ISRAEL JOURNAL OF MATHEMATICS, Vol. 47, Nos. 2-3, 1984 ON TSIRELSON'S SPACE BY P. G. CASAZZA,* W. B. JOHNSON*' AND L. TZAFRIRI" ABSTRACT A structure theory is developed for Tsirelson's example of a Banach space which contains no isomorphic copy of Ip or co. In particular, it is shown that this space is the first example, other than subspaces of lp and co, of a Banach space which embeds isomorphically into each of its infinite dimensional subspaces. Introduction In this paper we present some additional properties of the space introduced by Tsirelson in [11] and its variations discussed in [5], [6] and [3]. Further results on this interesting space will be presented in [4] and [2]. Tsirelson's original space (denoted here by T*) was the first example of a Banach space which contains no subspace isomorphic to Co or lp ; 1 < p < ~. The notation T for the dual of Tsirelson's original space as well as the analytic description of the norm in T were given in [5]. The convenience of working with a concrete formula for the norm made us prefer this particular notation. The main result in this paper asserts that T* is minimal in the sense that every infinite dimensional subspace of T* contains in turn an isomorphic copy of T*. (In fact, Theorem 14 says that T* embeds isomorphically into every infinite dimensional subspace of a quotient of T*.) Previously, only the spaces lp; 1 =<p < ~ , Co and their infinite dimensional subspaces were known to be minimal. The minimality of T* is a consequence of a simple criterion for the boundedness of operators on T and T* (Theorem 8) and an analysis of the structure of subspaces of quotients of T and T* which also yields, among other results, the fact that if Y is a quotient space of T or of T* then every infinite dimensional subspace of Y contains in turn a subspace E which is corn- t Supported in part by NSF MCS-8002221 and MCS-8102238. tt Supported in part by NSF MCS-7903042. Received August 3, 1983 81