A Banach Space With, Up To Equivalence, Precisely Two Symmetric Bases

It is well known that the classical sequence spaces co and Ip (1 <= p <) have, up to equivalence, just one symmetric basis. On the other hand, there are examplesof Orlicz sequence spaces which have uncountably many mutually nonequivalentsymmetric bases. Thus in [4], p. 130, the question is asked whetherthere is a Banach space with, up to equivalence, more than one symmetric basis,but not uncountably many. In this paper we answer the question positively, byexhibiting a Banach space with, up to equivalence, precisely two symmetricbases.