Barrelledness, Baire-like and (LF)-spaces

The purpose of the present book is to provide a modern approach to a development on the field of locally convex spaces that took place over the last twenty years. The roots of this subject lie in a number of weakened barrelledness concepts that are introduced in the first chapter. The methods and results derived therein will turn out to be of great value throughout this treatise. Another section on varieties of locally convex spaces was included mainly for aesthetic reasons (that is, because we think it's a beautiful theory). The concept of varieties is intended to be a unifying element in this work and to present other lines of research that grow out of the same origins as the subjects we are essentially concerned with. A major part of the book deals with several classes of spaces that are "more than barrelled" but "less than Baire". In a chapter on linear Baire spaces at the beginning of these considerations we present some characterizations of the Baire property that are usually not included in textbooks on locally convex spaces. The final section of the present book is dedicated to the study of inductive limits of locally convex spaces, particularly (LF)- and (LB)-spaces. Apart from a classification of these spaces by means of the weakened Baire properties introduced in the chapters before, it presents several quite recent results on metrizable and normable (LF)-spaces. For instance, we prove here that many of the classical Banach spaces contain dense (LF)-subspaces. Despite the fact that certain aspects of the subject of this book are also treated in the excellent monographs [56] by M. Valdivia and [5] by J. Bonet and P. Perez Carreras (to which we refer the reader for further study), we believe that this book is the first self-contained introduction to this theory, especially concerning a modern exposition of the theory of (LF)-spaces and the interrelation of generalized inductive limits, "almost barrelled" and "almost Baire" spaces.