Topological Vector Spaces Ii

In the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces im portant in analysis. It took me nearly twenty years to fulfill this promise, at least to some extent. To the six chapters of Volume One I added two new chapters, one on linear mappings and duality (Chapter Seven), the second on spaces of linear mappings (Chapter Eight). A glance at the Contents and the short introductions to the two new chapters will give a fair impression of the material included in this volume. I regret that I had to give up my intention to write a third chapter on nuclear spaces. It seemed impossible to include the recent deep results in this field without creating a great further delay. A substantial part of this book grew out of lectures I held at the Mathematics Department of the University of Maryland· during the academic years 1963-1964, 1967-1968, and 1971-1972. I would like to express my gratitude to my colleagues J. BRACE, S. GOLDBERG, J. HORVATH, and G. MALTESE for many stimulating and helpful discussions during these years. I am particularly indebted to H. JARCHOW (Ziirich) and D. KEIM (Frankfurt) for many suggestions and corrections. Both have read the whole manuscript. N. ADASCH (Frankfurt), V. EBERHARDT (Miinchen), H. MEISE (Diisseldorf), and R. HOLLSTEIN (Paderborn) helped with important observations.

It is the author's aim to give a systematic account of the most im portant ideas, methods and results of the theory of topological vector spaces. After a rapid development during the last 15 years, this theory has now achieved a form which makes such an account seem both possible and desirable. This present first volume begins with the fundamental ideas of general topology. These are of crucial importance for the theory that follows, and so it seems necessary to give a concise account, giving complete proofs. This also has the advantage that the only preliminary knowledge required for reading this book is of classical analysis and set theory. In the second chapter, infinite dimensional linear algebra is considered in comparative detail. As a result, the concept of dual pair and linear topologies on vector spaces over arbitrary fields are intro duced in a natural way. It appears to the author to be of interest to follow the theory of these linearly topologised spaces quite far, since this theory can be developed in a way which closely resembles the theory of locally convex spaces. It should however be stressed that this part of chapter two is not needed for the comprehension of the later chapters. Chapter three is concerned with real and complex topological vector spaces. The classical results of Banach's theory are given here, as are fundamental results about convex sets in infinite dimensional spaces.

"The most readable introduction to the theory of vector spaces available in English and possibly any other language."—J. L. B. Cooper, MathSciNet ReviewMathematically rigorous but user-friendly, this classic treatise discusses major modern contributions to the field of topological vector spaces. The self-contained treatment includes complete proofs for all necessary results from algebra and topology. Suitable for undergraduate mathematics majors with a background in advanced calculus, this volume will also assist professional mathematicians, physicists, and engineers.The precise exposition of the first three chapters—covering Banach spaces, locally convex spaces, and duality—provides an excellent summary of the modern theory of locally convex spaces. The fourth and final chapter develops the theory of distributions in relation to convolutions, tensor products, and Fourier transforms. Augmented with many examples and exercises, the text includes an extensive bibliography.Reprint of the Addison-Wesley Publishing Company, Reading, Massachusetts, 1966 edition.

With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn-Banach theorem. This edition explores the theorem's connection with the axiom of choice, discusses the uniqueness of Hahn-Banach extensions, and includes an entirely new chapter on v

This book provides an introduction to the theory of topological vector spaces, with a focus on locally convex spaces. It discusses topologies in dual pairs, culminating in the Mackey-Arens theorem, and also examines the properties of the weak topology on Banach spaces, for instance Banach’s theorem on weak*-closed subspaces on the dual of a Banach space (alias the Krein-Smulian theorem), the Eberlein-Smulian theorem, Krein’s theorem on the closed convex hull of weakly compact sets in a Banach space, and the Dunford-Pettis theorem characterising weak compactness in L1-spaces. Lastly, it addresses topics such as the locally convex final topology, with the application to test functions D(Ω) and the space of distributions, and the Krein-Milman theorem. The book adopts an “economic” approach to interesting topics, and avoids exploring all the arising side topics. Written in a concise mathematical style, it is intended primarily for advanced graduate students with a background in elementary functional analysis, but is also useful as a reference text for established mathematicians.

Topological Vector Spaces, Distributions and Kernels discusses partial differential equations involving spaces of functions and space distributions. The book reviews the definitions of a vector space, of a topological space, and of the completion of a topological vector space. The text gives examples of Frechet spaces, Normable spaces, Banach spaces, or Hilbert spaces. The theory of Hilbert space is similar to finite dimensional Euclidean spaces in which they are complete and carry an inner product that can determine their properties. The text also explains the Hahn-Banach theorem, as well as the applications of the Banach-Steinhaus theorem and the Hilbert spaces. The book discusses topologies compatible with a duality, the theorem of Mackey, and reflexivity. The text describes nuclear spaces, the Kernels theorem and the nuclear operators in Hilbert spaces. Kernels and topological tensor products theory can be applied to linear partial differential equations where kernels, in this connection, as inverses (or as approximations of inverses), of differential operators. The book is suitable for vector mathematicians, for students in advanced mathematics and physics.