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In this paper we extend two interpolation theorems in the setting of rearrangement invariant p-spaces, for 0^ p>< CU Some applications of these theorems are given, particularly we extend Theorem 2. C. 6-[4] proving that the Haar system is an unconditional basis in a rearrangement invariant p-space X iff the Boyd indices px and qx verify the relations 1< py and q^< oo• Some non locally convex Lorentz fonction spaces are examples of such rearrangement invariant p-spaces, while in [3] NJ Kalton proved that only the locally convex Orlicz spaces have a Schauder basis.

In the sequel we assume all the vector spaces to be real. p is a positive real number less than 1. Let X a topological complete vector space such that its topology is generated by a positive function||| L, called p-norm, which fulfills the following properties: 1)|| x|| x= 0 iff x= 0; 2)||^ x| U'=:| c) c|»|| x| L foroCeiR, X6X; 3)|| x+ y||£<|| x|| P+|| y| LP for x, ysX.(We recall that|| L generates the topology of X if IV={xeX;| lxllx^~ H~ 3> n€ IN; constitute a neighbourhood basis of origin for this topology). We say that X is a p-Banach space. If p= 1 we find the classical definition of a Banach space. A p-Banach space (X,||||) which is moreover a vector lattice, is called a p-Banach lattice if