Constants and Normal Structure in Banach Spaces Satit Saejung Department of Mathematics, Khon Kaen University, Khon Kaen 4000, Thailand Franco-Thai Seminar in Pure and Applied Mathematics October 9 31, 009
This talk is based on the following papers: A. Jimenez-Melado, E. Llorens-Fuster and S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces, Proc. Amer. Math. Soc. 134 (006), no., 355 364. S. Saejung, On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property. J. Math. Anal. Appl. 33 (006), no., 1018-104. S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals. J. Math. Anal. Appl. 330 (007), no. 1, 597 604.
S. Saejung, The characteristic of convexity of a Banach space and normal structure, J. Math. Anal. Appl., 337, (008) 13-19. E. Casini, P. L. Papini, and S. Saejung, Some estimates for the weakly convergent sequence coefficient in Banach spaces, J. Math. Anal. Appl. 346 (008) 177-18. J. Gao and S. Saejung, Normal structure and the generalized James and Zbaganu constants, Nonlinear Analysis Series A: Theory, Methods & Applications, 71(7-8) (009), 3047-305. J. Gao and S. Saejung, Some geometric measures of spheres in Banach spaces, Appl. Math Comp. 14 (009) 10-107.
This talk is organized as follows: Some definitions and historical remarks James and von Neumann Jordan constants parameterized James and von Neumann Jordan constants generalized James and von Neumann Jordan constants Quantitative result
Let X be a Banach space. The research theme of this talk: properties or conditions on a Banach space X normal structure of X
Recall that the Banach space X has normal structure 1 if every nonempty bounded closed convex subset C of X, with diam C > 0, contains a non-diametral point, that is, there exists x 0 C such that sup{ x x 0 : x C} < diam C. 1 Brodskĭı, M. S.; Mil man, D. P. On the center of a convex set. (Russian) Doklady Akad. Nauk SSSR (N.S.) 59, (1948). 837 840.
Kirk s fixed point theorem X has normal structure and is reflexive X has the fixed point property Recall that X has the fixed point property if for every bounded closed convex subset C of X and every nonexpansive self-mapping T : C C, that is, Tx Ty x y for all x,y C, there exists a point x 0 C such that that is, x 0 is a fixed point of T. x 0 = Tx 0, A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 7 (1965), 1004-1006.
Spaces with/without normal structure Spaces with normal structure Finite dimensional spaces Uniformly convex spaces (Clarkson, 1936) Uniformly smooth spaces Spaces without normal structure C[a, b] (the space of real-valued continuous functions on [a, b]) Bynum spaces (Bynum, 1980) To test whether a given Banach space has normal structure is not an easy task.
Two starting points: Gao and Lau 3 proved that J(X) < 3 X has normal structure. Kato, Maligranda, and Takahashi 4 proved that C NJ (X) < 5 4 X has normal structure. Recall that J(X) = sup{min{ x + y, x y } : x = y = 1} { x + y + x y } C NJ (X) = sup x + y : x + y 0. 3 On two classes of Banach spaces with uniform normal structure. Studia Math. 99 (1991), no. 1, 41 56. 4 On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces. Studia Math. 144 (001), no. 3, 75 95.
Some facts on J(X) and C NJ (X): J(X) X is a Hilbert space J(X) = 1 C NJ (X) X is a Hilbert space C NJ (X) = 1 J(X) C NJ (X) J(X) J(l p ) = J(L p [a,b]) = 1/p if 1 < p C NJ (l p ) = C NJ (L p [a,b]) = /p 1 if 1 < p Recall that l p = {(x n ) R : n=1 x n p < } L p [a,b] = {f : f is a real-valued function on [a,b] and [a,b] f p dµ < }
Let X be a Banach space. The research theme of this talk: conditions on a Banach space X in terms of J(X) or C NJ (X) normal structure of X
The strongest results so far: J(X) < 3 J(X) < 1 + 5 1.61 5 X has normal structure C NJ (X) < 5 4 C NJ(X) < 1 + 3 1.36 6 X has normal structure 5 Dhompongsa, Keawkhao and Tasena, J. Math. Anal. Appl. 85 (003), no., 419 435. 6 S. Saejung, J. Math. Anal. Appl. 33 (006), no., 1018-104.
The strongest results so far: J(X) < 3 J(X) < 1 + 5 1.61 5 X has normal structure C NJ (X) < 5 4 C NJ(X) < 1 + 3 1.36 6 X has normal structure Remark: Both sufficient conditions cannot be applied for l p or L p [a,b] where p is near 1. In fact, it is known that all l p or L p [a,b] where 1 < p < have normal structure and J(l p ) = J(L p [a,b]) = 1/p if 1 < p ; C NJ (l p ) = C NJ (L p [a,b]) = /p 1 if 1 < p. 5 Dhompongsa, Keawkhao and Tasena, J. Math. Anal. Appl. 85 (003), no., 419 435. 6 S. Saejung, J. Math. Anal. Appl. 33 (006), no., 1018-104.
Parameterized James and von Neumann Joradan constants We study these constants: 7 for 0 t 1 J t (X) = sup{min{ x + ty, x ty } : x = y = 1} CNJ(X) t 1 = (1 + t ) sup{ x + ty + x ty : x = y = 1}. Remark: J(X) = J 1 (X) C NJ (X) = sup{cnj t (X) : 0 t 1} 7 S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals. J. Math. Anal. Appl. 330 (007), no. 1, 597 604.
Better sufficient conditions: J(X) < 1 + 5 ( J(X) < 1 + 1 ) J(X) J t t (X) < 1 + J t for some 0 t 1 (X) + 1 t X has normal structure Remark: If X = l p or X = L p [a,b] where 1 < p <, then J t (X) < 1 + for some 0 t 1. t J t (X)+1 t
Better sufficient conditions: C NJ (X) < 1 + 3 (1 + t )C t NJ (X) < (1 + ts) (1 + s )C s NJ (X ) (1 + s) for some 0 t,s 1 X has normal structure Remark: If X = l p or X = L p [a,b] where 1 < p <, then (1 + t )C t NJ (X) < (1+ts) (1+s )C s NJ (X ) (1+s) for some 0 t,s 1.
Some improvement in terms of these constants In 006, Jimenez-Melado, Llorens-Fuster and Saejung 8 proved the following: Recall that 9 µ(x) = inf J(X) < 1 + 1 X has normal structure µ(x) C NJ (X) < 1 + 1 X has normal structure µ(x) r > 0 : lim sup n x + x n r lim sup n x x n for all x X and all weakly null sequences {x n } in X.. 8 Proc. Amer. Math. Soc. 134 (006), no., 355 364. 9 B. Sims, A class of spaces with weak normal structure, Bull. Austral. Math. Soc. 50 (1994), 53-58.
J(X) < 1 + 1 X has normal structure µ(x) C NJ (X) < 1 + 1 X has normal structure µ(x) Note: Both results are sharp in the sense that there is a Banach space X such that X fails to have normal structure and J(X) = 1 + 1 µ(x) and C NJ (X) = 1 + 1 µ(x).
We can prove the following results 10 : J(X) < 1 + 1 µ(x) J t (X) < 1 + t µ(x) X has normal structure for some 0 t 1 C NJ (X) < 1 + 1 µ(x) ( ) 1 + t CNJ(X) t µ(x) < 1 + t for some 0 t 1 X has normal structure 10 J. Gao and S. Saejung, Appl. Math Comp. 14 (009) 10-107.
Generalized James and von Neumann Joradan constants Let B X = {x x : x 1}. Based on the Hexagonal Lemma of Gao and Lau, 11 the following constants are introduced 113 : { J(a,X) = sup min{ x + y, x z } : x,y,z B X, } y z a x { x + y + x z C NJ (a,x) = sup x + y : x,y,z X, + z } x + y + z 0, y z a x. 11 On two classes of Banach spaces with uniform normal structure. Studia Math. 99 (1991), no. 1, 41 56. 1 Dhompongsa, Kaewkhao, and Tasena, J. Math. Anal. Appl. 85 (003), no., 419 435. 13 Dhompongsa, Piraisangjun, and Saejung, Bull. Austral. Math. Soc. 67 (003), no., 5 40.
{ J(a,X) = sup min{ x + y, x z } : x,y,z B X, } y z a x { x + y + x z C NJ (a,x) = sup x + y : x,y,z X, + z } x + y + z 0, y z a x. Remark: J(0,X) = J(X) C NJ (0,X) = C NJ (X)
Improved sufficient conditions: 15 The following is an improvement of 14 : J(a,X) < 3 + a for some 0 a 1 J(a,X) < 1 a + (1 a) + 4(1 + a) X has normal structure for some 0 a 1 Remark: For all 0 a < 1, 3 + a < 1 a + (1 a) + 4(1 + a). 14 Dhompongsa, Kaewkhao, and Tasena, J. Math. Anal. Appl. 85 (003), no., 419 435. 15 J. Gao and S. Saejung, Nonlinear Analysis Series A: Theory, Methods & Applications, 71(7-8) (009), 3047-305.
Remark: We also obtain an improvement for the generalized NJ-constant.
An answer of Llorens-Fuster s question 16 Llorens-Fuster proved that where C Z (X) < 16 X has normal structure, 13 { } x + y x y C Z (X) = sup x + y : x + y 0. He asked this question: Is 16/13 sharp in this situation? Remark: C Z (X) C NJ (X) and there is a Banach space X such that C Z (X) < C NJ (X). 16 E. Llorens-Fuster, Zbăganu constant and normal structure, Fixed Point Theory 9 (008) 159 17.
The main tool that Llorens-Fuster used in his paper is the modified Hexagonal Lemma 17. A careful application of this lemma gives the following result: C Z (X) < 16 13 1.3 C Z (X) < 1 + 3 1.36 X has normal structure Remark: It seems to be unknown whether 1+ 3 is sharp in this situation. 17 S. Saejung, J. Math. Anal. Appl. 330 (007), no. 1, 597 604.
Quantitative results: Bynum 18 defined the weakly convergent sequence coefficient of X by { lim k sup{ x n x m : n,m k} } WCS(X) = inf inf { lim sup n x n y : y co({x n }) } where the infimum is taken over all weakly convergent sequences {x n } which are not norm convergent. It is clear that for reflexive spaces X WCS(X) > 1 X has normal structure. 18 Normal structure coefficients for for Banach spaces. Pacific J. Math. 86 (1980), 47-436.
Suppose that a Banach space X fails the Schur property, that is, X contains a weakly convergent sequence which is not norm convergent. Then 19 1 WCS(X). J(X)+ 1 5 ( 1 + 1 WCS(X) 1 J(X) WCS(X) 1 C NJ (X) µ(x) ). ( 1 + 1 µ(x) ). In particular, X has normal structure if J(X) < 1+ 5 or J(X) < 1 + 1 µ(x), or C NJ(X) < 1 + 1 µ(x). 19 E. Casini, P. L. Papini, and S. Saejung, J. Math. Anal. Appl. 346 (008) 177-18.