ALMOST ISOMETRIC L p SUBSPACES OF L p (0,1) GIDEON SCHECHTMAN I. Introduction The main result of this paper is the following theorem which answers a question raised by Enflo and Rosenthal [3] and by Dor [2]. THEOREM 1. For every p satisfying 1 < p < oo there exist a number A o > 1 and a function (j)(X), defined for 1 < X < X o , such that (f)(X) -> 1 as X -> 1 and whenever x l5 ..., x n are elements of L p (0,1) which satisfy ( n \l/p II n || / n \l/p Z W p ^ Z^UA Skl p i=l / II>=1 II V=l / for every sequence a 1 ,...,a n ofscalars, [*,-]"= x is complemented in L p (0,1) by means of a projection of norm at most A simple compactness argument (cf. e.g. [2]) shows then that ifX is a S£ p>x subspace ofL p (0,l)with\ < X < X o then X is complemented in L p (0,1) by means of a projection of norm (f)(X) at most (see [5] for the definition of S£ p x spaces). The corresponding theorem for p = 1 was previously proved by L. E. Dor [2]. Theorem 1 is a simple consequence of the following. THEOREM 2. Let 1 < p < oo,p # 2. There exists a function a(e) such that a(e) -> 0 as E -> 0 + and, ifx lt ..., x n is a sequence of norm one elements ofL p (0,1) which satisfy _ MP for all choices ofscalars a i} ..., a n , then there exist disjoint sets A h i = I,..., n of[0,1] such that h ' J ) forallscalarsa Xi ...,a n . (Here A\ — [0,1'] —A h and x\ A is thefunction which is equal to x on A and to zero elsewhere.) The main tool of the proof of Theorem 2 is the following theorem of Dor which enabled him to prove Theorem 1 for p = 1. Received 12 April, 1978. [J. LONDON MATH. SOC. (2), 20 (1979), 516-528]