A GEOrlETRI CALLY ABERRANT BANACH SPACE WITH?JOWL STRUCTURE

Transcription

1 BULL. AUSTRAL. MATH. SOC. VOL. 31 (1985), A GEOrlETRI CALLY ABERRANT BANACH SPACE WITH?JOWL STRUCTURE An example is given of a Banach space with normal structure which does not satisf'y the geometrical conditions commonly expected to be related to normal structure. A Banach space is said to have nomz structure if for each non- trivial bounded convex subset K there exists a point p e K such that sup{llp-xi( : x Kl < diam K. A Banach space is said to have mfformzy nomz structure if there exists a 0 < k < 1 such that for each bounded convex subset K there exists a point p e K such that sup{llp-x/( : x e K] i k diam K Normal structure was introduced by Brodski i significant in the development of fixed point theory. and Mi lrnan [2] and has been results on normal structure has been given by Swarninathan [11]. A recent survey of Considerable research has been directed into finding geometrical conditions which imply normal structure. A Banach space is said to be unfformzy rotund if for any given E > 0 there exists a 6 > 0 such that, for X, y, 1 = y = 1, 11x-y1/ < E Received 7 August The research of the third author was supported by NSERC Grant A-5615 as a visiting professor at the University of Newcastle, New South Wales. Copyright Clearance Centre, Inc. Serial-fee code: /85 S~

2 7 6 J.R. Giles, Brailey Sims and S. Swaminathan when 1lx+yl > 2-6 ; such a space has uniformly normal structure, C51. A Banach space is said to be uniformly rotund in every direction if for any given z # 0 and E > 0 there exists a 6(E, z) > 0 such that /A( < E for X, y, llxli = liyil = 1 and x - y = Az when llx+yll > 2-6 ; such a space has normal structure, [dl, but not necessarily uniformly normal structure, [bl. A Banach space is said to be weakly uniformly Kadec-Klee if there exists an E < 1 and a 6 > 0 such that for every sequence (xn!, Ixnll 5 1 which converges weakly to x and inf{llxn-xmcm)( : m # n} Z E we have ilxll Van Dul s t and S ims have recently shown that such a space has weak norm2 structure, that is, the normal structure property holds for weakly compact convex sets, [721. A Banach space is said to be ZocalZy uniformzy rotund if for any given x, 11x1 = 1 and E > 0 there exists a 6 ( ~ x), > 0 such that 12-yll < E for ilyll = 1, when Ilx+yll > 2-6. Smith and Turett have recently provided an example of a reflexive locally uniformly rotund space which does not have normal structure. [lo]. In this paper we give an example of a reflexive Banach space which lacks all of these geometrical properties but which does have normal structure. In order to produce an example of a discontinuous metric projection Brown devised a geometrically interesting equivalent renorming of Hilbert sequence space z2, [31. Given natural basis e n and writing and Mk 5 sp{el, ek} for k z 3, l2 can be given an equivalent rotund norm ( such that its restriction 2 to M remains the original 2 -norm / and its restriction to Mk is an ~~(~-norrn here p(k) + as k +.

3 An aberrant Banach s~ace Brown's space is not uniformly rotund in every direction. xk (e +e )/lie +e I and yk = (-e +e )/li-e +e 1 1, 1 k 1 k 1 k 1 k For 2 xk- yk ==- e and l/xk-yk/l + 2 as k + m 21/p(k) 1 but xk - Brow's space is not weakly uniformly Kadec-Klee. (el+ek)/ Iel+eklJ and any y 5 1 %en, For SO the sequence { X converges weakly to e 1 ' But C However, 'f2 as k,l+rn. (l/df)/(x(i25 l(xll 5 (/x/(~ for a11 x C l2 so Therefore, for every 0 < E < 1, lim inf Ilx -x k,zk 2 but J(el(( = 1. lim inf{/x -x ( 1 : k # I} 2 E ; k 2 Bmwn's space is not locally uniformly mtmd. For xk ' ( e l f e k ) / l e l + e k / l 3

4 J.R. Gi les, Brai ley Sirns and S. Swarninathan But Nevertheless, as a reflexive Banach space containing a Hilbert subspace of codimension one, Brown's space does have normal structure as is evident from the following general result. LEMMA. If a Banach space X contains a closed subspace M of finite codimension with wliformly nom2 structure then X has norma2 structure. Proof. Since M has uniformly normal structure it is reflexive, [9], and since X contains a reflexive subspace of finite codimension it too is reflexive. Suppose that X does not have normal structure. Then by the characterisation theorem of Brodskii and Milrnan [21, there exists a weakly compact convex subset K containing a sequence {xn} such that d ( ~ ~ CO{X~, + ~,..., xn]) + diam K as n + m. Subsequences of {xn} satisf'y this property so we may, by weak compactness assume that {xn} converges weakly; by translation we may assume that {xn} is weakly convergent to 0 ; by scaling we may assume that diam K = 1. Consider a linear projection P from X onto M. Since {xn} converges weakly to 0 so finite dimensional complement of M. {xn-~xn} is convergent to 0 in the Given 0 < k < 1 the constant associated with the uniformly normal structure of M, choose 0 < E < (I-k)/b(l+k). Then there exists a v such that

5 An aberrant Banach space Consider K' : co{x : n > V}. Now K' has the diarnetmz property that, n for any x C K', sup(llx-yll : y C K') = diam K' = 1. Since I ~ X - P X ~ ~ < E for all x C K', it follows that diam P(K') E. From the uniformly normal structure of M there exists a p C K' such that 2ut then, for all x C K', IjPp-PxII 5 k(1+2~) for all x C K'. xnd this contradicts the diametral property of K'. BernaI and Sul I ivan have recently provided a condition under which an squivalent renorming of Hilbert space has normal structure, [I 1. Given a... :~llbert space (1, I * 112) and a norm ( 1 ' 1 on X such that I IlxIl2 5 llxl r 1 1 ~ 1 1 ~ for all x C -*-here 1 5 B < fi, then the Banach space (x, I I a I ( ) has normal structure. '!-wevery Brown's renorming of Hilbert space has X B = fi and is therefore -1 example which shows that, for equivalent renormings of Hilbert space the L ::erna I-Su l l i van condition is not necessary for normal structure. As a reflexive Banach space containing a closed subspace with :iscontinuous metric projection it can be deduced indirectly from Fan and /I icksberg C61 that Brown's space lacks a variety of geometrical roperties. Brown's space has also been used by Gi 1 es C71 to demonstrate :ne relationship between geometrical properties used by V ~ ~ S Oin V the ~3nvexity of the Chebychev set problem.

6 80 J.R. Gi les, Brai ley Sirns and S. Swaminathan References Javier Berna 1 ard Francis SU 1 I i van, "~anach spaces that have normal structure and are isomorphic to a Hilbert space", Proc. Amer. Math. Soc. 90 (1981r), M.S. Brodski i and D.P. Mi Iman, "On the centre of a convex set1', Dokl. Akad. Nauk SSSR 59 (l948), A. L. Brown, "A rotund reflexive space having a subspace of codimension two with a discontinuous metric projection", Michigan fifath. J. 21 (1974), M.M. Day, R.C. James and S. Swami nathan, "Normed linear spaces that are uniformly convex in every direction", Canad. J. Math. 23 (1971), M. Edelstein, "A theorem on fixed points under isometrics", Amer. Math. Monthly 70 (1963), Ky Fan and l r vi ng GI i cksberg, "Some geometric properties of the spheres in a normed linear space", &kc ~ath. J. 25 (1958), J.R. Gi les, "Uniformly weak differentiability of the norm and a condition of ~lasov", J. Austml. Math. Soc. Ser. A 21 (1976), A.A. Gillespie and B.B. Will yarns, "Fixed point theorem for non- expansive mappings on Banach spaces with unifornily normal structure", Applicable Anal. 9 (1979), E. Ma l uta, "uniformly normal structure and related coefficients", Pacific J. Math. (to appear). Mark A. Smith and Barry Turett, "A reflexive LUR Banach space that lacks normal structure1', preprint. S. Swami nathan, "~ormal structure in Banach spaces and its generalisationsl', Contemp. Math. 18 (1983),

7 An aberrant Banach soace , 3. Van Du I st and Brai ley S i ms, "Fixed points of non-expansive mappings and Chebychev centres in Banach spaces with norms of type (KK)", Banach space theory and its app2ication.s, Bucharest 1981, (~ecture Notes in Mathematics, 991. Verlag, Berlin, Heidelberg, New York, 1983). Springer- :---ent of Mathematics, --si ty of Newcastle, ~---, " ~ Z, le,. :-'n Wales 2308, --:,. - la; :.~--sr: of Mathematics, - - -,'+y of New England, --.= -- I of Mathematics, - '-2 -ni versi ty,