Functional Analysis

Thepresent book is based on lectures given by theauthor at the University ofTokyo during the past ten years. It is intended as a textbooktobestudiedbystudents ontheirownortobeusedinacourse on Functional Analysis, i. e. , the generaltheory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessaryprerequisitesforthereadingofthisbookaresummarized, with or without proof, in Chapter 0 under titles: Set Theory,Topo- logical Spaces,MeasureSpacesand Linear Spaces. Then,startingwith the chapter on Semi-norms, a general theory of Banach and Hilbert spacesispresentedinconnectionwiththetheoryofgeneralizedfunctions ofS. L. SOBOLEVandL. SCHWARTZ. Whilethebookisprimarilyaddressed tograduatestudents,itishopedit mightproveusefultoresearchmathe- maticians, both pure and applied. The reader may pass e. g. from ChapterIX(AnalyticalTheoryofSemi-groups)directlytoChapterXIII (ErgodicTheoryandDiffusionTheory)andtoChapterXIV(Integration of the Equation of Evolution).

Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented inthe form of theappendices to ChapterVand ChapterX, respectively. These mightbeskipped forthefirst readingbythosewho areinterestedratherintheapplicationoflinearoperators. In the preparation of the present book, the author has received valuable advice and criticism from many friends. Especially, Mrs, K. HILLEhaskindlyreadthroughthemanuscript aswellasthegalley and page proofs. Without her painstaking help, this book could not have been printed in the present style in the language which was not spoken to the author in the cradle. The author owes very much to his old friends, Professor E. HILLE and Professor S. KAKUTANI of YaleUniversityandProfessorR. S. PHILLIPSofStanfordUniversityfor thechanceto stay in their universities in 1962, which enabled him to polishthegreaterpartofthemanuscript ofthisbook,availing himself of their valuable advice. Professor S. ITOand Dr. H.