Topological Vector Spaces I

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.

One Fundamentals of General Topology.- § 1. Topological spaces.- § 2 . Nets and filters.- § 3. Compact spaces and sets.- § 4. Metric spaces.- § 5. Uniform spaces.- § 6. Real functions on topological spaces.- Two Vector Spaces over General Fields.- § 7. Vector spaces.- § 8. Linear mappings and matrices.- § 9. The algebraic dual space. Tensor products.- § 10. Linearly topologized spaces.- § 11. The theory of equations in E and E*.- § 12. Locally linearly compact spaces.- § 13. The linear strong topology.- Three Topological Vector Spaces.- § 14. Normed spaces.- § 15. Topological vector spaces.- § 16. Convex sets.- § 17. The separation of convex sets. The Hahn-Banach theorem.- Four Locally Convex Spaces. Fundamentals.- § 18. The definition and simplest properties of locally convex spaces.- § 19. Locally convex hulls and kernels, inductive and projective limits of locally convex spaces.- § 20. Duality.- § 21. The different topologies on a locally convex space.- § 22. The determination of various dual spaces and their topologies.- Five Topological and Geometrical Properties of Locally Convex Spaces.- § 23. The bidual space. Semi-reflexivity and reflexivity.- § 24. Some results on compact and on convex sets.- § 25. Extreme points and extreme rays of convex sets.- § 26. Metric properties of normed spaces.- Six Some Special Classes of Locally Convex Spaces.- § 27. Barrelled spaces and Montel spaces.- § 28. Bornological spaces.- § 29. (F)- and (DF)-spaces.- § 30. Perfect spaces.- § 31. Counterexamples.- Author and Subject Index.