An example concerning reflexivity

Abstract

.v.~:WMp~. 288 W. Ruckle Proof of Theorem 3.3. In the case of 6' associated with a symmetric basis of an F-space it is necessary to prove that S” = S”, and this fol- lows from 1.3 and 2.6. If S is a o‘-perfect FK-space, then either 6' = s in which case we are finished or S S in [4]. In the second case for each u in 6'“ define P§’,(s) = sup |u,,(i)s,-|: :2 is a permutation on the natural numbers} q 1'.=-1 and proceed as in the y-perfect case. The seminorms Pi defined in the course of the argument will all be symmetric by 2.10 so that 6 will be a symmetric basis for its closed linear span by Definition 1.2.q The following is a generalization of (SB1) ¢ (SB,,) of Singer [8], 6 Theorem 5.3. , 3.4. COROLLARY. A basis {an} of a locally conoea F-space X is a sym- metric basis if and only if every permutation {an,,(.,,)} of {son} is a basis of X equivalent to {an}. Proof. Without loss of generality we may restrict our attention to 6 a basis for a locally convex space S. The necessity of the condition was given in Lemma 1.3. If every permutation of c‘ is a basis for 8' equivalent to c’, then sefl implies (s,,(,-,)s;S' so that S is symmetric. Therefore, 5”“ =8“ so that, by 3.2 and 3.3, 6’ is a symmetric basis for S. 1 References [1] M. G. Arsove, Similar bases and isomorphisms in Frechet spaces, Math. Ann. 135 (1958), p. 283-293. [2] M. M. Day, Nor-med li/near spaces, 1957. [3] N. Dunfo rd. and J. T. Schwartz, Linear operators, 1958. [4] G. Kiithe and 0. Toeplitz, Lineare Rdume mit unendbioher oielen .Koor- dinaten and Ringo unenzllicher Matrizen, J. fiir Math. 171 (1934), p. 193-226. [5] W. Ruckle, On the construction of sequence spaces that have Schauder bases, Canadian Math. J. (to appear). [6] — Symmetric coordi/natc spaces and symmetric bases, ibidem (to appear). [7] — On perfect symmetric BK spaces, Math. Ann. (to appear). [8] 1. Singer, Some characterizations of symmetric bases, Bull. Acad. Pol. Soi., Serie des 50- math-. aetr. et phys., 10 (1962), p. 135.192. [9] A. Wilansky, Functional analysis, Blaisdell 1964. Raw par la Redactim le 9. 8. 1966 M STUDIA MATHEMATICA, T. XXVIII. (1967) An example concerning reflexivity by ' R. HERMAN and R. VVHITLEY (Ma.ry1a.nd)* The spaces co and l not only fail to be reflexive but contain no infin- ite-dimensional reflexive subspace [7, 12]. It is natural to conjecture that each non-reflexive space contains an infinite-dimensional closed sub- space with this property; this conjecture is false. Here we gve an example of a Banach space which is not reflexive (or even quasi-reflexive [4]) yet has the property that each of its closed infinite-dimensional subspaces contains a subspace isomorphic to the Hilbert space Z’. We also discuss this type of non—reflexive space and show that it has some properties in common with reflexive and quasi-reflexive spaces. LEMMA 1. Let X be the quasi»reflc.m‘oc space constructed by R. C’. James ([8], also see [9], p. 198). Every infin.ite—olimen.sional closed sub- space of X contains a subspace isomorphic to Z3. Proof. We recall that the space X consists of vectors so = (a1, a2, ...), at a sequence of scalars, where Q? is in X if and only if lim an = 0 and it urn = sup[ZIa..,._,—a..2.12+:a.,.,..,I=]1” i=1 is finite, where the supremum is taken over all finite increasing (or one term) sequences. The vectors ca, with a one in the i-th place and zeros elsewhere, constitute a Schauder basis for X. Using a theorem due to Bessaga and Pelczyfiski ([2], 0.2, p. 157), each infinite-dimensional closed subspace M of X contains a sequence {y,,} which is basic ({y,,} is a Schauder basis for its closed linear span [y,.]) and is equivalent to a block basis [an], with respect to acq, i.e. each .2” is given by 4n+1 211. = 2 win i=qn+1 * The first author was supported by a National Defense Education Act fellowship and the second by National Science Foundation grant GP 5424. Studia Mathematics, t. XXVIII, z. 3 5