Introduction to Operator Theory in Riesz Spaces by Adriaan C. Zaanen (PDF)

Description

Since the beginning of the thirties a considerable number of books on func­ tional analysis has been published. Among the first ones were those by M. H. Stone on Hilbert spaces and by S. Banach on linear operators, both from 1932. The amount of material in the field of functional analysis (in­ cluding operator theory) has grown to such an extent that it has become impossible now to include all of it in one book. This holds even more for text­ books. Therefore, authors of textbooks usually restrict themselves to normed spaces (or even to Hilbert space exclusively) and linear operators in these spaces. In more advanced texts Banach algebras and (or) topological vector spaces are sometimes included. It is only rarely, however, that the notion of order (partial order) is explicitly mentioned (even in more advanced exposi­ tions), although order structures occur in a natural manner in many examples (spaces of real continuous functions or spaces of measurable function~). This situation is somewhat surprising since there exist important and illuminating results for partially ordered vector spaces, in . particular for the case that the space is lattice ordered. Lattice ordered vector spaces are called vector lattices or Riesz spaces. The first results go back to F. Riesz (1929 and 1936), L. Kan­ torovitch (1935) and H. Freudenthal (1936).

User’s Reviews

Editorial Reviews: From the Back Cover The book deals with the structure of vector lattices, i.e. Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible. Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the oridinary Lebesgue integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces, Banach lattices and positive operators are mentioned only briefly, or even not at all. The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were not known ten years ago.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Seems like a reasonable text as an introduction to Riesz spaces. The book jacket advertises that it contains new developments in the field, but the structure of the book makes it difficult to determine where the new results are located.Also, the short list of other books on Riesz spaces included at the back of the book is very short and seems not to include any in-depth resources for the new results.

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