1 SYMMETRIC STRUCTURES HT SUBSPACES OF L (0,l) p 2 The main result of this section is the following theorem on the classi- fication of subspaces of L (0,l) p 2 with a symmetric basis. ir Theorem 1.1: For every p 2 and every K1 there exists a number D = D(p K) OO SO that if {x.}3? ^s a finite normalized basic sequence in L (0 lVp 2 whose symmetry constant K({x.}1? ) ^s K then n n ' ^ x i'l D"1!! S a.x.!| max{( S |a. p ) 1 ^, - ^ _ ^ ( S |a.I2)1/2} D || S a.xj, i=l i=l y/ Q i=l i=l /or every choice of scalars {a.}1? This yields that every finite or infinite normalized K-symmetric basic sequence in L (0,l) p 2 is D(p,K)-equivalent to some symmetric X -basis. y P In the proof of 1.1 we use the notion of exchangeable random variables. A finite sequence {f.} of random variables (functions) over a probability space is called symmetrically exchangeable if, for every permutation j of the integers {1,2,... ,n} and for every choice of signs e. = + 1 , the probability distribution in R of {£.}._-, is the same as that of {e.f /.\}._1 The classical definition of exchangeability, as considered in probability theory, involves only invariance under permutations. The following lemma is known (see e.g. [12]). Lemma 1.2: A K-symmetric normalized basic sequence {x.}1? , in L (0,l) 1 p o o is K- equivalent to a normalized sequence {f.}._1 of symmetrically exchangeable random variables in L (0,l). Proof: Let H be the family of all distinct pairs (TT, {e.)._..) , where JT is a permutation of the integers {l,2,...,n} and {e.}._, a sequence of 3k
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