0 INTRODUCTION The main object of this section is to present the contents of the paper and its organization. In addition, wedescribe here many notions, methods and results which are used often throughout the work. The notions and results of local interest are discussed in the text only in the place where their use is required. The reader interested in a detailed account of the material used in the background of the paper is referred to [k-9] and [50]. The notations used here are quite standard and in most cases follow the books mentioned above. Since the well-known inequality of Khintchine A i l la.f2)1/2 (J1 I I a . r . ( u ) | V ) l / p B ( r jaj 2 ) 1 / 2 , p i= l 0 i=l p 1=1 - 00 where 1 p ° ° and lr.J._, denotes the sequence of Rademacher functions, is used frequently throughout the paper, we keep the symbols A andB exclusively for the constants appearing in Khintchinefs inequality. The next section of the article i.e., Section 1, is devoted to a quite thorough analysis of those subspaces of L(0,l) p2 which have some kind of symmetric structure. The starting point is the well-known and trivial fact that each of the spaces JL 1 p » has, up to equivalence, a unique symmetric basis. D. R.Lewis raised the question whether this fact has a local analogue i.e., whether each of the spaces I 1 p ° ° n = 1,2,..., ir has a unique symmetric basis. In order to state this problem as well as its solution in a precise manner, we first recall that the symmetry constant K({x.}) of a finite or infinite basis fx.} is the smallest number K such that the inequality K"1)! Z a x | | | | S e.a^.xx.U K | | S a.x.|| i i v ' i 5

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