Banach m8ücxêo# { ò ) (4ï ŒÆêÆ OŽÅ ÆÆ, 4ï 4² ) Á : ïä8ühausdorff(fractal) êž, écxêo '. ± Banach m8ücxêoñ ÏLEåÓÄ ª5 y. ÏLEBanach mx vžf ê P = 1 Ý

Bản ghi

1 Banach m8ücxêo# { ò ) (4ï ŒÆêÆ OŽÅ ÆÆ, 4ï 4² ) Á : ïä8ühausdorff(fractal) êž, écxêo '. ± Banach m8ücxêoñ ÏLEåÓÄ ª5 y. ÏLEBanach mx vžf ê P = 1 ÝKŽfP, é8üš ÝK ), (Ü È8Ü5Ÿ, Banach mx f8b F r 2 (0), ±Œ»( r 2) ŠCX ê «# O, dd Banach mx8ücxê Ñ *O {. Ù B F r 2 (0) B r2 (0) F, B r2 (0) X : %!Œ» r 2, F Banach mxk f m. ' c: Banach m; Œ»; Žf MR(2010)ÌK a: 37L30; 46B99 ã aò: O A new estimate of covering number for set in Banach space ZHONG Yan-sheng (School of Mathematics and Computer Science, Fujian Normal University, Fuzhou , China) Abstract: The covering number is one important basis for the researching on the Hausdorff(or Fractal) dimension of set. By constructing the isometry operator, it used to calculate the covering number in Banach space. However, a new estimate of the minimal number of the covering balls with radius for the subset Br F 2 (0) X was obtained, by constructing the project operator P with norm P = 1 in Banach space X, and then making a projecting decomposition on the set with the property of covering number on product set. It supports a more intuitive approach to estimate the covering number of the set in Banach space. Here Br F 2 (0) B r2 (0) F and F is a finite dimensional subspace of X and B r2 (0) is a ball centered in 0 with the radius r 2 in X. Keywords: Banach space; radius; idempotent operator MR(2010) Subject Classification: 37L30; 46B99 Chinese Library Classification: O ÝvFϵ Ä7: I[g, ÆÄ7]Ï 8(NO ); 4ïŽg, ÆÄ7]Ï 8(2012J05002); Æ Ä7] Ï(2011M501074) Ï&Šö: zhyansheng08@163.com. 1

2 1 Úó 3k'à ÄåXÚáÚf±9 êáúf êïä, CX½nåX ' Š^, [1-6]. 1981c, R. MañéÏLy²R n Banach mxn f mf 3åÓ, Äg Ñ8ÜB F r 2 (0) CXê «O( [7]Ún2.1). d, ÏLEåÓy ²Banach m8ücxê, X C2010 c, A. N. Carvalho( [2]Ún2.3), ÏL(Ü Banach mbanach-mazurål `þ.y²åó35, Ñ CX½n. ±þñ 3y²3åÓÄ:þ, åóvgwä. 1998c, R. Temam3Hilbert m( [6]P367Ún3.1)ÏL ÝK {, CXêO ½n. Édéu, dk. 5 ¼(ÜHahn-Banach½nEBanach m vžf ê P = 1ÝKŽfP, /dé8üšýk ), (Ü È8Ü5Ÿ8ÜCXê «#O, l Banach m8ücxê Ñ «*O {. PX Banach m, P ÝKŽf, Ú ). Ù 11!, EÝKŽfP ; 12!, y²cx½n. 2 EÝKŽf Äk, Ñ Ú ) Žf½Â. ½Â1 [8] eé? x X, 3 )ª x = x 1 + x 2, x 1 X 1, x 2 X 2. K Banach mxk Ú )X = X 1 X2. ½Â2 [8] NP : X X, ep 2 = P, K P Žf. 'u Žfke 5Ÿ. Ún1 [8] P : X X k. 5 Žf, KX = X 1 X2, Ù X 1 {x : P x = x}, X 2 {x : P x = 0}. dd, ÑÝKŽfP : ½n1 X 1 X? f m, K3ÝKŽfP : X X 1 v P X X1 = 1, Ó žki P : X X 2 v I P 2 X = X 1 X2. Ù IL«ðŽf, L«Žf 2

3 ê. y² š"x 0 X 1, KX 1 {αx 0 : α R}. ½Âx X 1 þ k. 5 ¼ x (αx 0 ) α x 0 X, α R K x X 1 = 1 9x (x 0 ) = x 0 X. dhahn-banach½n, x Œòÿ Xþk. 5 ¼(E P x ) x X = e5, y²þãp P (x) = x (x) x 0, x 0 x 0 x 0, ÝKŽf. dún1œd yp k. 5 Žf. þ, é? x 1, x 2 X, k P (αx 1 + βx 2 ) = x (αx 1 + βx 2 ) x 0 = αx (x 1 ) x 0 + βx (x 2 ) x 0 = αp (x 1 ) + βp (x 2 ). d lim y n y 0 x (y n ) x (y 0 )Œ P (y n ) = x (y n ) x 0 x (y 0 ) x 0 = P (y 0 ). ±P k. 5Žf. qdué? y X 1, 3k, y = k x 0 9 P.? Ú, é x X, k P (y) =x (y) x 0 = x (k x 0 ) x 0 = kx ( x 0 ) x 0 = kx ( x 0 x 0 ) x 0 =k 1 x 0 x 0 x 0 = k x 0 = y. P 2 (x) = P (P (x)) = P (x (x) x 0 ) = x (x (x) x 0 ) x 0 = x (x)x ( x 0 ) x 0 = x (x) x 0 = P (x). 3

4 nþœ, P k. 5 Žf. X = X 1 X2, Ù X 2 {x X P (x) = 0}, X 1 {x X P (x) = x} = {αx 0 : α R}. Ún2 P X X1 = 1 d Ï P (x) P = sup x 0 x x (x) x 0 = sup x 0 x x X x x 0 sup x 0 x = 1 (1) x 0 = x ( x 0 ) x 0 = P ( x 0 ) P x 0, Œ P 1 (2) (Ü(1), (2), k P X X1 = 1, k I P : X X 2 9 I P I + P = 2 =½n1y. 51 : eþã½n1 X 1 X? n f m, K3ÝKŽfP : X X 1 v P X X1 n. þ, X 1 Ä. {e 1, e 2,, e n }, ŠâAuerbach½n( [9]½n2.1.16), X 13 ¼{f i } n i=1, v f i (e j ) = δ ij f i X 1 = 1, i = 1,, n dhahn-banach ½n, {f i } n i=1þœòÿ Xþk. 5 ¼(EP f i ) v f i X = 1. Žf P (x) = w,p Xþk. 5Žf, P Š f k (P (x)) = f j (x)e j, x X. R(P ) = X 1. f j (x)f k (e j ) = f k (x) k=1 4

5 é?ûx X, P 2 (x) = P (P (x)) = f k (P (x))e k = k=1 f k (x)e k = P (x). k=1 ŠâÚn1, X = X 1 X2, Ù X 2 {x X P (x) = 0}, X 1 {x X P (x) = x}. qd P = sup P (x) X = sup x X =1 x X =1 = f j (x)e j X sup f j (x) e j X x X =1 sup f j (x) x X =1 f j = n, = P X X1 n. 3 CX½n e, (Üþ!ÝKŽf ÑBanach m8ücxê «# *O {: ½n2 F X? m f m, Ké? r 2 > 0, k N r1 (B F r 2 (0)) 2 m (m+1) (1 + r 2 ) m (3) Ù B F r 2 (0)L«B r2 (0) F, N r1 (B F r 2 (0))L«± Œ» CX8ÜB F r 2 (0) I ê. y½n2, IXe{üÚn Ún3 X = X 1 X2 ( v½n1), B X : %. e8üb X 1 ±ɛ Œ» CXê N(X 1, ɛ), 8ÜB X 2 ±ɛ Œ» CXê N(X 2, ɛ), KN(B, 2ɛ) N(X 1, ɛ) N(X 2, ɛ), Ù N(B, 2ɛ) ±2ɛ Œ» CX8ÜB ê. 52: eún3 X 1 ê n, K(JU N(B, (n + 1)ɛ) N(X 1, ɛ) N(X 2, ɛ), 5

6 y² P È8 E F = {(x, y) x E, y F }. w,, BŒ CXB X 1!B X 2 k È8Ü CX, ê õ N(X 1, ɛ) N(X 2, ɛ). qdu P = 1, 9 I P 2, lù È8Ü ŒŒ»Ø L2ɛ, Ïd, N(B, 2ɛ) N(X 1, ɛ) N(X 2, ɛ) y.. yy½n2: y² æ^êæ8b{. Äk, m = 1, (3)ªw, á. džb F r 2 (0) Ý 2r 2 ã, I(1 + r 2 ) Œ» =ŒCX, é2 2 (1 + r 2 )g, á. Ùg, (3)ªém 1 á, y'um á. š":x 0 B F r 2 (0) PX 1 {αx 0 : α R, x 0 B F r 2 (0)}, (ܽn1, Œ3ÝK ŽfP v P X X1 = 1. Ïd, 'uf mf, kƒa Ú ): Ún4 F = F 1 F2, Ù F 1 X 1 F, F 2 X 2 F 9dim F 1 = 1, dim F 2 = m 1.? Ú, é? x B F r 2 (0), k x = P x + (I P )x = x 1 + x 2 x 1 X = P x X P L (X,X1 ) x X r 2 (4) x 2 X = x x 1 x X + P x X 2r 2. (5) d(3) (4)ªŒŒ» /2 CXx 1 I ê (1 + r 2 /2 ), é(1 + 2r 2 /2 ) g, á. du(3)ªbém 1 á, ddim F 2 = m 1 (5)ªŒ, ±Œ» /2CXx 2 I ê 2 (m 1)m (1 + 2r 2 /2 )m 1. (ÜÚn3, ŒŒ» CX8ÜB F r 2 (0) 6

7 ên r1 (B F r 2 (0)) v N r1 (Br F 2 (0)) (1 + 2r 2 2 ) 2 (m 1) m (1 + 2r 2 = 2 (m 1) m (1 + 2r 2 2 ) m < 2 (m 1) m 4 m (1 + r 2 ) m = 2 (m 1) m+2m (1 + r 2 ) m = 2 m (m+1) (1 + r 2 ) m, 2 ) m 1 (3)ªém 4 (Š á, y.. þãy²l ^ ¼ Û Hahn-Banach½nék. 5 ¼Šòÿ, EÑ ÝKŽf, ddé8üš Ú )5CXêO, ØIïÄåÓŽf5Ÿ, Ï dy² {w *. 3y²L é È8ÜCXê, AO CXŒ» ÝKŽ f ê'x, #nø(j(=ún3, 52), l LCX½ny² {9 Ùƒ'Sº. ë z: [1] Robinson J.C. Dimensions, Embeddings, and Attractors[M], Cambridge Tracts in Math., vol. 186, Cambridge University Press, Cambridge, [2] Carvalho A N, Langa J A, Robinson J C. Finite dimensional global attractors in Banach spaces[j]. Journal Differential Equations, 2010, 249(12): [3] Dung L, Nicolaenko B. Exponential attractors in Banach spaces[j]. Journal of Dynamics and Differential Equations, 2001, 13(4): [4] Efendiev M, Zelik S. Finite and infinite-dimensional attractors for porous media equations[j]. Proceedings London Mathematical Society, 2008, 96(1): [5] Efendiev M, Miranville A. The dimension of the global attractors for dissipative reactiondiffusion systems[j]. Applied Mathematics Letters, 2003, 16(3):

8 [6] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics[M]. New- York: Springer-Verlag, 1997: [7] Mañé R. On the dimension of the compact invariant sets of certain non-linear maps[j]. Springer Lecture Notes in Mathematics, 1981, 898: [8] ôlj, šõ. ¼ Û[M]. : p Ñ, 1992: [9] c. Banach maûnø[m]. þ : uà ŒÆÑ, 1984: