Mathematische Zeitschrift

Math Z 194 321 329 (1987) МЭШС1Т13С18СПв

Zeitschrift

© Springer-Verlag 1987

Sequence Spaces with Small j8-Duals

Grahame Bennett

Department of Mathematics Indiana University Bloomington IN 47405 USA

§ 1. Introduction

It IS a famihar fact that the jS-dual of a sequence space provides a rough estimate of Its size if £^F, then E^^F^, E^ must be "small" whenever E is "large" Our purpose here is to attempt to make sense of the converse statement - if E^çF^, then £^F - at least when E and F are FX-spaces

As it stands, the converse is hopelessly wrong, for there exist Bi^-spaces E, F with E^ = F^, yet Er\F= {0} (An interesting example of this phenomenon, due to Wilansky, is obtained by setting E = Co and F = q, the closure, m m, of the space of periodic sequences ) To avoid such cases we shall always take E^F, so that E^ = F^, and then ask whether E = F Still the answer is negative (take £ = Co, F = m), so we add the hypothesis that E be dense in F

We are thus led to ask which FK-spaces F have the following property

( W ) If E IS an FK-space, contained and dense in F, with E^ = F^, then E = F

We shall see that, of the "classical" sequence spaces, Cq, P, l<p<oo, с s and с have (W), whereas l^,hvo, bv, hs,m and со do not

The letter "W" is m honor of Wilansky, who first studied problems of this type (see [18], 8 6 8, 16 3 10, 19 5 1, and Theorem 4 below) Snyder ([14], Sect 4) considers a property very similar to (W), but he imposes additional hypotheses on E and F Our analysis has little in common with any of their results

We begin by considering the simplest case, F = <:o, m Sects 3 and 4 A result on matrix mappings (Lemma 1), which extends the celebrated Silverman-Toep- litz theorem, is used to establish several results, for Cq, that are apparently stronger than (W) These new properties are discussed in a general setting in Sect 5 and are shown to hold also for F, 1 <p< oo The spaces с s and с require different methods and these are described in Sect 6 Finally, in Sect 7, we list some problems for further research