, f(2) = 3, f(3) = 4, f(4) = 1 (6/7) 2,3 2 ( ) K2-2S04-07 : C

32 2S III IV K208-1 : June 11, 2004 Version : 1.2 (inner product) (1) (2) (3) (4) A B (5) ( Otis Aretha ) (6) (1) R 3 R 3 a b a, b ( a1 ) ( b1 ) a = a 2, b = b2 R 3 a 3 b 3 a, b := a b cos θ = a 1 b 1 + a 2 b 2 + a 2 b 3 θ a b (0 θ π) a, b =(b a ) (b ) ( ) ( ) ( ) 1. R e 1 =, e 2 =, e 3 = a R 3 e 1, e 2, e 3 ( a, e 1, a, e 2, a, e 3 ) K2-2S04-08 : C

33 2S III IV K208-2 ( ) ( 2. R ) ( ) 00 u 1 =, u 2 = 1, u 3 = a R 3 u 1, u 2, u 3 ( a, u 1 u 1, a, u 2 2 u 2, a, u 3 2 u 3 ) 2 ( ) ( ) ( ) 3. R u 1 =, u 2 =, u 3 = 0 0 a R 3 u 1, u 2, u 3 ( a, u 1 u 1, a, u 2 2 u 2, a, u 3 2 u 3 ) 2 1 U R R 3 a, b U a, b R a U, α R (1) a + a, b = a, b + a, b (2) αa, b = α a, b (3) a, b = b, a (4) a, a 0 (5) a, a =0 a = 0 (4) a, a 0 a a a, b =0 a b a b U = R 3 a, b = a b cos θ 4. Schwarz a, b a b (1) α R a + αb 2 = a + αb, a + αb a, b (2) b =0 b 0 α = b 2 0 a 2 a, b 2 b 2 K2-2S04-08 : C

34 2S III IV K208-3 Schwarz 1 a, b a b 1 cos θ = a, b a b θ a b R 3 a, b = a 1 b 1 + a 2 b 2 + a 2 b 3 a, b = a b cos θ 5. a + b a + b (1) a + b 2 = a + b, a + b (2) Schwarz a + b 2 ( a + b ) 2 a a U C R a, b U a, b C a U, α C (1) a + a, b = a, b + a, b (2) αa, b = α a, b (3) a, b = b, a (4) a, a 0 (5) a, a =0 a = 0 a, a 0 a a a, b =0 a b a b K2-2S04-08 : C

35 2S III IV K208-4 R C pre-hilbert 6. a,αb =ᾱ a, b 7. (1) Schwarz a, b a b (2) a + b a + b a, b 1 1 a b a b a, b C n (a 1,...,a n ), (b 1,...,b n ) b1 a, b := a T b =(a1,...,a n ). = a 1 b1 + + a n bn bn 8. U =Poly C f = f(x), g = g(x) U f, g := 1 1 f(t)g(t)dt (1) f = f(x) =x 2 + i f (2) f 1 (3) U u 1,...,u n (orthogonal system) i j = u i, u j =0 u i u j u 1,...,u n U (orthogonal basis) i =1,...,n u i =1 (orthonormal system) (orthonormal basis) K2-2S04-08 : C

36 2S III IV K208-5 R n, C n 9. u 1,...,u n u 1,...,u n span{u 1,...,u n } U = span{u 1,...,u n } u 1,...,u n U a U u 1,...,u n a 1 a =(u 1,...,u n ). a n = a 1 u a n u n a i = a, u i u i 2 = a, u i u i, u i. a i = a, u i u 1,...,u n a, b U a, u i = b, u i i =1,...,n a = b Gram-Schmidt U u 1,...,u n (n<dim U ) u n+1 U u 1,...,u n, u n+1 dim U< n +1 = dimu 12. dim U = N (1) n<n span{u 1,...,u n } v U (2) v u n+1 0 (3) u i u n+1 (1 i n) u n+1 = v v, u 1 u 1 2 u 1 v, u n u n 2 u n K2-2S04-08 : C

37 2S III IV K208-6 u 1,...,u n u 1 u 1,..., u n u n 13. U = R v 1 = 0, v 2 = 3, v 2 = U =Poly C 2 PolyC (1) (2) f = f(x) =a 1 + a 2 x + a 3 x 2, g = g(x) =b 1 + b 2 x + b 3 x 2 U f, g a i,b i b1 f, g =(a 1,a 2,a 3 )A b2 b3 A U W W W (orthogonal subspace) W 15. (1) W U (2) a U a 1 W, a 2 W a = a 1 + a 2 U = W W K2-2S04-08 : C

38 2S III IV K209-1 ε-n ε-δ : June 18, 2004 Version : 1.2 (limit) a n 2 Jimmy Robert Robert Jimmy a n =1/n 0 Robert: a n 0 n a n 1 0 Jimmy: a 1,a 2,... 1 > 1 2 > 1 3 > 1 4 > 1 5 > n = 1000 a n =0.001 =1 n >1000 a n 1 a n 0 0 a n Robert: a n 0 n a n 1 Jimmy: n =10 6 a n =10 6 =1 n>10 6 a n 1 Robert: n a n Jimmy: N = a N = n>n= a n Robert: n a n Jimmy: Rob n a n =1/n Robert: a n 0 Jimmy: K2-2S04-09 : C

39 2S III IV K209-2 ε-n {a n } n=1 α (converge) ε>0 N n >N a n α <ε α {a n } n=1 (limit) lim a n = α a n α (n ) n ε>0 ε>0 1. (1), (2) {a n } n=1 (3) {a n } n=1 2. {a n } n=1 a n = n2 + n 1 a n 1 <ε N n 2 ε n >N (1) ε =5 (2) ε =1 (3) ε =0.1 (4) ε = (5) ε = ε 0 > 0 a n 1(n ) Robert ε>0 a n α {a n } n=1, {b n } n=1 lim n a n = α, lim n b n = β (1) lim n (a n + b n )=α + β (2) lim n (a n b n )=αβ (3) β 0 lim n a n b n = α β 3. (2) K2-2S04-09 : C

40 2S III IV K209-3 (2) C lim n (Ca n )=Cα. 4. (1), (2) (3) 5. lim n a n = α a n + a n+1 lim n 2 {a n } n=1 lim n an a n+1 = α 6. {a n } n=1, {b n} n=1 lim n (a n + b n ) lim n a n, lim n b n = α (1) lim n a n = α, lim n b n = β (2) lim n a n = α, lim n b n = β 7. {z n } n=1 z n := x n + y n i {x n } n=1, {y n } n=1 (1) {x n } n=1, {y n } n=1 {z n} n=1 (2) {z n } n=1 {x n} n=1, {y n } n=1 Cauchy {a n } n=1 (Cauchy sequence) ε>0 N n, m > N a n a m <ε {a n } n=1 lim n a n = α R C (completeness) Q K2-2S04-09 : C

41 2S III IV K {a n } n=1 lim n a n = α {a n } n= {a n } n=1 a n = 1 n a n = in n 10. {a nn } n=1 a 1 =1,a 2 =4,a n+2 = a n+1 + a n 2 (1) n 3 a n a n 1 =( 1/2)(a n 1 a n 2 ) (2) a 200 a 100 ( ) a a 99 (3) a 100 a a 98 2 a 1 = (4) {a n } n=1 1. {a n } n=1, {b n } n=1 lim n a n b n lim n a n, lim n b n 2. {a n } n=1, {b n} n=1 lim n a n = α, lim n b n = β 0 a n lim n = α b n β 3. {z n } n=1 z n := r n (cos θ n + i sin θ n ) 0 θ n < 2π z n =0 r n =0,θ n =0 {r n } n=1, {θ n } n=1 (1) {r n } n=1, {θ n } n=1 {z n} n=1 (2) {z n } n=1 {r n} n=1 {θ n} n=1 K2-2S04-09 : C

42 2S III IV K210-1 ε-n ε-δ : June 25, 2004 Version : 1.1 {a n } n=1 or (series) n a n := lim a k = lim (a a n ). n n n=1 k=1 a n = n n=1 a n n=1 a n 0 a 1 a 2 a 3 1. {a n } n=1 n=1 a n (1) a n =( 1) n (2) a n = ( 1)n n (3) a n = in n (4) a n = in 2 n = = π 4 2. a n n=1 a n n=1 a n n=1 a n a n n=1 a n n=1 a n =log2 log 2 K2-2S04-10 : C

43 2S III IV K θ sin θ = θ θ3 3! + θ5 5! + cos θ =1 θ2 2! + θ4 4! + exp(θ) =1+θ + θ2 2! + θ3 3! + (1) θ R n =0, 1, 2,... a n := (iθ)n n! n=0 a n θ 0 (a) b n := a n b n+1 /b n (b) N r (0, 1) n N b n+1 /b n r (c) k =0, 1, 2,... b N+k r k b N (d) n=0 b n (2) e iθ =exp(iθ) := n=0 (iθ)n /n! e iθ =cosθ + i sin θ 1. {a n } n=1 n=1 a n (1) b n := n k=1 a k n=1 a n {b n } n=1 ε-n (2) {b n } n=1 ε-n (3) s n := n k=1 a k {s n } n=1 n=1 a n ε>0 N n m>n s n s m = a m a n <ε b n K2-2S04-10 : C

44 2S III IV K {z n } n=1, {w n} n=1 z n = z, n=1 w n = w n=1 α, β n=1 (αz n + βw n ) αz + βw 3. {a n } n=1, {b n } n=1 n=1 a2 n n=1 b2 n n=1 a nb n (Hint (A B) 2 = (A 2 + B 2 ) 2AB 0) {z n } n=1, {w n } n=1 n=1 z nw n K2-2S04-10 : C

45 2S III IV K211-1 ε-n ε-δ : July 2, 2004 Version : n a n {S n } n=1 M>0 n S n S n+1 <M lim n S n 1. {a n } n=1 n=1 a n lim n a n =0 lim n a n =0 n=1 a n 2. S n = , (n =1, 2,...) n lim n S n = 1 n=1 n α + 1 n α + α 1 α > <α<1 1 <α (n i) 2 n=1 (n + i) 5 4. (1) n=1 (2n) 1 3 (n +1) 2 (2) n=1 (2n) 1 3 n +1 (3) ( ) n n + i n=1 2n i K2-2S04-11 : C

46 2S III IV K n=1 a n n=1 a2 n 6. n=1 a n n=1 a2 n 1. p>1 n=1 a n n=1 ap n {a n } n=1 n=1 a n n=1 sin a n 2. (1) n n=1 (2n +1) 3 (2) 1 n=1 2n +1 (3) x x 2n n=0 (2n)! (4) x < 1 n=1 ( x)n n 3. f X Y X P, P 1,P 2 B Q, Q 1,Q 2 (1) Q 1 Q 2 = f 1 (Q 1 ) f 1 (Q 2 ) (2) f(p 1 P 2 )=f(p 1 ) f(p 2 ) (3) f 1 (Q 1 Q 2 )=f 1 (Q 1 ) f 1 (Q 2 ) (4) P f 1 (f(p )) (5) Q = f(f 1 (Q)) Q Y f 1 (Q) :={x X : f(x) Q} Y Q X f 1 (Q) Q 1 K2-2S04-11 : C

47 2S III IV K212-1 ε-n ε-δ : July 9, 2004 Version : 1.1 ε-n ε-δ C D C D f : D C f(z) =z 2 + z 1 D, f(d) R f : D C Otis Aretha Otis C D Aretha C D = C f f : D C z 0 D z 0 D {z n } n=1 {f(z n)} n=1 f(z 0) lim z n = z 0 = lim f(z n )=f( lim z n )=f(z 0 ) n n n z n z f(z) =z z = z 0 ε-n f : D C z D {z n } n=1 ε>0 N n >N z n z <ε K2-2S04-12 : C

48 2S III IV K212-2 ε > 0 N n >N f(z n ) f(z) <ε f : D C z = z 0 D ε>0 δ>0 z z 0 <δ f(z) f(z 0 ) <ε f D f D f(z) =z 2 z =1 ε z 1 <δ f(z) f(1) <ε δ>0 (1) ε =1 (2) ε = 1 (3) ε = 1 (4) ε> f(z) =z 2 z =1 ε-n f(x) =x 2, (x R) f(z) =z 2 x =1 4. f(x) =x 3 x = x 0 R f R f(x) =x 3 R C 5. f : R [0, 1] f ε-δ f(x) = x (x 0), f(x) =0(x =0) x ε>0 δ>0 z z 0 <δ f(z) f(z 0 ) <ε z z 0 δ f(z) f(z 0 ) ε δ z 0 ε 6. f(x) =x 2 x x 0 =0 δ =0.1 f(x) f(x 0 ) x x 0 = 100 δ =0.1 K2-2S04-12 : C

49 2S III IV K212-3 f : D C D ε>0 δ>0 z 0 D z z 0 <δ f(z) f(z 0 ) <ε 7. f(x) =x 3 [0, 1] ( 100, 100) 8. f :(0, 1] R f(x) =1/x D =(0, 1] f 1. f(z) =a n z n a 1 z + a 0 C n =2, 3 2. f(z) =z 2 D = {z C : z 1} z : R C, t z(t) R (1) z : t z(t) :=3t +4ti (2) z : t z(t) :=t + t 2 i (3) z : t z(t) :=e it =cost + i sin t 4 3 [0, 1] R R K2-2S04-12 : C

50 2S III IV K213-1 ε-n ε-δ : July 16, 2004 Version : 1.2 (1) (continuous) (2) (uniformly continuous) (3) (pointwise convergence) (4) (uniform convergence) (compact uniform convergence) (equicontinuous) (compact) (1) D C f : D C f : D C z = z 0 D ε>0 δ>0 z z 0 <δ f(z) f(z 0 ) <ε f D f D ε-n ε-δ 1 ε-n ε-δ (2) D C f : D C K2-2S04-13 : C

51 2S III IV K213-2 f : D C D ε>0 δ>0 z 0 D z z 0 <δ f(z) f(z 0 ) <ε f : D C D D D z f f(z) D f f(z) =1/z z 1 0 f(z) 1 z f(z) z δ f(z) ε f : D C D f(z) =1/z D = C {0} [ , 1] (3) I R f n : I R (n =1, 2,...) f n (x) =x n, f n (x) =sinnx, f n (x) = sin x n, f n(x) = n k=0 xn etc. x 0 I a n := f n (x 0 ) x {f n (x)} n=1 lim n f n (x) f(x) f : I R f n I f f n : I R I f : I R x I ε > 0 N = N(x, ε) n > N f n (x) f(x) <ε 1. f n :[0, 1] R (1) f n (x) = x n (2) f n(x) =nx (3) f n (x) =x n (4) f n (x) = x n x (x 0),f n(0) = 0 K2-2S04-13 : C

52 2S III IV K213-3 (4) I R f n : I R (n =1, 2,...) f n : I C I f : I R ε>0 N = N(ε) n>n x I f n (x) f(x) <ε f n 4. f n :[0, 1] R (1) f n (x) = x n (2) f n(x) =nx (3) f n (x) =x n (4) f n (x) = x n x ( x 0),f n(0) = 0 5. (1) f n :[0, 1] R (2) f n :[0, 1] R I f n (x) f(x) f(x) f n (x)dx f(x)dx. I I K2-2S04-13 : C

53 2S III IV K213-4 D C f n : D C K D f : D C f(z) f n K D f K D D K2-2S04-13 : C

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D 1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00