Barrelled Locally Convex Spaces
This book is a systematic treatment of barrelled spaces, and of structures in which barrelledness conditions are significant. It is a fairly self-contained study of the structural theory of those spaces, concentrating on the basic phenomena in the theory, and presenting a variety of functional-analytic techniques.
Beginning with some basic and important results in different branches of Analysis, the volume deals with Baire spaces, presents a variety of techniques, and gives the necessary definitions, exploring conditions on discs to ensure that they are absorbed by the barrels of the space.
The abstract theory of barrelled spaces is then presented, as well as local completeness and its applications to the inheritance of the Mackey topology to subspaces. Further discussed is the abstract study of bornological and ultrabornological spaces; B- and Br-completeness; inductive limits; strong barrelledness conditions; characterizations of barrelled, bornological and (DF)-spaces in the context of spaces of type C(X); the stability of barrelledness conditions of topological tensor products and the related questions of commutability of inductive limits and tensor products; and the holomorphically significant properties of locally convex spaces as developed by Nachbin and others.
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CHAPTER 8 INDUCTlVE LIMIT TOPOLOGIES
CHAPTER 9 STRONG BARRELLEDNESS CONDITIONS
CHAPTER 10 LOCALLY CONVEX PROPERTIES OF THE SPACE OF CONTINUOUS FUNCTIONS ENDOWED WITH THE COMPACTOPEN T...
ABBREVIATIONS and SYMBOLS
0-n ghb absolutely convex subset absorbing According B-complete Baire space Banach disc Banach space barrel led barrelled space basis of 0-nghbs bornivorous bornological space bounded set bounded subset canonical injection canonical surjection Cauchy Clearly closed absolutely convex closed graph theorem closed subspace closed unit ball closure coarser codimension coincides compact subset complemented complete space conclusion follows contains continuous linear continuous norm contradiction converges Corollary countable defined denote equicontinuous exists a positive finite Fréchet space h-bornological Hausdorff hence homomorphism hyperplane increasing sequence inductive limit infinite isomorphic Lemma Let E,t Let F LF)-space linear form linear mapping linear space linearly independent locally bounded locally complete locally convex space locally convex topology Mackey space normed space null sequence numbers Observation positive integer Proof Proposition quasibarrelled quotient resp satisfies seminorm separable space E,t space F strong dual suppose surjection tensornorm ultrabornological vector WALDIVIA