PREFACE The most important and most interesting class of Banach spaces is un- doubtedly that of L -spaces 1 p o o . These spaces, which appear in many problems arising naturally in analysis, have been studied thoroughly for a long time. As a result of this intensive research, the discrete case of JL - ' P spaces is relatively well understood by now while that of L -spaces over an atomless measure space, though in a quite satisfactory stage, still raises many questions of importance. The observation that the unit vector basis of JL : 1 p » is equiva- lent to any of its permutations (i.e., & , as a normed space of sequences, is invariant under permutations) led to the study of a more general class, namely that of spaces with a symmetric basis. Besides the X -spaces 1 p o o 9 the simplest examples of spaces with a symmetric basis are the Orlicz sequence spaces i y , where F(t) is a non-decreasing convex function on [0,oo) with F(0) = 0 and lim F(t) = » (the function F(t) plays t - oo about the same role for & as t does in the case of & ). In recent years, a great deal of research went into the study of spaces with a symmetric basis and, in particular, into that of Orlicz sequence spaces and, by now, there is quite clear picture of their structure. A survey of this topic can be found e.g. in [lj-91* The present paper has only relatively little bearing on the subject of infinite dimensional spaces with a symmetric basis but, instead, it attempts to use the methods and the knowledge accumulated in this field as a starting point in the investigation of some other classes of spaces with a symmetric structure. One of our main targets is the class of L -spaces over an atom- less measure space or, more generally, that of rearrangement invariant (r.i.) Received by the editor March 27, 1978. 1

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