A 21 A.1 L p A A.3 H k,p () A

Transcription

1 Analysis III Functional Analysis III (10:40-12:10) n R n or C n ( ) (Linear sp. (Vector sp.)) (Normed Spaces) (Norm) (Banach Spacses) Banach (Examples of Banach sps) (Continuous function space) L p (L p -sp.) (Separable & equivarent norms) (Completion) (Hilbert Spaces) ( ) (Pre-Hilbert sp. (Inner prod. sp.)) (Hilbert sp.) (Projection theorem) (ONS=orthonormal system) (Linear Operators) (Examples of bounded operators) (Inverse operators) (,, ) (Uniform bounded principle) (Open mapping theorem) (Closed graph theorem) (Linear Functionals) (Dual spaces) (Hahn-Banach s extension thoerem)

2 A 21 A.1 L p A A.3 H k,p () A A A A.7 A 2 () B, 26 B B C 27 D 28 1,,,,,,.,. R n, C n l p, C([a, b]), L p () ( R n ),,., (= ),,,.,,.,,,,.,,.,,,,,,.,,.,,,,.,.

3 Functional Analysis n R n or C n n N. x = (x 1,..., x n ), y = (y 1,..., y n ) R n (or C n ), : x + y = (x 1 + y 1,..., x n + y n ), : α R (or C), αx = (αx 1,..., αx n ). R n (or C n ). (x, y) = x j y j in R n or (x, y) = x j y j in C n j=1 j=1,, x x = (x, x) 1/2. R n, C n, n (Euclid spaces). ( Hilbert sp., Banach sp..) 1.2 ( ) (Linear sp. (Vector sp.)) K = R or C. 1.1 ( ) X K ( ), i.e., x, y X, x + y X, α K, x X, αx X;. (i) ( ) (x + y) + z = x + (y + z) (x, y, z X) (ii) ( ) x + y = y + x (x, y X) (iii) ( ) θ X; x X, x + θ = x (θ = 0 ) (iv) ( ) x X, x X; x + x = 0 (x = x ) (v) ( ) α(x + y) = αx + αy, (α + β)x = αx + βx (x, y X, α, β K) (vi) ( ) (αβ)x = α(βx) (x X, α, β K) (vii) ( ) 1x = x ( x X) 1.1. ( θ X; x X, x + θ = x θ ) [ ] θ X; x X, x + θ = x θ = θ + θ = θ + θ = θ, (x X, x X; x + x = 0 x ) [ ] x X, x X; x + x = 0 0 = x + x = x + x,, x = x + (x + x ) = (x + x) + x = (x + x ) + x = x. (1) X x 1,..., x n X ( ) (linear independent) [α 1 x α n x n = 0 (α 1,..., α n K) = α 1 = = α n = 0] ( )(linear dependent), i.e., (α 1,..., α n ) 0; α 1 x α n x n = 0. X n (n-dimensional) n, n + 1. dim X = n. X (infinite dimensional) n N, n. x X x 1,..., x n X (linear combination) α 1,..., α n K; x = α 1 x α n x n

4 Functional Analysis X n n,, i.e., dim X = n = x 1,..., x n X; lin. indep., x X, α 1,..., α n K; x = α 1 x α n x n. (2) X K. Y X (subspace) x, y Y, x + y Y,, α K, αx Y. 2 (Normed Spaces), (Norm) 2.1 ( ) X : x x x X (norm). (i) x 0 (x X) ( ) (ii) x = 0 x = 0 ( ) (iii) αx = α x (α K, x X) (iv) x + y x + y (x, y X) ( ) (X, ) (normed space). (X, ) d(x, y) = x y.,. (d.1) d(x, y) 0 (x, y X) ( ) (d.2) d(x, y) = 0 x = y ( ) (d.3) d(x, y) = d(y, x) (x, y X) ( ) (d.4) d(x, z) d(x, y) + d(y, z) (x, y, z X) ( ) (X, d). {x n } X, x n x (n ) x n x 0 (n ) x {x n }. 2.1, i.e., (i) x n x, y n y = x n + y n x + y, (ii) α n α, x n x = α n x n αx. 2.2, i.e., x n x = x n x. 3 (Banach Spacses) 3.1 (Banach sp.).,,., (X, ), {x n } X : (Cauchy sequence) x n x m 0 (m, n ) X (complete) Cauchy {x n } X, i.e., x X; x n x.

5 Functional Analysis Banach (Examples of Banach sps) 3.1 R n, C n Banach sp.,,,, R 1, Cauchy,, Cauchy P n : n (n N) x(t) = a n t n + a n 1 t n a 0 P n (a k C) (x + y)(t) = x(t) + y(t), (αx)(t) = αx(t),, y(t) = b n t n + b n 1 n b 0, (x + y)(t) = (a n + b n )t n + + (a 0 + b 0 ), (αx)(t) = αa n t n + + αa 0 P n. dim P n = n + 1 P n x(t) = n j=0 a jt j, x = n j=0 a j. 3.1 P n Banach. ({1, t, t 2,..., t n } ) (Continuous function space) 3.3 R n C() [(x + y)(t) = x(t) + y(t), (αx)(t) = αx(t)]. dim C[0, 1] =. x = sup t x(t), Banach sp. (x(t) = t n 1 (n 1) ) {x n } Cauchy in C(). t, x n (t) x m (t) x n x m 0(m, n ), {x n (t)} R Cauchy., x (t) R; x n (t) x (t). n, x (t) x m (t) lim n x n x m t, sup t, m lim sup x (t) x m (t) lim x n x m = 0 m m,n t x {x n } x. x n x in C(). C b (R) = {x C(R); x < }., (C b, ) Banach sp P n x(t) n j=0 a jt j, x = max α j, x(t) C([0, 1]), x L 1 = x(t) dt.. [0,1] 3.2 P n [0, 1]: P n [0, 1] C[0, 1] ( ). ( x = t [0,1] x(t) )..

6 Functional Analysis L p (L p -sp.) p <, R n, L p () u ( 1/p u L p := u(t) dt) p <. u = v a.e., L p () Banach sp.. p or L p. 3.5 R n, L () u α < ; u(t) α a.e.. u ess.sup t u(t) := inf{α; u(t) α a.e.}. u u a.e.. u = v a.e., L () Banach sp.. or L. 2,, normed sp.. [ (Hölder s inequality)] 1 p, 1 q 1/p + 1/q = 1, p = 1 q =, p = q = 1 (q p ). uv L 1 u L p v L q., ( ) 1/p ( 1/q u(t)v(t) dt u(t) p dt v(t) dt) q (1 < p < ), u(t)v(t) dt u(t) dt v (p = 1, q = ). p = 1,. 1 < p <. u L p = 0 or v L q = 0 uv = 0 a.e., u L p 0 and v L q 0. ab a p /p + b q /q (a, b 0). (log, log(a p /p + b q /q) (log a p )/p + (log b q )/q = log a + log b = log(ab)).) a = u(t) / u L p, b = v(t) / v L q,. uv L 1 ( u L p v L q) u p L p p u p + v q L q L q v q = 1 p L p + 1 q = 1 q [ (Minkovsky s inequality)] ( ) 1 p <. u, v L p () u + v L p (), u + v L p u L p + v L p. p = 1. p = ( ). 1 < p <.. (u + v L p ().) u + v p ( u + v ) u + v p 1 Hölder., 1/q = 1 1/p = (p 1)/p, i.e., q = p/(p 1) u + v p L p ( ) 1/q u u + v p 1 dt + v u + v p 1 dt ( u L p + v L p) u + v p dt ( ) 1/q u + v p dt = u + v p/q L p u + v p dt = 0, 0 p/q = p 1,.

7 Functional Analysis u + v u + v. L p (). Banach. 3.3 (X, ), {u n } X: Cauchy. {u nk } {u n }; u nk u in X u n u in X. u k u u k u nk + u nk u 0 (n k k ). 3.4 Lebesgue, Lebesgue. [L p () ] {u n } Cauchy in L p (). {u nk }; u nk+1 u nk L p < 1/2 k., m u nj+1 u nj = lim m u nj+1 u nj u nj+1 u nj L p 1 <. j=1 L p j=1 L p j=1 u nj+1 u nj L p (). j=1 k < m, u nj+1 (t) u nj (t) < for a.e. t. j=1 u nm (t) u nk (t) m 1 j=k u nj+1 (t) u nj (t) 0 (m > k ) a.e. a.e. t, {u nk (t)} R Cauchy, u nk (t) u (t). u {u n } L p ()., k 1 u nk (t) u n1 (t) + u nj+1 (t) u nj (t) u n1 (t) + j=1 u nj+1 (t) u nj (t) L p (), g(t), k a.e. t, u (t) g(t) L p (), i.e., u (t) L p (). k < m, u nm u nk L p m 1 j=k j=1 u nj+1 u nj L p j=k 1 2 j = 1 2 k 1. u nm (t) u nk (t) 2g(t) L p () Lebsgue, m u u nk L p 1 0 (k ). 2k 1 u nk u in L p (), u n u in L p (). (X, F, µ) (X a set, F σ-field on X, µ = µ(dx) on X), f L p (X) or L p (X, dµ) or L p (X, F, µ) f p L := f p dµ = f(x) p µ(dx). p L (X). Hölder, Minkovsky, L p (X) Banach.. (.) X

8 Functional Analysis p <. x = (x 1, x 2,... ) x lp = l p Banach sp.. ( ) 1/p x n p < 3.7 x = (x 1, x 2,... ) x = sup{ x n ; n 1} < l Banach sp.. n=1 3.2 (Separable & equivarent norms) X Banach sp.. L X X (dense) {x n } L; x n x). X (separable) L = X, L L (L ; x L, i.e., L X; L = X, L ℵ 0 = N. 3.8 R n Banach sp., Q n. 3.9 R n, C(). C[0, 1] L [0, 1],,, L = C[0, 1]. X, 2 1, ; < c, c < ; c x 2 x 1 c x 2. normed sp. (X, 1 ) (X, 2 ).,. (X, 1 ) (X, 2 ). 3.5 X X 2. ( ) 3.3 (Completion) X., X, X, X = X., X X Cauchy, {x n }, {y n } X, {x n } {y n } x n y n 0 (n )., x = [{x n }] X := X/, x = [{x n }] X x := lim n x n ({ x n } R Caushy, ), ( X, ), i.e., Banach sp.. (.) x X [{x n x}] X, X X X = X. X X (completion) X 0 := {x = (x 1, x 2,..., x n, 0, 0,... ); x i R, n N} ( 0 ). x = x l p, X 0. X 0 l p dense,., x (n) = (1, 1/2, 1/ , 1/2 n, 0, 0,... ) m > n, 1 p < x (n) x (m) l p = ( m k=n+1 2 kp ) 1/p 0 (n ), Cauchy

9 Functional Analysis 7, x = (1, 1/2, 1/ , 1/2 n,... ) / X 0. X 0. (p =.) X 0 X 0 l p, i.e., l p. X 0 l p. ) (X, X ), (Y, Y ), f : X Y ;, f(x) Y = x X X Y. ( X Y.) 3.11 [0, 1] X 0, i.e., X 0 = n 1 P n[0, 1]. x X 0, x = sup t [0,1] x(t), X 0. X 0 C[0, 1]. 4 (Hilbert Spaces) (inner product) x, y or (inner prod. sp. or pre-hilbert sp.), x = x.x, (Hilbert sp.). R n C n,,. 4.1 ( ) (Pre-Hilbert sp. (Inner prod. sp.)) 4.1 X C, x, y X, x, y C (inner product) ( ) x, x 0 (x X). x, x = 0 x = 0. ( ) x, y = y, x (x, y X) ( ) x 1 + x 2, y = x 1, y + x 2, y, αx, y = α x, y (x 1, x 2, y X, α C). 4.2 X or (X,, ) (pre-hilbert sp.) or (inner prod. sp.). 4.1 X, x := x, x 1/2 (x X),. ( ) ( ) (Schwartz ) x, y x y (x, y X). y = 0 y = 0, x, y = 0 ( ),, y = 0. α C, 0 x + αy, x + αy, α := x, y / y 2., 0 x + αy, x + αy = x 2 + α x, y + α x, y + α 2 y 2 = x 2 x, y 2 / y 2. y = 0 x, y = 0 Schwartz. ( x, y = x, 0y = 0 x, y = 0) [ 4.1 ]., x + y 2 = x 2 + x, y + y, x + y 2 x x y + y 2 = ( x + y ) 2.

10 Functional Analysis X x = x, x 1/2,. x + y 2 + x y 2 = 2( x 2 + y 2 ).. x, y = 1 ( x + y 2 x y 2 + i x + iy 2 i x iy 2). 4 X,, x, y,., X..,. 4.1 x, y x, y, i.e., x n x, y n y x n, y n x, y. Schwartz. 4.2 (Hilbert sp.) 4.3 X (Hilbert sp.). K = R, K = C. [Hilber sps ] 4.1 R n, C n :, Hilbert. 4.2 l 2 : x = (x n ), y = (y n ) l 2, x, y = x n y n,, n 1, x 2 2 = x n 2 = x, x. Hilbert. 4.3 L 2 () ( R n open): u, v L 2 (), u, v = u(t)v(t)dt Hilbert. 4.4 A 2 () ( C n open, f A 2 (), f(z) 2 dxdy < (z = x + iy)) f, g A 2 (), f, g = f(z)g(z)dxdy Hilbert. ( ) 4.5 C() ( R n ) L 2 (), Hilbert. 4.3 (Projection theorem) H Hilbert sp. x, y H, A, B H, x y x, y = 0. A B a A, b B, a b. ( x B L H, {x} B.) L := {x H; x L} L (orthogonal complement).

11 Functional Analysis L H, L H. x y = x + y 2 = x 2 + y 2 ( ) L 1, L 2 H, L 1, L 2 (direct sum) L 1 L 2 := L 1 +L 2 ; L 1 L 2 = {0}. [L 1 L 2 = {0} x = x 1 + x 2 L 1 + L 2 ].. [ ] ( ) x 1 + x 2 = x 1 + x 2 (x i, x i L i) x 1 x 1 = x 2 x 2 L 1 L 2 = {0}, x i = x i. ( ) x L 1 L 2 x = x + 0 = 0 + x L 1 + L 2, x = ( ) L H H L L ; H = L L, i.e., x H, y L, z L ; x = y + z,. y L x H L or or. P L x = y, P L L ( ). x L L x, x = 0, x = 0, i.e., L L = {0}... x H, δ := inf y L x y. inf {y n } L; x y n δ. 2( x y n 2 + x y m 2 ) = (x y n ) + (x y m ) 2 + (x y n ) (x y m ) 2 = 2x (y n + y m ) 2 + y n y m 2. (y n + y m )/2 L, δ x (y n + y m )/2. y n y m 2 = 2( x y n 2 + x y m 2 ) 4 x y n + y m 2 2 2( x y n 2 + x y m 2 ) 4δ 2 ( ) 0 (m, n ). H y H; y n y. L closed y L. δ = x y. z = x y ( z = δ). z L. ξ L, γ = z, ξ, φ(t) = z γtξ 2 = x (y + γtξ) 2 (t R). y + γtξ L, φ(t) δ 2 = φ(0) (δ ). φ(t) = z 2 γt z, ξ γt ξ, z + γ 2 t 2 ξ 2 = δ 2 2 γ 2 t + γ 2 t 2 ξ 2 = δ 2 γ 2 t(2 t ξ 2 ). γ 0, t > 0 0, 2 t ξ 2 > 0, φ(t) < φ(0) = δ 2. γ = H = L 2 () ( R n : bdd open), u L u H; u(t)dt = 0, L, P L u(t) = u(t) 1 u(t)dt. L = { }. 4.7 H = L 2 ( 1, 1) u L, L = {v H; v( t) = v(t)}. u H; u( t) = u(t) L (, M = { }, L M, L M., [0, 1), u(t) = v(t) + v( t).)

12 Functional Analysis (ONS=orthonormal system) {x k } H: (ONS) x j, x k = δ jk. 4.8 L 2 (0, 1) { 2 sin(πkt)}, {e2πkti }. 4.2 {x k } H: ONS x H, k x, x k 2 x 2 (Bessel ). x H, α k = x, x k. n N, 0 x α k x k 2 = x 2 α k x, x k α k x k, x + α j α k x j, x k = x 2 α k 2 j, α k 2 x 2. n. { n } {x k } H, {x k } := α k x k ; α k K, n N, L := {x k } {x k }. ( upper bar.) 4.4 {x k } H: ONS, L = {x k } : {x k }.. (i) L = H (ii) x H, x = k x, x k x k ( Fourier ) (iii) x, y H, x, y = k x, x k y, x k ( ). (iv) x H, x 2 = k x, x k 2 (Perseval ). (v) k, x, x k = 0 x = 0. (i) L = H L = {0} (by Proj. Th.). [(i) (ii)] x H, α k = x, x k Bessel, k α k 2 x 2,,. m > n, m α k x k 2 α k x k = m 2 m α k x k = α k 2 0 (m > n ). k=n+1 k=n+1 { α k x k } Cauchy in H. y = α k x k H. x y, x k = x, x k α n x n, x k = α k α n x n, x k = α k α k = 0. n n (x y) {x k },, (x y) {x k } = L = H. x y = 0, i.e., x = y. [(ii) (iii)] x, y H, α k = x, x k, β k = y, x k. Schwartz Bessel ( n ) 1/2 ( n ) 1/2 α k β k α k 2 β k 2 x y.

13 Functional Analysis 11 n α k β k. [(iii) (iv)], [(iv) (v)]. x, y = lim n α k x k, β k x k = α k β k. n [(v) (i)] x L k 1, x, x k = 0. x = 0, L = {0}. H = L. 4.4 H {x k }, (complete ONS=CONS). 4.9 l 2 e j = (δ j,n ) n 1 (j 1, 0) {e j } CONS H = L 2 ( π, π), x H, { 1 2π, 1 π sin nt, } 1 cos nt CONS. π n=1 x(t) = 1 2 a 0 + (a n cos nt + b n sin nt) n=1 ( a n = 1 π x(t) cos ntdt, b n = 1 π ) x(t) sin ntdt π π π π, Fourier, a n, b n Fourier [Schmidt ] {y k } H. n 1 e 1 x 1 := y 1 / y 1, e n = x n / x n with x n = y n y n, e k e k (n 2). {e k } ONS. Schmidt. {e k } ONS. 4.5 Hilbert H CONS. H {z k } {y n }. Schmidt, {e n }., n, x, e n = 0 x = 0, CONS H. H. x, y H, x n, y n H; x n x, y n y. { x n, y n } Cauchy (Schwartz { x n }, { y n } ). x, y := lim x n, y n,. (, {x n }, {y n } H, H.)

14 Functional Analysis 12 5 (Linear Operators) 5.1 X, Y, D X, T : D Y (linear) (i) T (x 1 + x 2 ) = T x 1 + T x 2 (x 1, x 2 D), (ii) T (αx) = αt x (α K, x D) (linear operator). D(T ) := D T (domain), R(T ) := T (D) T (range). Y = X T X. 5.2 X, Y normed sps, lin. op. T : D(T ) X Y (i) (bounded operator) M; T x M x (x D(T )). (ii) (conti. operator) x n x in D(T ) T x n T x. 5.1 X, Y normed sps, T : D(T ) X Y T ( )., n 1, x n D(T ); T x n > n x n y n := x n /( n x n ), y n = 1/ n, y n 0., T y n = T x n /( n x n ) > n, T. T. ( ) x n x in D(T )., T x n T x = T (x x n ) M x n x 0, T x n T x, T. X, Y normed sps, T L(X, Y ) D(T ) = X T : X Y ; bdd lin. op. L(X) := L(X, X).,, T : X Y D(T ) = X. X := L(X, K) X (conjucate sp.), (lin. conti. functional). T L(X, Y ), (operator norm). T := T x sup x X\{0} x = sup T x. x =1 T x T x. ( :.) 5.2 X normed sp., Y Banach sp.. L(X, Y ) T Banach sp.. {T n } L(X, Y ) Cauchy, i.e., T n T m 0 (m, n ). x X, T n x T m x T n T n x 0, {T n x} Cauchy in Y. Y :, y Y ; T n x y. y x T x = y,., ε > 0, n, m, T n T m < ε, T n x T m x T n T n x ε x, m T n x T x ε x., T x T x T n x + T n x (ε+ T n ) x, T L(X, Y ). T n T ε, T n T in L(X, Y ). L(X, Y ).

15 Functional Analysis (Examples of bounded operators) 5.1 R n, k(t) L (), 1 p. x L p (), (T x)(t) = k(t)x(t) (t ) T L p (), T = k (, k = 0. > 0. T k. ε > 0, ε = { k > k ε}, ε, ε > 0, x(t) := ε 1/p 1 ε (t)., ε =, { t n}; n:,.) 5.2 R n, k(t, s) L 2 ( 2 ), i.e., k(t, s) 2 dtds <. x L 2 () 2 ( ) 1/2, (T x)(t) = k(t, s)x(s)ds T, T k(t, s) 2 dtds. 2 k, T. Schwartz, (T x)(t) (T x)(t) 2. ( ) 1/2 ( 1/2 k(t, s) x(s) ds k(t, s) 2 ds x(s) ds) 2. k(t, s) 2 ds x 2 L 2. t T x 2 L 2, 5.3 ( ) ρ L 1 (R n ). 1 p, x L p (R n ), (T x)(t) = (ρ x)(t) := ρ(t s)x(s)ds = R n ρ(s)x(t s)ds R n T L p (R n ) L p (R n ), T x L p T = ρ x ρ (convolution op.). ρ L 1 x L p. p = 1,. 1 < p < q p (1/p + 1/q = 1). ρ(t s)x(s) = ( ρ(t s) 1/p x(s) ) ρ(t s) 1/q Hölder, ( (T x)(t) ) 1/p ( ρ(t s) x(s) p ds 1/q ( ρ(s) ds) = ) 1/p ρ(t s) x(s) p ds ρ 1/q L. 1 p, t, T x p L p ρ p/q L 1 dt ρ(t s) x(s) p ds = ρ p/q+1 L 1 x p L p. p/q + 1 = p(1/q + 1/p) = p,. 5.2 (Inverse operators) T L(X, Y ), S L(Y, X); T S = I Y, ST = I X S T (inv. op.), T 1. T L(X) y X, (I T )x = y x X, (I T ) 1.

16 Functional Analysis X: Banach, T L(X). T < 1 R(I T ) = X, (I T ) 1 L(X); (I T ) 1 = T n = I + T + T 2 + (T 0 = I). (Neumann n=0 series), L(X). (I T ) 1. T < 1. (I T ) 1 T < 1 n=0 T n 1/(1 T ) n=0 T n <,, n=0 T n <. n=0 T n T n = 1/(1 T ) <, n=0. X, L(X). m T k T k T k 0 (n > m ), k=m+1 T k L(X) Cauchy, S = T k L(X). n=0 T S = ST = T n+1 = T n I = S I, (I T )S = S(I T ) = I, i.e., (I T ) 1 = S = n=0 T n. (I T ) ( ) < a < b <, y C[a, b]. y(t) = x(t) b a n=0 k(t, s)x(s)ds x C[a, b]. k(t, s) C([a, b] 2 ), M := max t,s [a,b] k(t, s), M(b a) < 1. (X, ) := (C[a, b], ) x X, (Kx)(t) = b a k(t, s)x(s)ds, K L(X), Kx M(b a) x, i.e., K M(b a) < 1. y = (I K)x, (I K) 1 ; x = (I K) 1 y = y + Ky + K 2 y +. k 1 (t, s) = k(t, s), k n (t, s) = b k n (t, s) M n (b a) n 1, K n y(t) = a k 1 (t, r)k n 1 (r, s)dr (n 2) b a k n (t, s)y(s)ds. h(t, s) := n 1 k n(t, s),, h(t, s) C([a, b] 2 ), x(t) = y(t) + n=1 b a k n (t, s)y(s)ds = y(t) + b a h(t, s)y(s)ds T n n=0

17 Functional Analysis x C[a, b] Kx C[a, b]. 5.5 ( ), t. y(t) = x(t) t a k(t, s)x(s)ds x C[a, b]., M(b a) < 1. x., (Kx)(t) = t k 1 (t, s) = k(t, s), k n (t, s) = a k(t, s)x(s)ds t s k 1 (t, r)k n 1 (r, s)dr (n 2) n (t s)n 1 n (b a)n 1 k n (t, s) M M K n y(t) = (n 1)! (n 1)! n=1 n=1 t a k n (t, s)y(s)ds. K n n (b a)n 1 M <, (n 1)! (I K) 1 ; x = (I K) 1 y = y + Ky + K 2 y +. h(t, s) := n 1 k n(t, s), h(t, s) C([a, b] 2 ), x(t) = y(t) + t a h(t, s)y(s)ds. 5.3 k n (t, s) M n (t s) n 1 /(n 1)!. 6 (,, ) 3,,, ( (Baire s category theorem)) (X, d), X n X (n 1). X = n=1 X n, 1 X n X, i.e., X B X n.. X n. X 1 X 1 X (X ). x 1 X \ X 1. X 1 closed, d 1 := d(x 1, X 1 ) = inf x X1 d(x 1, x) > 0. ρ 1 := 1 (d 1 /2) 1, B 1 := B(x 1, ρ 1 ) B 1 X 1 =. X 2, x 2 B 1 \ X 2, d 2 := d(x 2, X 2 ) > 0. x 2 / X \ B 1 ( ), d 2 := d(x 2, X \ B 1 ) > 0. ρ 2 := min{1/2, d 2 /2, d 2/2} 1/2, B 2 := B(x 2, ρ 2 ) B 2 B 1, B 2 X 2 =. B 1 B 2, B k X k =, ρ k 1/k {B k }. B k x k k < m d(x k, x m ) ρ k 1/k, Cauchy. X x X; x k x. k, x B k B k X k =, x / X k, i.e., x / X k, X k = X.

18 Functional Analysis (Uniform bounded principle) 6.2 ( ) X Banach sp., Y normed sp.. {T λ } λ Λ L(X, Y ), x X, sup T λ x < = sup T λ <. λ Λ λ Λ X n := λ Λ {x X; T λx n} T λ X n, n=1 X n = X. X Baire, X n0, i.e., x 0 X, ρ 0 > 0; B(x 0, ρ 0 ) X n0. y B(0, ρ 0 ), y + x 0 B(x 0, ρ 0 ), y = (y + x 0 ) x 0 T λ y T λ (y + x 0 ) + T λ x 0 2n 0. x X, µ = 2 x /ρ 0 µ 1 x = ρ 0 /2 < ρ 0 µ 1 x B(0, ρ 0 ), T λ x = µ T λ (µ 1 x) 2n 0 µ = 4n 0 x /ρ 0., sup λ T λ 4n 0 /ρ (Banach-Steinhaus theorem) X Banach sp., Y normed sp. {T n } L(X, Y ). x X, {T n x}, T x := lim n T nx T L(X, Y ) T lim n T n. γ := lim n T n. {T n x}, sup n T n x <. X Banach, sup n T n <. γ <. ε > 0, {n k }; T nk γ + ε. x X, T x = lim k T nk x (γ + ε) x. T L(X, Y ), T γ + ε. ε T γ. 6.2 (Open mapping theorem) 6.4 ( ) X, Y Banach sps. T L(X, Y ), R(T ) = Y T, i.e., U X; open, T (U) Y ; open. (1st Step) ρ > 0; B Y (0, ρ) T B X (0, 1). R(T ) = Y, Y = T (X) = T B X (0, n) = T B X (0, n), Y Baire, n=1 n=1 n 1, a Y, δ > 0; B Y (a, δ) T B X (0, n). y B Y (0, δ). y + a, a B Y (a, δ) T B X (0, n), y k, y k T B X(0, n); y k y + a, y k a. y k y k T B X(0, 2n),, y = (y + a) a = lim(y k y k ) T B X(0, 2n). B Y (0, δ) T B X (0, 2n). ρ = δ/(2n) T B Y (0, ρ) T B X (0, 1). (2nd Step) ρ, η = ρ/2 > 0 B Y (0, η) T B X (0, 1). B Y (0, ρ) T B X (0, 2), i.e, y B Y (0, ρ), x B X (0, 2); y = T x. ε k = 2 k (k 0), B Y (0, ε k ρ) T B X (0, ε k ). y B Y (0, ρ) T B X (0, 1) x 0,n B X (0, 1); T x 0,n y, x 0 B X (0, 1); y T x 0 < ε 1 ρ. y T x 0 B Y (0, ε 1 ρ), x 1 B X (0, ε 1 ); y T x 0 T x 1 < ε 2 ρ., k m m m x k B X (0, ε k ); y T x j < ε k+1 ρ. x j x j ε j 0 (m > j=0 j=k k k ), { x j } Cauchy in X, x = x k X. T j=k j=0 j=k

19 Functional Analysis 17 T x = T x k. x x 0 + x k < x 0 + ε k < = 2. x B X (0, 2). k y T x j < ε k+1 ρ k y = T x k = T x. j=0 (3rd Step) T, i.e., U X; open, T (U) Y ; open. T, α > 0, B Y (0, αη) T B X (0, α). y 0 T (U). x 0 U; y 0 = T x 0. U open, δ > 0; x 0 + B X (0, δ) = B X (x 0, δ) U. y B Y (y 0, δη), y := y y 0 B Y (0, δη). B Y (0, δη) T B X (0, δ) x B X (0, δ); y = T x. x 0 + x U, y = y 0 + y = T x 0 + T x = T (x 0 + x ) T (U). B Y (y 0, δη) T (U), T (U) open. 6.5 ( (range theorem)) X, Y Banach sps.. T L(X, Y ), R(T ) = Y T 1 to 1 T 1 L(Y, X). T 1, Y X.. U X; open, S := T 1 S 1 (U) = {y Y ; Sy = T 1 y U} = {y Y ; y T (U)} T (U)., T (U) open., S 1 (U) open, S = T 1. T 1 L(Y, X). 6.3 (Closed graph theorem), T D(T ) X. 6.1 (X, X ), (Y, Y ) normed sps.. T X Y (closed op.) T : D(T ) X Y, T G(T ) = {(x, T x) X Y ; x D(T )} (x, T x) G = x X + T x Y, x n D(T ) x in X, T x n y in Y (x, y) G(T ), i.e., x D(T ) y = T x. D = D(T ), T T D. x n x in D T x n T x in Y, 6.6 (i) D(T ) closed, T : D(T ) X Y (= ) T. D(T ) = X. (ii) Y Banach, T : D(T ) X Y, T D(T ) T,, T. (iii) X, Y Banach, T D(T ) x G := x X + T x Y. (i) x n D(T ) x X, T x n y Y D(T ) closed, x D(T ), y = T x, T. (ii) x n D(T ) x X. {T x n } Y Cauchy, Y T x n y Y. T x := y, T D(T ), T = T on D(T ), T = T. (i), T. (iii) ( ) {x n } D(T ); x n x m G 0 (m, n ). X, Y x n x in X, T x n y in Y. x D(T ), y = T x. x n x G = x n x X + T x n T x Y 0, D(T ).

20 Functional Analysis 18 ( ) x n D(T ) x X, T x n y Y {x n } D(T ) G Cauchy. x D(T ); x n x G 0. x = x, T x = T x = y, T closed. D(T ) closed, Y Banach, T. 6.1 X = C[0, 1], D(T ) = C 1 [0, 1] (T x)(t) = x (t),,. 6.7 ( ) X, Y Banach sps, T X Y. D(T ) = X T L(X, Y ). Z = G(T ) = {(x, T x) X Y ; x D(T ) = X}. T X, Y Banach, (x, T x) Z := x + T x Banach. S : Z X; S(x, T x) = x S L(Z, X), R(S) = X S 1 to 1. X Banach, S 1. T x (x, T x) Z = S 1 x Z M x. T, i.e., T L(X, Y ). 7 (Linear Functionals) normed sp X K = R or C (bdd lin. functional)., (conti. lin. functional). X := L(X, K), X (dual sp.). K = R or C, X. f X, i.e., f : X K; f(αx + βy) = αf(x) + βf(y) (α, β K, x, y X), f <. 7.1 (Dual spaces) X Hilbert sp. H, H = H. R n, C n, L 2 (), l 2 dual sps ( (Riesz s representation theorem)) X = H Hilbert sp.. f H, 1 y H; f(x) = x, y ( x H). f = y. H = H. f 0 y = 0. f 0. N = {x H; f(x) = 0}. (, x N, α K f(αx) = αf(x) = 0, αx N. x, x N f(x + x ) = f(x) + f(x ) = 0, x + x N.. x n N x H f(x) = lim f(x n ) = 0 x N. N.) H = N N. f 0, N. y 0 N ; y f(y 0 ) 0. y := (f(y 0 )/ y 0 2 )y 0, y N,., x H, f(y 0 )x f(x)y 0 N, 0 = f(y 0 )x f(x)y 0, y = f(y 0 ) x, y f(x) y 0, y = f(y 0 )( x, y f(x))

21 Functional Analysis 19, f(x) = x, y. y H; f(x) = x, y = x, y x H, x, y y = 0, x = y y y y = 0, i.e., y = y. y. x = 1, Schwartz, f(x) = x, y x y = y. f y. x 0 = y/ y f(x 0 ) = x 0, y = y y = f(x 0 ) sup f(x) = f. f = y. x =1 Banach sp. X, X, p <. q p, i.e., 1/p + 1/q = 1 (, p = 1 q = ). (i) R n (L p ()) = L q (). (ii) (l p ) = l q. 7.2 l 0 := {(x n ) l ; lim n x n = 0} l 0 l Banach, (l 0 ) = l (Hahn-Banach s extension thoerem) 7.2 ( ) X, L X. f L. p : X R;, p(λx) = λp(x) (λ > 0, x X), p(x + y) p(x) + p(y) (x, y X), f p on L, F X = L(X, R); F = f on L, F p on X., f X, f p. L = X, L X. x 0 X \ L, L 1 = L + Rx 0.. x = y + tx 0 L 1. f L 1, c R, F (x) = F (y + tx 0 ) := f(y) + tc. F L 1. c F p on L 1. Zorn f X F p. L 1, x = y+tx 0 = y +t x 0 (y, y L, t, t R) 0 = (y y )+(t t )x 0, i.e, (t t )x 0 = y y L. t t x 0 = (y y)/(t t ) L,. t = t, y = y. x i = y i + t i x 0 L 1 (y i L, t i R) α i R, F (α 1 x 1 + α 2 x 2 ) = F ((α 1 y 1 + α 2 y 2 ) + (α 1 t 1 + α 2 t 2 )x 0 ) = f(α 1 y 1 + α 2 y 2 ) + (α 1 t 1 + α 2 t 2 )c = α 1 (f(y 1 ) + t 1 c) + α 2 (f(y 2 ) + t 2 c) = α 1 F (x 1 ) + α 2 F (x 2 ). y, y L, f(y) + f(y ) = f(y + y ) p(y + y ) = p(y + x 0 + y x 0 ) p(y + x 0 ) + p(y x 0 ), f(y ) p(y x 0 ) p(y + x 0 ) f(y). β 1 := sup y L(f(y ) p(y x 0 )), β 2 := inf y L (p(y + x 0 ) f(y)), β 1 β 2 β 1 c β 2 c, f(y) + c p(y + x 0 ), f(y ) c p(y x 0 ) (y, y L). t > 0 F (x) = F (y + tx 0 ) = f(y) + tc = t(f(y/t) + c) tp(y/t + x 0 ) = p(y + tx 0 ) = p(x). t < 0 F (x) = F (y + tx 0 ) = f(y) + tc = ( t)(f( y/t) c) tp( y/t x 0 ) = p(y + tx 0 ) = p(x). t = 0 F (x) = F (y) = f(y) p(y) = p(x). F p on L 1. g Φ g : L g R;, L L g, g = f on L, g p on L g. Φ. Φ ( ). g, g P hi, g g L g L g, g = g on L g. Φ {g λ }. L λ := L gλ

22 Functional Analysis 20. L 0 := L λ, g 0 on L 0 g 0 = g λ on L λ, g 0 Φ {g λ }. Φ. Zorn, F Φ, i.e, g Φ; F g g = F. F X.,, F Φ. F. 7.3 ( ) X, L X. f L. p : X C;, p(λx) = λ p(x) (λ C, x X), p(x + y) p(x) + p(y) (x, y X), f p on L, F X = L(X, C); F = f on L, F p on X., f X, f p.. f(x) = g(x) + ih(x). g, h L. g, h f p on L. g X G; G p on X. G(x) = G( x) p( x) = p(x) G p., g(ix) + ih(ix) = f(ix) = if(x) = ig(x) h(x), h(x) = g(ix). F (x) := G(x) ig(ix),., F = f on L F (x 1 + x 2 ) = F (x 1 ) + F (x 2 ), F (ix) = G(ix) ig( x) = i( ig(ix)+g(x)) = if (x) a R F (ax) = af (x), α C F (αx) = αf (x). F (x) = re iθ F (e iθ x) = e iθ F (x) R, G(e iθ x), F (x) = e iθ F (x) = G(e iθ x) p(e iθ x) = e iθ p(x) = p(x). 7.1 X ( ), L X. f L ( ). F X ; F = f on L, F X = f L. p(x) = f L x,. 7.2 X. x 0 X, 0, g X ; g(x 0 ) = x 0, g = 1. L := x 0 = {tx 0 ; t K} f(x) = f(tx 0 ) := t x 0 (x = tx 0 L),. f(x) = t x 0 = tx 0 = x f = X, L X. x 0 X \ L, d := inf y L x 0 y > 0. f X ; f = 0 on L, f(x 0 ) = 1, f 1/d. L 1 := L + Rx 0 g(x) = t (x = y + tx 0 L 1 ) g = 0 on L, g(x 0 ) = 1, g L1 1/d..,. 2, Banach sp.,,,,, (= compact ),,.,,,,.

23 Functional Analysis 21 A,. A.1 L p (X, F, µ) (X a set, F σ-field on X, µ = µ(dx) on X), Hölder, Minkovsky. f L p (X) (or L p (X, dµ) or L p (X, F, µ) ) f L p <., ( 1/p f L p = f(x) µ(dx)) p (1 p < ), X f = ess.sup x X f(x) := inf{α; f(x) α µ-a.e}., 1 p, 1 q 1/p + 1/q = 1., p = 1 q =, p = q = 1. Hölder. fg L 1 f L p g L q, fg L 1 f L 1 g ( fg L 1 f g L 1). Minkovsky f + g L p f L p + g L p, L p (X). (X, F) = (N, 2 N ) µ = n 1 δ n counting measure ( ), N f(n), f(n)µ(dn) = f(n). x = (x 1, x 2,... ), N n 1 f(n) = x n p x n p. l p Banach sp. n 1. A.2 A.1 X, X Banach sp., J : X X ; Jx = x (x X), J(X) dense in X J(X) X, X X X = X. X X (completion). X X Cauchy. {x n }, {y n } X, {x n } {y n } x n y n 0 (n )., x = [{x n }] X := X/. x = [{x n }], ỹ = [{y n }] X, α K, α x := [{αx n }], x + ỹ := [{x n + y n }], X. x = [{xn }] X, x n x m x n x m 0, { x } R Cauchy, lim x n =: x. X n ( ). x X, x n = x, Jx := [{x n = x}] x, X X., J : X X, Jx = x. x X,

24 Functional Analysis 22 {x n } ε > 0, N; n, m N, x m x n < ε, n N, Jx n X, x Jx n = [{x m x n } m 1 ] x Jx n = lim m x m x n ε x Jx n 0 (n )., J(X) dense in X. X. { x n } X Cauchy. x n {x (n) k } k 1 Cauchy k n ; m > k n, x (n) m x (n) k n 1/n. x := [{x (n) k n }], x X, { x n }., x X {x (n) k n } X (i.e., X Cauchy ). (A.1) x n Jx (n) k n = lim m x(n) k m x (n) k n 1 n. (A.2) x (n) k n x (m) k m = Jx (n) k n Jx (m) k m Jx (n) k n x n + x n x m + x m Jx (m) k m x n x m + 1 n (n, m ). m {x (n) k n } X. (A.1), (A.2),., 2, x x n x Jx (n) k n + Jx (n) k n x n x Jx (n) k n + 1 n. x Jx (n) k n = lim p x(p) k p x (n) k n lim x p x n + 1 p n lim x x n lim x Jx(n) n n k n lim x p x n = 0, n,p x n x in X, X. A.3 H k,p () R n, C k () k. α = (α 1,..., α n ) (multi-index ), α := α α n, x α := α1 1 α n n., j = / xj. C k,p () := u Ck (); u k,p := α; α k x α u(x) p dx 1/p <. (C k,p (), k,p ) H p,k (), Sobolev sp.. {u n } Cauchy in C k,p () α; 1 α k, x α u n (x) x α u m (x) p dx 0 (m, n ). L p (), u α L p (); x α u n (x) x α u α (x) p dx 0 (n ). (u α ) α k H p,k (), u α = α x u.

25 Functional Analysis 23 A.4 A.2 [a, b] f(t) P n (t), i.e., {P n (t)}; P n f on [a, b], i.e., lim sup n t [a,b] P n (t) f(t) = 0. t = (t a)/(b a), [a, b] [0, 1], ( ), [a, b] = [0, 1]. t [0, 1], ( ) n (k nt) 2 t k (1 t) n k = nt(1 t) 1 k 4 n. ( t(1 t) (t + (1 t))/2 = 1/2.),, ( ) n x k y n k = (x + y) n, k ( ) n t k (1 t) n k = 1 k, x, x, ( ) n ( ) n k x k y n k = nx(x + y) n 1, k 2 x k y n k = nx(nx + y)(x + y) n 2 k k x = t, y = 1 t (k nt) 2 = k 2 2ntk + n 2 t 2. P n (t) = f ( k n ) ( ) n t k (1 t) n k k P n [0, 1], f. ( ) ( ) f(t) P n (t) k n f(t) f t k (1 t) n k n k, f [0, 1], ε > 0, δ > 0; t, t [0, 1]; t t < δ, f(t) f(t ) < ε. t, k t k/n < δ t k/n δ, S 1, S 2, f(t) P n (t) S 1 + S 2,, S 1 ε ( ) n t k (1 t) n k = ε. k M = max t [0,1] f(t) f(t) f(k/n) 2M, t k/n δ 1 nt k /(nδ), S 2 2M ( nt k nδ ) 2 ( ) n t k (1 t) n k k M 2nδ 2. f(t) P n (t) ε + M 2nδ 2. t [0, 1], sup t [0,1], n lim sup n t [0,1], ε > 0, ( )= 0. f(t) P n (t) ε

26 Functional Analysis 24 A dim X = n {x 1,..., x n } basis x X, (α 1,..., α n ); x = α i x i. x = max α i 1 norm. x. x α i x i max α i x i = c 1 x (c 1 = x i )... k 1, y k ; y k > k y k, (y k / y k ) < 1/k 0., z k = y k / y k, z k = 1, z k < 1/k 0, i.e., z k 0 under., z k = n x i, 1 = z k = max β (k), β (kj) β i in K, i=1 β(k) i z kj z under, under by ( ). z = 0, 1 = z kj z. c 0 > 0; x c 0 x (x X).. [ ] c 0 > 0; x c 0 x (x X), i.e., 0 < c 0 x / x = (x/ x ). y = x/ x, f(y) := y, min y; y =1 f(y) > 0. y k y under under, f(y) conti. under. S := {y; y = 1}, compact., y = n i=1 β ix i, 1 = y = max β i, y (β 1,..., β n ) K n, S = {(β 1,..., β n ); max β i = 1} K n. S ( ) compact., y S, f(y) > 0, compact set, y 0 S; min S f = f(y 0 ) = y 0 > 0, c 0 = y 0. i i A.6 A.3 X, x + y 2 + x y 2 = 2( x 2 + y 2 ), x, y = 1 ( x + y 2 x y 2 + i x + iy 2 i x iy 2) 4,.. x, x = x 2, y, x = x, y, ix, y = i x, y, x, y = x, y, x, 0 = 0, x = 0.,,. (1) Re x 1, y + Re x 2, y = 1 2 Re x 1 + x 2, 2y = Re x 1 + x 2, y. (2) Re. (3) α R αx, y = α x, y. γ C, γx, y = γ x, y. (1), i.e., x, y = Re x, y. x 1, y + x 2, y = 1 { ( x1 + y 2 + x 2 + y 2 ) ( x 1 y 2 + x 2 y 2 ) } 4 = 1 { ( x1 + x 2 + 2y 2 + x 1 x 2 2 ) ( x 1 + x 2 2y 2 + x 1 x 2 2 ) } 8 = 1 8 ( x 1 + x 2 + 2y 2 x 1 + x 2 2y 2 ) = 1 2 x 1 + x 2, 2y.

27 Functional Analysis 25 x 1, y + x 2, y = x 1 + x 2, 2y /2. x 2 = 0, 0, y = 0, x, y = x, 2y /2., x 1, y + x 2, y = x 1 + x 2, 2y = x 1 + x 2, y /2. Re x 1, y + Re x 2, y = Re x 1 + x 2, y. (2) x 1 + x 2, y + y, x 1 + x 2 = ( x 1, y + y, x 1 ) + ( x 2, y + y, x 2 ) y iy, x 1 + x 2, y y, x 1 + x 2 = ( x 1, y y, x 1 ) + ( x 2, y y, x 2 ), x 1 + x 2, y = x 1, y + x 2, y. (3) y X α R, f(α) := αx, y.. (2), n N, f(1) = f(n/n) = nf(1/n), i.e., f(1/n) = (1/n)f(1). f(m/n) = (m/n)f(1) (m, n N). f( α) = f(α),, r Q, f(r) = rf(1)., α R, f(α) = αf(1), i.e., αx, y = α x, y. β R, iβx, y = i βx, y = iβ x, y. γ C, γx, y = γ x, y. A.7 A 2 () C n open, f A 2 () f on, f(z) 2 dxdy < (z = x + iy). f, g A 2 (), (f, g) = f(z)g(z)dxdy Hilbert.. {f n } A 2 () Cauchy. z = x+iy C (x, y) R 2, f n L 2 () Cauchy f L 2 (); f n f L 2 0. f. z, Cauchy, f n (z) = 1 f n (ζ) dζ (0 < r ε) 2πi ζ z ζ z =r, ε > 0 ; { ζ z < ε}. r, r 0 ε (A.3) 1 2 ε2 f n (z) = 1 2πi ε 0 f n (ζ) drr ζ z =r ζ z dζ = 1 f n (ζ)dξdη 2π ζ z ε (ζ = ξ + iη). ( ζ z = r ζ z = r(cos θ + i sin θ), dζ = r( sin θ + i cos θ)dθ = ir(cos θ + i sin θ)dθ = i(ζ z)dθ, ε 0 dζ drr ζ z =r ζ z = ε 0 drr 2π 0 idθ = i dξdη.) ζ z ε f n L 2 () (A.3) z. ( K compact 1, ε > 0 K ε := {ζ; ζ z ε, z K}. sup f n (ζ) f(ζ) dξdη f n (ζ) f(ζ) dξdη f n f L 2 K ε 1/2 0). z K ζ z ε K ε (A.3) 2/ε 2 f. f n f in L 2 (), f = f a.e. f A 2 ().

28 Functional Analysis 26 B, X, X := (X ) X 2. X X. X = X X Banach sp.. 1 < p <, L P () l p Banach sps. B.1 X normed ps..,. x n x (strong) in X x n x 0. s-lim n x n = x. x n x (weak) in X f X, f(x n ) f(x). w-lim n x n = x.. B.1 x n x (weak),. [ ] x n x (weak). x x, f X ; f(x x ) = x x = 0, f = 1. f(x x ) = f(x) f(x ) = w- lim(f(x n ) f(x n )) = 0,. qed B.1 X = l 2 f l 2, 1 y = (y k ) l 2 ; f(x) = x k y k (x = (x k ) l 2 ). x (n) = (x (n) k = δ n,k ) l 2 f(x (n) ) = y n 0, x (n) 0 (weak). x (n) x (m) l 2 = 2 (m n). {x (n) }. B.1 X, x n x (weak) in X. { x n } x lim inf x n. f X, T n (f) := f(x n ) T n X., {T n (f)}. X Banach,, sup T n <. T (f) := f(x) = lim f(x n ) = lim T n (f) T X, T = x. T n = x n, Banach-Steinhaus, x = T lim inf T n = lim inf x n. B.2 H Hilbert sp.. x n x (weak) in H,, x n x x n x (strong) in H. x n x (weak) in H Riesz, y H, x n, y x, y. x n x 2 = x n 2 + x 2 2R x n, x x 2 + x 2 2R x, x = 0. X. f n f (weak*) in X : ( * ) x X, f n (x) f(x). B.3 X, f n f (weak*) in X. f lim inf f n. B.2.

29 Functional Analysis 27 B.2 B.1 X, Y. T : D(T ) X Y, D(T ) dense in X. g D(T ) Y f X ; g T = f on D(T ), T : D(Y ) Y X T (g) = f., T (g) = g T. T T (adjoint op.). g T = f f g,., g T = f, f = f on D(T ), D(T ) dense, f, f f = f on X. T, T X, D(T ) = X. g Y, f := g T f X, D(T ) = Y. X = H, Y = H Hilbert x, T y = T x, y (x D(T ), y D(T )). D(T ) = D(T ) T = T on D(T ) T (self-adjoint op.). D(T ) D(T ) T = T on D(T ),, x, T x = T x, y (x, y D(T )) T (symmetric op.). B.2 X = Y = R n, T R n R n t j,k := T e j, e k, T = (t j,k ). T = t T ; T. X = Y = C n T = t T ;. B.3 H = L 2 (0, 1). k(t, s) [0, 1] 2, x H, T x(t) = k(t, s)x(s)ds T H, D(T ) = D(T ) = H. [0,1] T y(t) = k(t, s)y(s)ds. [0,1] C T n, λ C, x R n ; x 0, T x = λx λ T, x T. λ det(λi T ) = 0. λ,, (λi T )x = 0 x 0 (Ker(λI T ) = {0}, i.e., λi T 1 to 1), det(λi T ) 0, (λi T ) 1., T X, z ρ(t ) z C; Ker(zI T ) = {0} (zi T 1 to 1) (zi T ) 1 L(X)., ρ(t ) T (resolvent set). R(z) := (zi T ) 1 T. σ(t ) := C \ ρ(t ) T (spectrum). x D(T ); x 0, T x = zx, i.e., (zi T )x = 0 z C T, σ p (T ), (point spectrum). N(zI T ) := Ker(zI T ) z, z. σ p (T ) σ(t ). z 1, z 2 ρ(t ) R(z 1 ) R(z 2 ) = (z 1 z 2 )R(z 1 )R(z 2 ) (resolvent eqution). ρ(t ). σ(t ). R(z): holmorphic on ρ(t ). T L(X), r(t ) := lim sup n T n (spectral radius).

30 Functional Analysis 28 (i) z > r(t ) = z ρ(t ), R(z) = k 0 (ii) z σ(t ); z = r(t ). 1 z k+1 T k = 1 z + 1 z 2 T + 1 z 3 T 2 +. D X, Y Banach sps. T : X Y compact (or ) {x n } X: bdd, {x nk }; T x nk y Y. compact op.. (, x n X; x n = 1, T x n n, compact ) T : compact = x n x (weak) in X T x n T x (strong) in Y. D.1 H: Hilbert, T : H H: compact self adj. op., {λ n } R ONS {x n }, x H, c k K, x H; T x = 0, x = c k x k + x, T x = λ k c k x k. λ n 0 dim N(λ n I T ) < ( n).