- PDF Free Download

J J l 2 J T l 1 J T J T l 2

l 1 J J l 1 c 0 J J J J J l 2 l 2 J J J T J T l 1 J J T J T J T J

{e n } n N {e n } n N x X {λ n } n N R x = λ n e n {e n } n N {e n : n N} e n 0 n N k 1, k 2,..., k n N λ k1, λ k2,..., λ kn R λ k1 e k1 + λ k2 e k2 +... + λ kn e kn n = 0

{e n } n N { } D = r i e i : {r i } n Q, n N D D X x = λ n e n X ϵ > 0 N N {r i } i N Q λ n e n x < ϵ 2 n > N r i e i x X n > N λ i r i e i < ϵ 2 i+1 i N < λ i r i e i + ϵ 2 i+1 + ϵ 2 < ϵ λ i e i x X {e n } n N x = λ n e n X { } λ i e i < n N x = n N { } λ i e i

X n N P n : X X P n (x) = P n ( λ n e n ) = λ i e i n N P n(x) = x x X n { P n (x) } = x x X K > 0 x x K x x X x = n P n (x) x (X, ) { y (k) } k N (X, ) ϵ > 0 k 0 N y (i) y (j) < ϵ i, j > k 0 P n (y (i) ) P n (y (j) ) < ϵ i, j > k 0, n N { P n (y (k) ) } k N < e 1,..., e n > { P n (y (k) ) } k N n N z n = P n (y (k) ) {z n } n N k ϵ > 0 k, N N P n (y (k) ) P n (y (j) ) < ϵ 3 j > k, n N P n (y (k) ) z n < ϵ 3 n N P n (y (k) ) P m (y (k) ) < ϵ 3 n, m > N

n, m > N z n z m z n P n (y (k) ) + P n (y (k) ) P m (y (k) ) + P m (y (k) ) z m < ϵ {z n } n N (X, ) {z n } n N z = z n m > n N n P n (z m ) = P n ( P m (y (k) )) = P n (P m (y (k) )) = P n (y (k) )) = z n k k k P n < e 1,..., e n > {α i } i N z n = a i e i n N z 1 < e 1 > α 1 R z 1 = α 1 e 1 P 1 (z 2 ) = z 1 {e n } n N α 2 R z 2 = α 1 e 1 +α 2 e 2 α 1,...α n α n+1 z = n z n P n (z) = α i e i = z n k 0 N k > k 0 n N ϵ > 0 { z n P n (y (k) ) : n N } < ϵ { P n (z) P n (y (k) ) : n N } < ϵ z y (k) < ϵ k y (k) = z (X, ) n N x X P n (x) x K x P n < n N bc({e n } n N ) = P n bc({e n } n N ) n N bc({e n } n N ) 1 x X, x 0 P n (x) P n x bc({e n } n N ) x

n n N e n : X R e n(x) = e n( λ k e k ) = λ n x = k=1 e n(x)e n {e n} n N n N x = λ n e n X e n(x) = e n(x)e n e n = n 1 λ i e i λ i e i e n 2 x e n 2K e n x {e n} n N {e n } n N {e n } n N X {e n } n N e n 0 n N [e n : n N] = X [e n : n N] = < e n : n N > K > 0 m > n N λ 1,..., λ m R λ i e i K λ i e i

m > n N λ 1,..., λ m R λ i e i = P n (x) = P n (P m (x)) P n P m (x) bc({e n } n N ) λ i e i {e n : n N} n N λ 1,..., λ n R λ 1 e 1 +... + λ n e n = 0 i i 1 λ i e i = λ j e j λ j e j 2K λ i e i = 0 λ i = 0 j=1 j=1 i = 1,..., n n N p n :< e n : n N > < e n : n N > p n ( λ i e i ) = {n,m} p n {e n : n N} {n,m} p n ( λ i e i ) = λ i e i K λ i e i p n p n K n N < e n : n N > X p n p n : X < e n : n N > p n K n N m > n x X p n (x) = p n (p m (x)) x X {λ i } i N p n (x) = λ i e i n N p n(x) = x ϵ > 0 z = n x z < ϵ K + 1 m > k p m(z) = z λ i e i k β i e i < e n : n N > x p m (x) x z + z p m (z) + p m (z) p m (x) (1 + p m ) x z < ϵ

1 n N λ 1,..., λ m R λ i p ) 1/p n 0 λ i e i = ( λ i p ) 1/p ( λ i p ) 1/p = λ i e i c 0 {e n } n N K 0 m > n N λ 1,..., λ m R λ i e i K λ i e i bc({e n } n N ) K x X x 1 n N P n (x) K P m (x) m > n n P n (x) K x K

x X, x 0 {x n} n N X x n = M < n N S X x n(s) k x (s) s S x = w x n n x (x) = x n n(x) x X x X {s n } n N S x = s n ϵ > 0 N, n 0 N n s N x < ϵ 3 {M, x } x n(s N ) x (s N ) < ϵ 3 n n 0 n n 0 x n(x) x (x) x n(x) x n(s N ) + x n(s N ) x (s N ) + x (s N ) x (x) < M s N x + ϵ 3 + x s N x < ϵ 3 + ϵ 3 + ϵ 3 = ϵ {e n } n N x X, x 0 {x k } k N X x k < k N x k (e n) k x (e n ) n N x = w k x k S =< e n : n N > {x k } k N x

{x n } n N {x n } n N [x n : n N] {x n } n N [x n : n N] {x n } n N x n : [x n : n N] R x n( α k x k ) = α n k=1 {x n } n N X {x n } n N x n 0 n N K > 0 m > n N λ 1,..., λ m R λ i x i K λ i x i {e n } n N x [e n : n N] x = x (e n )e n m > n N λ 1,..., λ m R x X x 1

λ i e i (x) = λ i e i (P n (x)) = Pn( λ i e i )(x) Pn λ i e i λ i e i K λ i e i K λ i e i {e n} n N x [e n : n N] {α n } n N x = λ n e n x (e n ) = λ n n N ϵ > 0 x X x = 1 y (1 + ϵ) y + mx y F m R F S F k k N y 1,..., y k S F S F S(y i, ϵ ) S(y, ϵ) 2 y ϵ j {1,..., k} k yi S X yi (y i ) = y i = 1 Ker(yi ) X x X x = 1 yi (x) = 0 i = 1,..., k y S F 0 < ϵ < 1 j {1,..., k} y y j < ϵ 2

m R y + mx y j + mx y y j yj (y j + mx) ϵ 2 = y j (y j ) ϵ 2 = y j ϵ 2 = 1 ϵ 2 1 0 < ϵ < 1 1 + ϵ ϵ > 0 y S F m R x S X (1 + ϵ) y + mx 1 y F y 0 ϵ > 0 m R x S X 1 (1 + ϵ) y y + mx y y (1 + ϵ) y + mx ϵ > 0 (1 + ϵ) {ϵ n } n N (1 + ϵ n ) n N 2 n (1 + ϵ n ) (1 + ϵ) n N (1 + ϵ n ) 1 + ϵ x 1 X x 1 = 1 F 1 =< x 1 > x 2 X x 2 = 1 y (1 + ϵ 2 ) y + mx 2 y F 1, m R F 2 =< x 1, x 2 > x 3 X x 3 = 1 y (1 + ϵ 3 ) y + mx 3 y F 2, m R F n =< x 1, x 2,..., x n > x n+1 X x n+1 = 1 y (1 + ϵ n+1 ) y + mx n+1 y F n, m R {x n } n N X 1 + ϵ m > n N α 1,..., α m R y k = k α i e i F k k N

α i e i = y n (1 + ϵ n+1 ) y n + α n+1 x n+1 = (1 + ϵ n+1 ) y n+1 (1 + ϵ n+1 )(1 + ϵ n+2 ) y n+1 + α n+2 x n+2 = (1 + ϵ n+1 )(1 + ϵ n+2 ) y n+3 m (1 + ϵ i ) y m (1 + ϵ) α i e i i=n+1 {x n } n N 1 + ϵ {x n } n N X, {y n } n N Y {x n } n N {y n } n N n N α 1,..., α n R c α n x n α n y n C α n x n {x n } n N X, {y n } n N Y {x n } n N {y n } n N {α n } n N R α n x n α n y n T : [x n : n N] [y n : n N] T (x n ) = y n n N i) ii) {α n } n N α n x n { } α i y i ϵ > 0 N N n N

m > n > N i=n+1 i=n+1 α i x i < ϵ C α i y i < ϵ T : [x n : n N] Y T (x) = T ( α n x n ) = α n y n T {x n } n N n N T k : X < y 1,..., y n > T n ( α i x i ) = F n :< x 1,..., x n > < y 1,..., y n > F n ( α i x i ) = α i y i α i y i n N T n = F n P n P n (y n ) n N T n F n P n T T (x) = T n (x) x X n T y = α n y n x = α n x n T (x) = y T c = 1 C = T T 1 {x n } n N {y n } n N n N α 1,..., α n R

c α n x n α n y n C α n x n {y n } n N m > n N α 1,...α m R α i y i CK α i x i CK m c α i y i T : X X δ (0, 1) x X x T (x) δ x T T T (x) x δ x T (x) (1 + δ) x x T (x) δ x (1 δ) x T (x) T T (X) X T T (X) x 0 S X x 0(x) = 0 x T (X) 0 < δ < 1 x 0 X x 0 = 1 x 0(x 0 ) > δ x 0 T (x 0 ) = {x (x 0 T (x 0 )) : x = 1} x 0 T (x 0 ) x 0(x 0 T (x 0 )) = x 0(x 0 ) > δ = δ x 0

{x n } n N, {y n } n N {x n } n N K δ = { x n : n N} > 0 x n y n < n=i δ 3K {y n } n N {x n } n N T : X X T (x n ) = y n {y n } n N {x n } n N m > n α 1,..., α m R α i y i T α i x i T K α i x i T K T 1 α i y i {y n } n N n N α 1,...α n R 1 T 1 α i x i α i y i T n N x [x n : n N] x n(x) x n = α i x i n 1 x i (x)e i x i (x)e i x n 2K x x n 2K δ n N x n X x X x n(x)(y n x n ) X x n(x) y n x n 2K δ x y n x n < 2 3 x

T : X X T (x) = x + x n(x)(y n x n ) T T (x n ) = y n n N x T (x) = x n(x)(y n x n ) x x n y n x n < 2 3 x. T l 1 l l 1 l 1 {x n } n N X {x n } n N {x n } n N l 1 l 1 l 1 Y X B Y {s n } n N B Y l 1 l 1 X {s n } n N {s nk } k N

{ s nk+1 s nk }k N {s n k } k N θ > 0 n N k, m N : n < k < m s k s m θ ( ) ( ) {p n } n N N s p2n s p2n 1 θ n N w u n = s p2n s p2n 1 u n 0 u n θ n N {z n } n N z n = u n Y u n {e n } n N {u n } n N {α i } i N {n i } i N k N u k = k+1 i=n k +1 {e n } n N α i e i {u n } n N u n 0 λ 1,..., λ m R n N m > k N

k λ j u j = j=1 k j=1 K λ j n j+1 i=n j +1 n j+1 j=1 i=n j +1 α i e i = k λ j α i e i K n j+1 j=1 i=n j +1 λ j α i e i λ j u j j=1 {u n } n N K {e n } n N {x n } n N δ = x n > 0 e n N n k(x n ) = 0 k N ϵ > 0 {x n} n N {u n } n N {e n } n N x n u n 2 < ϵ 2 x n = 1 n N { } x n n N {b n } n N {e n } n N x n b n 2 < ϵ 2 P k (y n ) = 0 k N k N ϵ > 0 n N N i = 1,..., k e i (x n ) e i < ϵ 2 i n > N n > N k P k (x n ) = e i (x n )e i k e i (x n ) e i < ϵ ( ) 0 < ϵ < δ ( ) k 1 = 1 n 1 = 0 x k1 = α (1) i e i n 2 N n 2 > n 1 n=n 2 +1 α n (1) e n α (1) n e n < ϵ 2 n 2 α (1) n e n < ϵ 2

u 1 = n 2 n=n 1 +1 n 3 > n 2 u 2 = n 3 i=n 2 +1 α (1) n e n x k1 u 1 2 < ϵ2 2 ( ) k 2 > k 1 α (2) i e i x k2 = n 2 i=n 3 +1 n 2 α (2) i e i α (2) i e i < ϵ 4 α (2) i e i < ϵ 4 α (2) i e i n 3 i=n 2 +1 α k 2 i e i < ϵ 4 α (2) i e i x k2 u 2 < ϵ 2 x k 2 u 2 2 < ϵ2 4 {k n } n N {u n } n N {e n } n N x kn u n 2 < ϵ2 2 n n N u n > δ ϵ > 0 n N {x n} n N = {x kn } n N x n u n 2 < ϵ 2 (1 ϵ)ϵ x n = 1 n N 0 < ϵ < δ m = 2 0 < m < ϵ < δ {x n} n N {u n } n N 1 ϵ < 1 m < u n n N x n u n 2 < m 2 = (1 ϵ)2 ϵ 2 4 z n = u n u n z n = 1 n N n N

x n z n = x n u n u n x n u n u n u n 1 x u n 1 ϵ n 1 ϵ 2 u n 1 + x n u n u n 1 x n z n 2 2 1 ϵ 1 ϵ + x n u n 1 ϵ + 2 x n u n 2 (1 ϵ) 2 (1 ϵ) 2 x n u n = 0 u n = 1 n n { } 2 u n {u n N n} n N u n 1 < (1 ϵ)2 ϵ 2 { } x n 4 n N { } x n {b n N n} n N {z n } n N x n b n 2 (1 ϵ) 2 u n 1 2 + 2 (1 ϵ) 2 x n u n 2 < ϵ 2 {e n } n N {x n } n N δ = x n > 0 e n N n k(x n ) = 0 k N ϵ > 0 {x n} n N {u n } n N {e n } n N x n u n < ϵ ϵ = δ 3K K {e n} n N {x n} n N {u n } n N ϵ = δ 3K {x n } n N

{x n } n N l 1 {x n } n N l 1 l 1 X {y n )} n N X l 1 m, M > 0 m y n M n N k N x k (y n) x k M n N {x k (y n)} n N k N {y n} n N {y n } n N {x k (y n)} n N k N z n = y n+1 y n k N n x k(z n ) = 0 c, C > 0 m N α 1,..., α m R c α i α i z i C α i n N { z n } > 0 {z n } n N {z n} n N {u n } n N {x n } n N {u n } n N l 1 X {e n } n N X = [e n : n N] X {x n } n N X {x n } n N {λ n } n N n N λ i x i < λ n x n {e n } n N {e n } n N {e n } n N

w i) ii) {u n } n N u n = 1 n N u n 0 x X x = λ n e n ϵ > 0 k 0 k > k 0 n=k+1 λ n e n < ϵ u k = k+1 i=n k +1 n k > k 0 ( ) x (u k ) = n=k+1 λ i e i (u k ) = λ n e n(x) < ϵ x 1 ( ) α i e i m N n m > k 0 k > m i=n k +1 λ i e i (u k ) < ϵ u k w 0 ii) i) {e n } n N x / [e n : n N] x = 1 Pn(x ) = x (e i )e i {P n(x )} n N x ϵ > 0 {m k } k N x P m k (x ) 2ϵ k N ( ) n 1 = m 1 ( ) x 1 = x ( i=n 1 +1 n 2 = m k2 x i=n 1 +1 u 1 = λ (1 i )e i n 2 i=n 1 +1 λ (1) n e n X x 1 = 1 λ (1) i e i ) 2ϵ k 2 N m k2 > n 1 n 2 i=n 1 +1 λ (1) i e i < ϵ x ( λ (1) i e i x ( i=n 1 +1 i=n 1 +1 λ (1) i e i ) < x ( λ (1) i e i ) 2ϵ x (u 1 ) > ϵ n 2 i=n 1 +1 λ (1) i e i ) + ϵ

( ) x 2 = λ (2) n e n X x 2 = 1 x ( i=n 2 +1 k 3 N m k3 > n 2 n 3 = m k3 x i=n 2 +1 u 2 = λ (2) i e i n 3 i=n 2 +1 n 3 i=n 2 +1 λ (2) i e i < ϵ x ( λ (2) i e i x ( i=n 2 +1 i=n 1 +1 λ (1) i e i ) < x ( λ (2) i e i ) 2ϵ x (u 2 ) > ϵ n 3 i=n 2 +1 λ (1) i e i ) 2ϵ λ (2) i e i ) + ϵ {u k } k N x (u k ) > ϵ k N {u k } k N {u k } k N u k = P nk+1 (x k ) P nk (x k ) 2K x k = 2K z n = u n u n {e n } n N x X 1 K n N x (e i )e i x n N x = n n N { x (e i )e i = x (e i )e i x (e i )e i } x (e i )y (e i ) : y B X y B X y = x (e i )y (e i ) = x ( y (e i )e i ) x y (e n )e n y (e i )e i ) K x y K x

x (e i )e i K x n N 1 K n N x (e i )e i x x B X x = x (x ) = n N x (e n )x (e n) = n x (e i )e i x n N x (e n )e n x (e i )x (e i ) n N x (e i )e i x ( x (e i )e i ) c 0 {e nk } k N c 0 {e nk } k N k e ni = 1 < e ni k N {x n } n N c 0 T : c 0 X z n = T (e n ) {e n } n N c 0 M > 0 z n M n N w w e n 0 z n 0 {u n } n N X {z nk } k N {z n } n N {u n } n N

{x nk } k N T {e nk } k N {e n } n N Y = [e n, n N] J : X Y J(x)(y) = y(x) y Y J J x X J(x)(y) = y(x) x y y Y J(x) x J x = α n e n x n = α n e n x = x n n x n K J(x n ) n N J n N x X x = 1 x (x n ) = x n x P n < e 1,..., e n > Y x n = P n (x n ) x n = (x P n )(x n ) = J(x n )(x P n ) x P n J(x n ) K J(x n ) J(x) = x x X { J } y Y y (e i )e i X K 2 y n N y (e i )e i K J( y (e i )e i ) = K y (e i )J(e i ) n N

y (e i )J(e i ) K y y = y 1 y(e n )e n Y y (e i )J(e i )(y) = y ( y(e i )e i ) y y(e i )e i y K y K y y (e i )e i x = y (e n)e n y = J(x) {e n } n N {e n } n N i) ii) X = [e n, n N] J X X ii) i) x X P n(x ) = x (e i )e i w x n X w w X P n(x ) w x X = < e n, n N > w = < e n, n N >. { } {α n } n N R α i e i = M α n e n x n = n N α i e i X M B X w

X x M B X {x nk } k N {x n } n N x = w k x nk i N e i w e i (x) = e i (w k x nk ) = k e i (x nk ) = α i i N x = e n(x)e n = α n e n X ˆX T : X Y Y = T [X ] ˆX Y = X X = T [X ] ˆX T : X Y T (x )(y) = y(x ) y Y T (x ) = { T (x )(y) : y B Y } = { y(x ) : y B Y } { ˆx(x ) : x B X } = { x (x) : x B X } = x T (x ) = { y(x ) : y B Y } { x (x ) : x B X } = x T T [X ] Y X T [X ] Y Q : Y X Q(y ) = y : X X X X Q P : Y Y P = T Q P Q T = I Y I Y Y (Q T )(x )(x) = T (x )(ˆx) = ˆx(x ) = x (x) x X P 2 = T Q T Q = T Q = P P [Y ] = T [X ] Q Y x X y = ˆx Y Q(y ) = x T [X ] Y Y = T [X ] KerP

KerP = ˆX y ˆX P (y )(y) = T (Q(y ) = T (y )(y) = y(y ) = 0 y Y ˆX KerP y KerP T (y ) = 0 T 1 1 y = 0 y ˆX KerP ˆX Y = T [X ] ˆX Y = X (X/Y ) Y T : (X/Y ) Y T (ˆx )(x) = ˆx (x + Y ) x Y T (ˆx )(x) = ˆx (Y ) = 0 x X T (ˆx )(x) = ˆx (x + Y ) ˆx x + Y ˆx x T T x Y ˆx : X/Y R ˆx (x + Y ) = x (x) x X ˆx (x + Y ) = x (x) = x (x y) y Y x x y y Y ˆx (x + Y ) x x + Y ˆx (X/Y ) T (ˆx ) = x ˆx (X/Y ) T (ˆx ) = { T (ˆx )(x) : x B X } { T (ˆx )(x) : x + Y B X/Y } = { ˆx (x + Y ) : x + Y B X/Y } = ˆx x X x + Y 1 ˆx (x + Y ) = T (ˆx )(x) = T (ˆx )(x y) y Y T (ˆx ) x y y Y ˆx (x + Y ) T (ˆx ) x + Y T (ˆx ) T (ˆx ) = ˆx T

Y = [e n, n N] X = ˆX Y dim(x / ˆX) = dimy = dim(x /Y ) Y = T [Y ] Ŷ Y = J[X] Y = J [X ] J [ ˆX] = T [Y ] J [Y ] = Ŷ. X = ˆX Y (X / ˆX) = Y (X /Y ) dim(x / ˆX) = dimy = dim(x /Y )

J n, m N n m [n, m] = {k N : n k m} n N [n, ) = {k N : n k} { { m J = x = {x n } n N R : } } x n 2 < n I i {I i } m J { m x J x = } 1/2 x n 2 n I i {I i } m (J, ) x, y J {I i } m

( (x n + y n ) 2 ) 1/2 = [( x n + y n ) 2 +... + ( x n + y n ) 2 ] 1/2 n I i n I 1 n I 1 n I m n I m ( x n 2 ) 1/2 + ( y n 2 ) 1/2 x + y x + y x + y n I i n I i { x (n)} J ϵ > 0 N n N m > n > N x (m) x (n) < ϵ i N I i { = {i}} x (m) i x (n) i < ϵ m > n > N i N x i = x = {x i } i N x (n) i n N x (n) i n x J x = x (n) ϵ > 0 M N n m > n M x n x m < ϵ 2 {I i } k ( k ( j I i (x (m) j k x (n) j ) 2 ) 1/2 < ϵ 2 j I i (x (n) j x j ) 2 ) 1/2 ϵ 2 m > n M m < ϵ n M ( ) n = M ( ) x (M) x J x J x = x (n) n n N e n = X {n} x J k N s k = k x i e i s k 2 + x s k 2 x 2

{I i } m l {1,..., m 1} I i [1, n] i = 1,..., l I i [n + 1, + ) i = l + 1,..., m l s n 2 + (x s n ) 2 = x n 2 x 2 n I i n I i n I i i=l+1 s k 2 + x s k 2 x 2 {e n } n N = (X {n} ) n N J e n 0 n N e n = 1 n N J = [e n : n N] x = {x n } n N J s n = x i e i x = s n. ϵ > 0 n {I i } m x n 2 > x 2 ϵ 2 ( ) n I i n 0 = { n : n } m I i ( ) x 2 s n 2 < ϵ 2 n n 0 n n 0 x s n 2 = x s n 2 + s n 2 s n 2 x 2 s n 2 < ϵ 2 x = s n. k, n N n k > n α 1,..., α k R {I i } m I i [1, n] i = 1,..., m ( α j 2 )) 1/2 j I i k α i e i α i e i k α i e i

J c 0 J {α n } n N n N {e n } n N { α i e i < α i e i = α i e i n n N } α i e i 2 ϵ > 0 n N n 0 N m > n > n 0 α i e i 2 m α i e i α i e i 2 i=n+1 α i e i 2 < ϵ 2 α i e i 2 < ϵ J α n e n J < c 00 (N) > c 00 (N) J < c 00 (N) > J I I = e n I J n I I x J ϵ > 0 n 0 N m > n > n 0 e i (x) < ϵ i=n+1

n I e n(x) x J I : J R I (x) = n I e n(x) x J I I J I w = e n I = 1 n I I s w = e n s / [e n : n N] s. = e n 0 < ϵ < 1 n 0 N n 0 s (x) e x=e n0 +1 n(x) < ϵ x B J 1 < ϵ, I / [e n : n N] I J x = α n e n J x n = α i e i { m } 1/2 x n = Ii (x n ) 2 n N {I i } m suppx n n N n ( ) ( ) { m } 1/2 x = Ii (x) 2 {I i } m J J J = [e n, n N] [s ] J

{d n } n N J d 1 = e 1 d n = e n e n 1, n > 1 {d n } n N J < d n, n N >=< e n, n N > [d n, n N] = J. n N α 1,..., α n, α n+1 R {I i } m m [1, n] n / I i n+1 n+1 ( Ii ( α i d i ) 2 ) 1/2 = ( Ii ( α i d i ) 2 ) 1/2 α i d i n+1 α i d i α i d i I m = [i, n] i n ( Ii ( m 1 n+1 = ( Ii ( m 1 α i d i ) 2 ) 1/2 = ( n+1 α i d i ) 2 + αi 2 ) 1/2 α i d i Ii ( α i d i ) 2 + Im( α i d i ) 2 ) 1/2 n+1 α i d i α i d i n+1 α i d i α i d i {d n } n N {d n } n N {u n } n N {d n } n N M = u n u n n n N

u i m > k N i 2 5M 2 1 i2 i=k i=k J k+1 u k = λ i d i i=k i=n k +1 u i i = λ n k +1 k e nk + λ n k +1 λ nk +2 k e nk +1 +... + λ n k+1 1 λ nk+1 e nk+1 1 + k + ( λ n k+1 λ n k+1 +1 k k + 1 )e n k+1 + λ n k+1+1 λ nk+1+2 e nk+1 +1 + k + 1 +... + + ( λ n m 1 m 1 λ n m +1 m )e n m +... + λ n m+1 1 λ nm+1 e nm+1 1 + λ n m+1 m m e n m+1 {I j } l j=1 {1,..., l} F k = {j {1,..., l} : I j [n k, n k+1 1]} F s = {j {1,..., l} : I j [n s + 1, n s+1 1]}, s = k + 1,..., m 1 F m = {j {1,..., l} : I j [n m + 1, n m+1 ]} m F = i=k F i U = {j {1,..., l} : s 1 < s 2 {k + 1,..., m} : I j supp(u s1 ), I j supp(u s2 ) } F, U {1,...l} s = k, k + 1,..., m F s I u i j ( i ) 2 = 1 I s 2 j (u s ) 2 M 2 I u i s 2 j ( i ) 2 M 2 1 i 2 j F s j F s j F i=k j U i=k s j,1 = {i {1,..., m} : I j supp(u i ) } s j,2 = {i {1,..., m} : I j supp(u i ) } i=k I j ( i=k u i i ) 2 = Ij ( u s j,1 + u s j,2 ) 2 2M 2 + 2M 2 s j,1 s j,2 s 2 j,1 s 2 j,2 4M 2 s 2 j,1

U m k + 1 i=k Ij ( j U u i i 2 5M 2 i=k i=k u i i ) 2 4M 2 1 i 2 i=k 1 i 2 u n n x = (d n ) n N x X, ϵ > 0 (u n ) n N (d n ) n N x (u n ) > ϵ n N x x (u n ) 1 (x) = > ϵ = + n n u n n J = [e n, n N] < s > J l 1 J < d n, n N >=< e n, n N > < s > J = [d n, n N] = < e n, n N > < s > [e n, n N] < s > = [e n, n N] < s > dim < s >= 1 < + J = [e n, n N] < s > l 1 c 0 J J

J J J J J e J \ J ˆ w ê n e [e n, n N] = [e ] {e n } n N w x J x = λ n e n + λs x (e n ) = λ n + λ n λ n λ n 0 e J ê n e e / J ˆ e w e (s ) = e (e n) = 0 e (s ) = s (e n ) = 1 n [e n, n N] = [e ] x = e ( λ n e n) = w λ n e n [e n, n N] λ n e (e n) = 0 < e > [e n : n N] x [e n : n N] x J y [e n : n N] λ R : x = y + λs x (x ) = λx (s ) = x (s )e (x ) x = x (s )e [e nn : N] < e > [e n, n N] =< e >

J = J ˆ [e ] J dim(j / J ˆ ) = 1 {e n } n N J 1 2 J (J, 1 ) (J, 2 ) J J = { {x n } c 0 : { (x p1 x p2 ) 2 + (x p2 x p3 ) 2 +... + (x pm 1 x pm ) 2} < } m N p 1 <... < p m x J x 1 = { (x p1 x p2 ) 2 +... + (x pm 1 x pm ) 2 : m N, p 1 <... < p m N } 1/2 J (J, 1 ) {e n } n N e n = X {n} (J, 1 ) x 2 = { (x p1 x p2 ) 2 +... + (x pm 1 x pm ) 2 + (x pm x p1 ) 2} 1/2 m N p 1 <... < p m 1 x 2 x 1 x 2 2 (J, 2 ) (J, ) (J, 1 )

n N α 1,..., α n R α i d i = α i e i 1 m N 1 p 1 p 1 < p 2 p 2 <... < p m p m n I j = [p j, p j] j = 1,..., m α n+1 = 0 Ij ( α i d i ) 2 = (α p1 α p 1 +1) 2 + (α p2 α p 2 +1) 2 +... + (α pm α p m +1) 2 j=1 n+1 α i e i ) 2 1 = α i e i ) 2 1 α i d i ) α i e i ) 1 m N p 1 <... < p m n + 1 I j = [p j, p j+1 1], j = 1,..., m (α p1 α p2 ) 2 +... + (α pm 1 α pm ) 2 = j=1 α i e i 1 Ij ( α i d i ) 2 α i d i α i d i 2 {e n } n N (J, 1 ) (J, 2 ) α i d i = α i e i 1 α 1,..., α n R, n N 1 2 u n {u n } n N {e n } n N n

x J {x (e n)} n N x J 1 {e n } n N x 1 = x (e i )e i 1 n N x n = x (e i )e i N N n x 2 1 x N 2 1 < ϵ 2 ( ) k N p 1 <... n N + 2 x (e m ) x (e n ) < ϵ (x (e n)) n N m N x m 1 > x 1 x 1 = n N x (e i )e i 1 (J, 2 ) π : J J π(x ) = ( λ, x (e 1) λ, x (e 2) λ,...) λ = n x (e n) π α 1 = λ α n = x (e n 1) λ π(x ) = α n e n π(x ) 2 = α n e n 2 = n N α i e i 2 = n N x (e i )e i 2 = x 2 {e n } n N (J, 2 ) x = {x n } n N J (x i+1 x 1 )e i 2 < y n = (x i+1 x 1 )e i {y n } n N n N

J x J {y n} n N {y n } n N ŷ n w x x (e n) = x n+1 x 1 n N π(x ) = x J (J, 2 ) J (J, 2 ) J l 2 J l 2 l 2 J {u n } n N {e n } n N m N α 1,..., α m R ( αi 2 ) 1/2 α i u i m N α 1,..., α m R αi 2 = 1 1 α i u i u i = 1 i = 1,..., m {u n } n N {I j } l j=1 1 < l 1 <... < l m 1 < l l 0 = 1 l m = l {I j } l i j=l i 1 supp(u i ) i = 1,..., m l i j=l i 1 I j (u i ) 2 = 1 i = 1,..., m l Ij ( α i u i ) 2 = j=1 αi 2 = 1 1 α i u i

{u n } n N {e n } n N n s (u n ) = 0 ϵ > 0 {u n} n N m N α 1,...α m R (1 ϵ m 2 )( αi 2 ) 1/2 u k = k+1 i=n k +1 α i u i ( 5 + ϵ m 2 )( αi 2 ) 1/2 λ i e i y k = u k s (u k )e nk+1 y k 1 + s (u k ) k N u k y k = 0 k {y n} n N (y n ) n N {u n} n N {u n } n N k=1 u k y k 2 < ϵ2 16 y k < 1 + ϵ2 128 k N m N α 1,..., α m R αi 2 = 1 1 ϵ m 2 < 1 α i u i α i y i 5 + ϵ 4 {I j } l j=1 i = 1,..., m m F i = {j {1,..., l} : I j supp(y i)} F = F i i:f i Ij ( α i y i) 2 = Ij (α i y i) 2 j F j F i i:f i α 2 i (1 + ϵ2 128 ) < 1 + ϵ2 32 U = { j {1,..., l} : i 1 < i 2 {1,..., m} : I j supp(y i 1 ), I j supp(y i 2 ) } j U s j,1 = {i {1,..., m} : I j supp(y i) }

s j,2 = {i {1,..., m} : I j supp(y i) } Ij ( α i y i) 2 = I j (α sj,1 y s j,1 ) + I j (α sj,2 y s j,2 ) 2 j U j U j U (2α 2 s j,1 I j (y s j,1 ) 2 + 2α 2 s j,2 I j (y s j,2 ) 2 ) 4 + ϵ2 32 F, U {1,..., l} α i y i < 5 + ϵ2 16 < 5 + ϵ 4 α i u i α i u i y i + α i y i ( u i y i 2 ) 1/2 + 5 + ϵ 4 5 + ϵ 2 J l 2 J {x n } n N ϵ > 0 {x n} n N {x n } n N m N α 1,..., α m R (1 ϵ)( αi 2 ) 1/2 α i x i ( 5 + ϵ)( αi 2 ) 1/2 J l 1 {x n } n N J {u n } n N {x n} n N {x n } n N x n u n 2 < ϵ2 4

m N α 1,..., α m R αi 2 = 1 α i x i α i x i u i + α i u i 5 + ϵ α i u i α i x i u i α i x i 1 ϵ α i x i

J T J T J 2 <N = {σ = (σ 1,..., σ n ) : σ i {0, 1} i = 1,..., n, n N} { } 2 <N : 2 <N N {0} { 0 s = s = n s = (s 1,..., s n ) 2 <N s s 2 <N s, u 2 <N s, u, s t s t s i = t i i = 1,..., s 2 N = {(σ n ) n N R : σ n {0, 1} n N} σ 2 N n N σ n = (σ 1,..., σ n ) 2 <N σ 1,..., σ n 2 N N N σ i N σ j N i, j {1,..., n}

i j σ 1,..., σ n I 2 <N s, t I s t t s s, t I w 2 <N s w t w I I I n I = {s 1, s 2,..., s n } s 1 s 2,..., s n s 1 I s n I I in(i) end(i) in(s) end(s) S s, t I s t S I in(s) = s end(s) = t S, I in(s) in(i) end(s) = end(i) I S I σ(i) 2 N n N I = {σ(i) n+k : k = 0, 1,...} σ(i) n in(i) I 1,..., I n σ(i 1 ),..., σ(i n ) I 1,..., I n N σ(i i ) N σ(i j ) N i, j {1,..., n} i j N in(i i ) i = 1,..., n t 1 = t 2 = (0) n > 2 t n = (σ 1,..., σ m ) i = 1,...m σ i = 1 t n+1 = (σ 1,..., σ m, σ m+1) σ i = 0 i = 1,..., m + 1 i 0 = max {i {1,..., m} : σ i = 0} t n+1 = (σ 1,..., σ m) σ i = σ i i = 1,..., i 0 1 σ i 0 = 1 σ i = 0 i = i 0 + 1,..., m 2 <N = {t n : n N} 2 <N n N t n = [ 2 n] [ ] s 2 <N e s = X {s} e n = X {tn } {e s } s 2 <N {e n } n N { { m J T = x : 2 <N R : } } x(t) 2 < t I i

{I i } m J T { m x J T x = } 1/2 x(t) 2 t I i {I i } m (J T, ) x, y J T {I i } m ( (x(t) + y(t)) 2 ) 1/2 = [( x(t) + y(t)) 2 +... + ( x(t) + y(t)) 2 ] 1/2 t I i t I 1 t I 1 t I m n I m ( x(t) 2 ) 1/2 + ( y(t) 2 ) 1/2 x + y x + y x + y t I i t I i {x n } n N J ϵ > 0 N m > n N x m x n < ϵ t 2 <N I t = {t} x m (t) x n (t) < ϵ m > n > N {x n (t)} n N t 2 <N x : 2 <N R x(t) = x n (t) x J T x = x n n n ϵ > 0 M N m > n M x n x m < ϵ 2 {I i } k k ( x m (t) x n (t)) 2 ) 1/2 < ϵ 2 t I i k ( x n (t) x 2 ) 1/2 ϵ 2 t I i m > n M m < ϵ n M ( )

n = M ( ) x M x J T x J T x = x n n x J T k N s k = k x i e i s k 2 + x s k 2 x 2 {e n } n N J T e n 0 n N e n = 1 n N J T = [e n : n N] x J T s n = x(t i )e i x = s n. ϵ > 0 n {I i } m x(t) 2 > x 2 ϵ 2 ( ) t I i n 0 = { t : t } m I i ( ) x 2 s n 2 < ϵ 2 n 2 n 0+1 n 2 n0+1 x s n 2 = x s n 2 + s n 2 s n 2 = x 2 s n 2 < ϵ 2 x = s n. n N n α 1,..., α n, α n+1 R {I i } m m t n+1 / I i ( n+1 α j 2 )) 1/2 α i e i t j I i n+1 α i e i α i e i

{e n } n N c 0 J T J T < c 00 (2 <N ) > I I = e s J T I I w = e s J T s I s I I = 1 I σ 2 N σ w = e σ n I / [e n : n N] I {e n } n N J T x J T { m } 1/2 x = Ii (x) 2 {I i } m σ 2 N [e σ n, n N] J T :< e σ n, n N > J T ( λ i e σ i ) = (λ 1,..., λ n, 0,...) T N I 1 = [n 1, n 1],...I m =

[n m, n m] 2 <N S 1 = [σ n1, σ n 1 ],..., S m = [σ nm, σ n m ] Ij ( λ i e i ) 2 = j=1 Sj ( λ i e σ i ) 2 T (x) = x j=1 T [e σ n, n N] x = λ n e n J ϵ > 0 N N m > n > N i=n+1 λ n e σ n T ( λ n e σ n ) = λ i e σ i = λ i e i < ϵ i=n+1 λ n e n J T { σ : σ 2 N} 1 σ1 σ2 s 2 <N σ1(e s ) = 1 σ2(e s ) = 0 σ1 σ2 1 J T σ 2 N σ J T \ ˆ J T ê σ n w σ T : [e σ n : n N] J T : J [e σ n : n N] J = [e n, n N] [s ] x [e σ n : n N] (λ n ) n N λ R x = T ( λ n e n + λs ) = λ n (e n T ) + λ(s T )

(e σ k ) k N w x J T x [eσ n :n N] [e σ n : n N] k N x (e σ k ) = λ n e n(t (e σ k ))+λs (T (e σ k )) = λ n e n(e k )+λs (e k ) = λ k +λ k λ λ k k 0 σ J T σ = w k ê σ k σ / J ˆ T. σ (σ ) = σ (e σ n ) = 1 n σ ˆ J T σ w σ (σ ) = σ (e σ n ) = 0 l 1 J T l 1 J T J J J T {I n } n N 2 <N x J T In(x) 2 x 2 {α n } n N B l2 α n In(x) x α n In(x) x J T w α n In B J T K = { w } α n In : {α n } n N B l2 {I n } n N K B J T J T x = {k (x) : k K} x = { k (x) : k K} x J T

x J T K B J T {k (x) : k K} x n N x n = e i (x)e i {I i } m x n 1 n λ i = ( Ii (x n ) k = Ii (x n ) 2 ) 1/2 m < ( Ii (x n ) 2 ) 1/2 λ i Ii K k (x n ) = ( Ii (x n ) 2 ) 1/2 x n 1 < n k (x n ) {k (x n ) : k K} n x {k (x) : k K} w J T J T (B X, w ) K kn = w α i,n Ii,n K n N α i+1,n α i,n i N {I n } n N {I kn } n N I Ik w n I {I kn } n N { Ik n (e s ) } n N s 2 <N I = { } s 2 <N : n s N s I kn n n s I s, t I n 0 N s, t I kn0 s t t s s, t I w 2 <N s w t I n 0 N s, t I n n n 0 w I n n n 0 w I n I k n (s) = I (s) s 2 <N In = 1 n N

M [N] {α i } i N B l2 n M α i,n α i i N {I i } i N I i = w I n M i,n {I i } i N k = w α i Ii K k = w k n M n s 2 <N ϵ > 0 N N ( αi 2 ) 1/2 < ϵ 4 n 0 N n n 0 N i=n+1 α i,n Ii,n(e s ) α i Ii (e s ) < ϵ 4 α N+1,n α N+1 < ϵ n M 4 n n 0 + i=n+1 α i,n Ii,n(e s ) α i I i (e s ) < ϵ 4 + ( α i Ii (e s ) i=n+1 α 2 i ) 1/2 ( N α i,n Ii,n(e s ) α i Ii (e s ) + i=n+1 i=n+1 α i,n I i,n(e s ) + I i (e s ) 2 ) 1/2 + α j,n j N + 1 < ϵ 4 + ϵ 4 + α N+1,n ϵ 2 + α N+1,n α N+1 + α N+1 ϵ 2 + ϵ 4 + ϵ 4 = ϵ kn(e s ) n M k (e s ) s 2 <N (X, A, µ) f L 1 ( µ ) f µ f dµ = f dµ + f dµ X X X f n : (X, A) R f = f n g L 1 ( µ ) n f n g n N X f n dµ = f dµ n X X n X f n f dµ = 0 X µ X µ M r (X)

f C (X) µ M r (X) f (f) = f dµ f C(X) X C(K) w {x n } n N J T {I (x n )} n N I {x n } n N w M = { x n } T : J T C(K) T (x) = ˆx K n N T T (x) = ˆx K = { ˆx(x ) : x K} = { x (x) : x K} = x K J T T : C (K) J T x J T f C (K) x = f T x J T µ x M r (K) n N x (x n ) = ˆx n K, dµ x ˆx n K M n N K µ x (K) < x J T {x (x n )} n N x K x = w λ i Ii K α i = Ii (x n ) n ( αi 2 ) 1/2 M λ i α i M ( i=n+1 λ i α i ϵ > 0 N N λ 2 i ) 1/2 < ϵ 3M i=n+1 λ i α i < ϵ 3 n 0 N

N n n 0 λ i Ii (x n ) λ i α i < ϵ 3 n n 0 x (x n ) λ i α i N λ i Ii (x n ) λ i α i + λ i Ii (x n ) + λ i α i < ϵ 3 + ( λ 2 i ) 1/2 ( i=n+1 i=n+1 i=n+1 i=n+1 I i (x n ) 2 ) 1/2 + ϵ 3 < ϵ 3 + ϵ 3 + ϵ 3 = ϵ X f n : X R { f n (x) } < x X ϵ > 0 M [N] L n N [M] n L f n (x) n L f n(x) < ϵ x X {f n } n N {L k } k N N k N n L k {n k } k N n k L k f n (x) f n (x) < 1 n L k k x X k N L = {n k : k N} L k N L L k x X f n (x) n L n L k f n (x) = n L n L n L f n (x) (f n ) n L f n (x) f n (x) < 1 n L k k f n (x) k N k l 1 J T

l 1 J T J T l 1 l 1 J T {u n } n N M = n N { u n } {u nk } k N {I (u nk } k N I K [N] {S (u n )} n K S ϵ > 0 M [K] L [M] σ (u n ) n L n L σ (u n ) ϵ σ 2 N ϵ > 0 M [K] L [M] σ L 2 N n L σ L(u n ) n L σ L(u n ) > ϵ σ L2 (u n )+ n L n L σ L2 (u n ) > ϵ2 ( ) k N k ϵ2 2 4 > M 2 L 0 [M] σ 1 2 N ( ) σ 2 1 (u n ) σ 2 1 (u n ) ϵ2 n L 0 n L 0 4 L 1 [L 0 ] σ 2 1 (u n ) > ϵ2 n L1 4 σ 2 2 N ( ) σ 1 σ 2 L k L k 1,..., L 1 N σ 1,..., σ k 2 N σ 2 i (u n ) > ϵ2 n Li 4 i = 1,..., k σ 2 i (u n ) > ϵ2 n Lk 4 i = 1,..., k N L k i = 1,..., k σi 2 (u n ) > ϵ2 n L k : n N σ 1,..., σ k 4 {u n } n N k n 0 L k n 0 N u n0 2 σ 2 i (u n0 ) > k ϵ2 4 > M 2 J T w

l 1 J T l 1 J T l 1 c 0 J T l 1 w K X conv w (K) = conv. (K) J T J T = [I : I 2 <N ] = [I : I 2 <N ] K J T B J T = conv. (K) = conv w (K) K B J T conv w (K) B J T B J T w x B J T \ conv w (K) w x J T { k (x) : k conv w (K) } < x (x) {k (x) : k K} < x (x) x K J T

{ { } } l 2 (2 N ) = f : 2 N R : f(σ) 2 : F 2 N < σ F { 1/2 < f, g >= f(σ)g(σ) : F 2 } N σ F l 2 (2 N ) S : l 2 (2 N ) l 2(2 N ) S(f)(g) =< f, g > g l 2 (2 N ) f l 2 (2 N ) {λ n } n N l 2 {σ n } n N 2 N f = λ n X {σn } suppf = { σ 2 N : f(σ) > 1 } suppf n { σ 2 N : f(σ) > n} 1 n N n N k N σ 1,..., σ k 2 N f(σ i ) > 1 i = 1,..., k n k k n < f(σ 2 i ) 2 f 2 k suppf = {σ n : n N} k f(σ i )X {σi } f = n N { i=k+1 f(σ i ) 2 } = i=k+1 f(σ i ) 2 k 0 {f(σ n )} n N l 2 f = f(σ n )X {σn }

(X, d) (Y m ) m N P(X) Y = dist(x, Y ) = m dist(x, Y m) x X m=1 Y m x X (dist(x, Y m )) m N Y m Y m N dist(x, Y ) dist(x, Y m) m y Y {y k } k N Y m y {m k } k N y k Y mk k N m 1 N y 1 Y m1 m 2 N y 2 Y m 2 m 2 = {m 1, m 2} + 1 > m 1 y 2 Y m2 y k Y mk m k+1 N y k+1 Y m k+1 m k+1 = { m k, m k+1} + 1 {mk } k N dist(x, Y mk ) d(x, y k ) k N dist(x, Y m) = m dist(x, Y m k ) d(x, y k ) = d(x, y) dist(x, Y k k m) dist(x, Y ) m dist(x, Y ) = dist(x, Y m). m y k k m=1 Y = [e s : s N ] Q : J T J T /Y Q(x ) = x + Y J T /Y = Q[J T ] = Q( Q[] = [I + Y : I ] J T /Y = [I + Y : I ] J T /Y l 2 (2 N ) I, S

I + Y = S + Y σ(i) = σ(s) I S Y U : l 2 (2 N ) U( λ i Ii + Y ) = λ i X {σ(ii )} U I 1,..., I n λ 1,..., λ n R λ 2 i = 1 λ i Ii +Y = 1 λ i Ii (x) ( λ 2 i ) 1/2 ( λ i Ii 1 I i λ i Ii + Y 1 2 (x)) 1/2 x x J T N I 1,..., I n Y m =< e s : s m > Y = Y m m N x m = λ i e σ(ii ) m+1 x m = 1 y Y m λ i Ii (x m ) y (x m ) = 1 dist( λ i Ii (x m ) = 1 1 λ i Ii, Y m ) m N = dist( λ i Ii, Y m ) 1 m m=1 m N m N λ i Ii y y Y m λ i Ii + Y = dist( λ i Ii, Y ) = U J T /Y U k f = λ n X {σn} = U( λ n σn +Y ) k f U[J T /Y ] J T /Y U[J T /Y ] U[J T /Y ] l 2 (2 N ) f U[J T /Y ] U J

J T J T / ˆ J T J T dim(j T / J ˆ T ) = dim((j T /Y ) ) = dim(l 2(2 N )) = dim(l 2 (2 N )) = + J T = J ˆ T [σ : σ 2 N ] x J T x = ˆx + λ n σn x J T {λ n } n N l 2 {σ n } n N 2 N JˆT [σ : σ 2 N ] = {0} J T y Y y = λ n σ n {λ n } n N l 2 {σ n } n N 2 N T U S l 2 (2 N ) L = T U S : l 2 (2 N ) Y L L L(X {σ} ) = σ σ 2 N σ 2 N I 2 <N L(X {σ} )(I ) = T U S(X {σ} )(I ) = S(X {σ} )(U(I + Y )) = U(I + Y )(σ) = = X {σ(i)} (σ) = σ (I ) L(X {σ} ) = σ J T = [I : I ]. X X ˆX w X ˆX w X X w X l 1 J T

J ˆ T w J T x {x n } n N J T x = w ˆx n n J T x = ˆx + j=1 σ 1,..., σ n x n = x + λ j σ j x 1 = x n > 1 k n λ j e σj kn {x n } n N {λ n } n N l 2 I (x n ) n λ j σj (I ) I j=1 I i N σ(i) = σ i n n 0 I (x n ) = λ i = λ n σn (I ) I (x n ) = 0 = λ j σj (I ) n N I I (x n ) n j=1 j=1 j=1 λ j σ j (I ) J T l 2 J T l 2 l 2 M M (2) = {(n, m) : n, m M, n < m} [M] M

A 1,..., A k N N (2) = M (2) A i k A i M [N] i {1,..., k} L [N] n N {n} (2) L = {n, m) : m L, n < m} M 1 = N m 1 M 1 M 2 [M 1 ] i 1 {1,..., k} {m 1 } (2) M 2 A i1 m 2 M 2 m 2 > m 1 M 3 [M 2 ] i 2 {1,..., k} {m 2 } (2) M 3 A i2 {m n } n N N {M n } n N [N] {i n } n N {1,..., k} m p M n p n {m n } (2) M n+1 A in n N k { N = n N : {m n } (2) M n+1 A i } i {1,..., k} { } L = n N : {m n } (2) M n+1 A i M = {m n : n L} [N] m n, m p M m n < m p n 0 {x n } n M S in(s) = S (x n ) ϵ n M n(s) M S ϵ > 0 K = { x n } n N n N α n = {supp(x n )} Q n = { t 2 <N : t = α n St (x n ) > ϵ S t in(s) = t }

n N ϵ 2 Q n < St (x n ) 2 x n 2 K 2 Q n K2 ϵ 2 t Q n n N α N {0} L [N] Q n = α n L α = 0 (x n ) n L Q n = n L α 1 n L Q n = {t i,n : 1 i α} i, j {1,..., α} A i,j = {n, m L : n < m S t i,n t j,m S (x n ) > ϵ} A = N (2) \ 1 i,j α A i,j N (2) = A 1 i,j α A i,j M [N] M (2) A M (2) A i,j i, j {1,..., α} i, j {1,..., α} M (2) A i,j n, k M n+1 < k (n, k), (n+1, k) M (2) t i,n t i,n+1 S 1 t i,n S 2 t i,n+1 t j,k S1(x n ) > ϵ S2(x n+1 ) > ϵ t i,n+1 S 1 S n,n+1 t i,n t i,n+1 Sn,n+1(x n ) = S1(x n ) > ϵ I = S n,n+1 I I (x n ) = Sn,n+1(x n ) > ϵ n M I (x n ) = 0 M (2) A n (x n ) n M S in(s) = n, m M n < m S (x n ) > ϵ S (x m ) > ϵ t 1 S t 1 = α n t 2 S t 2 = α m Q n Q m i, j {1,..., α} t 1 = t i,n t 2 = t j,m (n, m) A i,j A A i,j =. 1 i,j α (x n ) n N J T ϵ > 0 M [N] S in(s) = S (x n ) ϵ n M n(s) ϵ {y n } n N J T n I (x n ) = 0 I ϵ > 0 {y n} n N n N

λ 1,..., λ n R ( λ 2 i ) 1/2 λ i y i 2(1 + ϵ)( λ 2 i ) 1/2 ϵ > 0 {ϵ k } k N R {M k } k N [N] {y n } n Mk ϵ k k N k N S in(s) = S(y n ) ϵ k n M k n(s) M k n N α n = {supp(x n )} m n = 2 α n {p n } n N {k n } n N p n M kn n N m n ( l=n+1 ϵ 2 k l ) < ϵ 2 k 1 = 1 p 1 M k1 ϵ n n 0 L 1 [N] r L 1 ϵ 2 r < ϵ2 2m 1 n L k 2 L 1 k 2 L 1 p 2 M k2 p 2 > p 1 ϵ 1 n 0 L 2 [L 1 ] ϵ 2 r < ϵ2 2 2 m 2 r L 2 k 3 L 2 k 3 > k 2 p 3 M k3 p 3 > p 2 {k n } n N {p n } n N p n M kn n N {L n } n N [N] k n L n ϵ 2 r < ϵ2 n N k 2 n l L n l n + 1 m n r L n l=n+1 ϵ 2 k l r L n ϵ 2 r < ϵ2 2 n m n n N m n ( l=n+1 ϵ 2 k l ) < ϵ 2 {k n } n N {p n } n N n N y n = y pn, b n = α pn, δ n = ϵ kn {y n} n k δ k k N k N

p n M k n k {y n} n k {y n } n Mk δ k m n ( δl 2 ) < ϵ 2 l=n+1 S i(s) = {n N : in(s) b n } in(s) = b i(s) S A(S) = {n N : S(y n) > δ n } A(S) λ(s) = mina(s) λ(s) = + S (y n) 2 n λ(s) n i(s) A(S) = {n 1 < n 2 <... < n j < n j+1 <...} λ(s) = n 1 i(s) n 1 j N (y n) n nj δ nj S (y n j ) > δ nj S (y n j+1 δ nj S (y n) 2 = S (y n) 2 = S (y n j ) 2 + S (y n) 2 n λ(s) n i(s):n n 1 j=2 n i(s):n/ A(S) δn 2 j + δn 2 = j=1 n i(s):n/ A(S) n N λ 1,..., λ n R δ 2 n n i(s) 1 δ 2 n λ 2 i = 1 λ i y i U S U S = S 0 S S 0 = { t S : t < b i(s) } S = { t S : b i(s) t } S 0, S S S = S1 S S 1 = { t S : b i(s) t < b i(s)+1 } S = { t S : b i(s)+1 t }

S S = S 1 S 2 S 3 { } S 1 = t S : b i(s)+1 t < b λ(s ) } S 2 = {t S : b ) λ(s t < b )+1 λ(s } S 3 = {t S : b λ(s )+1 t λ(s ) = + b λ(s ) = + S = S 0 S 1 S 1 S 2 S 3 x = λ i y i S (x) 2 = S U S U 4 S U S 0(x) + S 1(x) + S 1 (x) + S 2 (x) + S 3 (x) 2 ( S0(x) 2 + S1(x) 2 + S 2 (x) 2 )) + 4 S 1 (x) + S 3 (x) 2 S U R S 0, S 1, S 2 R(x) = λ i R(y i) i {1,..., n} R(x) 2 = λ 2 i R(y i) 2 S U ( S 0(x) 2 + S 1(x) 2 + S 2 (x) 2 )) = S U = λ 2 i ( S U ( S 0(y i) 2 + S 1(y i) 2 + S 2 (y i) 2 )) (λ 2 i ( S 0(y i) 2 + S 1(y i) 2 + S 2 (y i) 2 )) λ 2 i y i = λ 2 i = 1 S U S 1 (x) + S 3 (x) 2 =,i λ(s ) n λ(s ) λ i S (y i) 2 ( S (y n) 2 n i(s ),i λ(s ) δ 2 n λ 2 i )(,i λ(s ) S (y i) 2 ) k N U k = {S U : i(s) = k} S U k i(s ) = k + 1

S U = S 1 (x) + S 3 (x) 2 = k=1 S U k n i(s ) δ 2 n = S 1 (x) + S 3 (x) S U k δn 2 m k ( k=1 k=1 S U k n k+1) k=1 n=k+1 δ 2 n) < ϵ 2 U S (x) 2 < 4 + 4ϵ 2 < 4(1 + ϵ) 2 S U λ i y i 2(1 + ϵ) 1 λy i 2(1 + ϵ) J T l 2 Y {x n } n N Y ϵ > 0 {x n} n N n N λ 1,..., λ n R (1 ϵ)( λ 2 i ) 1/2 λ i x i (2 + 3ϵ)( λ 2 i ) 1/2 Y J T l 1 J T {x n } n N Y ϵ > 0 {x n} n N (y n ) n N x n y n 2 < ϵ 2 ( )

I (x n) n 0 ( ) x n y n n 0 I I (y n ) I (x n ) + I x n y n n N I (y n ) = 0 n ϵ > 0 {y n} n N n N λ 1,..., λ n R ( λ 2 i ) 1/2 λ i y i 2(1 + ϵ)( λ 2 i ) 1/2 {x n} n N {x n} n N n N λ 1,..., λ n R λ i x i λ i x i y i + λ i y i ( λ 2 i ) 1/2 ( x i y i 2 ) 1/2 + 2(1 + ϵ)( (2 + 3ϵ)( λ 2 i ) 1/2 λ 2 i ) 1/2 λ i y i λ i x i y i ( λ 2 i ) 1/2 ( λ 2 i ) 1/2 ( x i y i 2 ) 1/2 (1 ϵ)( λ 2 i ) 1/2 λ i x i λ i x i λ i x i (1 ϵ)( λ 2 i ) 1/2 λ i x i (2 + 3ϵ)( λ 2 i ) 1/2

l 1 l 1