This text uses Banach lattices and Riesz spaces as tools to study the classical ideas and theorems of functional analysis. It is suitable for students who have completed a standard course in functional analysis and for mathematicians interested in the systematic application of such techniques.

This book is mainly concerned with the theory of Banach lattices and with linear operators defined on, or with values in Banach lattices. Moreover we will always consider more general classes of Riesz spaces so long as this does not involve more complicated constructions or proofs. In particular, we will not treat any phenomena which occur only in the non-Banach lattice situation. Riesz spaces, also called vector lattices, K-lineals, are linear lattices which were first considered by F. Riesz, 1. Kantorovic, and H. Freudenthal. Subse- quently other important contributions came from the Soviet Union (L.V. Kan- torovic, A.J. Judin, A.G. Pinsker, and B.Z. Vulikh), Japan (H. Nakano, T. Oga- sawara, and K. Yosida), and the United States (G. Birkhoff, H.F. Bohnenblust, S. Kakutani, and M.l\f. Stone). In the last twenty-five years the theory rapidly increased. Important con- tributions came from the Dutch school (W.A.J. Luxemburg, A.C. Zaanen) and the Tiibinger school (lI.lI. Schaefer). In the middle seventies the research on this subject was essentially influenced by the books of H.H. Schaefer (1974) and W.A.J. Luxemburg and A.C. Zaanen (1971). More recently other impor- tant books concerning this subject appeared, A.C. Zaanen (1983), H.U. Schwarz (1984), and C.D. Aliprantis and O. Burkinshaw (1985).