Moscow Lectures 4 Vladimir I. Bogachev Oleg G. Smolyanov Real and Functional Analysis Moscow Lectures Volume 4 SeriesEditors LevD.Beklemishev,Moscow,Russia VladimirI.Bogachev,Moscow,Russia BorisFeigin,Moscow,Russia ValeryGritsenko,Moscow,Russia YulyS.Ilyashenko,Moscow,Russia DmitryB.Kaledin,Moscow,Russia AskoldKhovanskii,Moscow,Russia IgorM.Krichever,Moscow,Russia AndreiD.Mironov,Moscow,Russia VictorA.Vassiliev,Moscow,Russia ManagingEditor AlexeyL.Gorodentsev,Moscow,Russia Moreinformationaboutthisseriesathttp://www.springer.com/series/15875 Vladimir I. Bogachev • Oleg G. Smolyanov Real and Functional Analysis 123 Vladimir I. Bogachev Oleg G. Smolyanov Department of Mechanics Department of Mechanics and Mathematics and Mathematics Moscow State University Moscow State University Moscow, Russia Moscow, Russia Higher School of Economics Moscow Institute of Physics National Research University and Technology Moscow, Russia Dolgoprudnyi, Russia This book is an expanded and revised version of the work first published in Russian in 2009 (1st edition) and 2011 (2nd edition) with the publisher Regular and Chaotic Dynamics, Moscow - Izhevsk, under the title Действительный и функциональный анализ: университетский курс. ISSN 2522-0314 ISSN 2522-0322 (electronic) Moscow Lectures ISBN 978-3-030-38218-6 ISBN 978-3-030-38219-3 (eBook) https://doi.org/10.1007/978-3-030-38219-3 Mathematics Subject Classification (2010): 28-01, 46-01, 46A03, 46Bxx, 46Fxx, 47-01 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: https://www.istockphoto.com/de/foto/panorama-der-stadt-moskau-gm490080014- 75024685, with kind permission This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Book Series Moscow Lectures YouholdavolumeinatextbookseriesofSpringerNaturededicatedtotheMoscow mathematical tradition. Moscow mathematics has very strong and distinctive fea- tures.Thereareseveralreasonsforthis,allofwhichgobacktogoodandbadaspects of Soviet organization of science. In the twentieth century, there was a veritable galaxyofgreatmathematiciansinRussia,whileitsohappenedthattherewereonly few mathematicalcenters in which these experts clustered. A major one of these, andperhapsthemostinfluential,wasMoscow. There are three major reasons for the spectacular success of Soviet mathemat- ics: 1. Significant support from the governmentand the high prestige of science as a profession.Bothfactorswererelatedtotheprocessofrapidindustrializationin theUSSR. 2. Doing research in mathematics or physics was one of very few intellectual activities thathad nomandatoryideologicalcontent.Many would-becomputer scientists,historians,philosophers,oreconomists(andevenartistsormusicians) becamemathematiciansorphysicists. 3. TheIronCurtainpreventedinternationalmobility. These are specific factors that shaped the structure of Soviet science. Certainly, factors(2)and(3)aremoreonthenegativesideandcannotreallybecalledfavorable but they essentially came together in combination with the totalitarian system. Nowadays, it would be impossible to find a scientist who would want all of the threefactorstobebackintheirtotality.Ontheotherhand,thesefactorsleftsome positiveandlonglastingresults. An unprecedented concentration of many bright scientists in few places led eventuallytothedevelopmentofaunique“Sovietschool”.Ofcourse,mathematical schools in a similar sense were formed in other countries too. An example is the Frenchmathematicalschool,whichhasconsistentlyproducedfirst-rateresultsover alongperiodoftimeandwhereanextensivedegreeofcollaborationtakesplace.On the other hand, the British mathematicalcommunitygave rise to many prominent successes but failed to form a “school” due to a lack of collaborations. Indeed, a v vi PrefacetotheBookSeriesMoscowLectures school as such is not only a large group of closely collaborating individuals but alsoagroupknittightlytogetherthroughstudent-advisorrelationships.IntheUSA, whichiscurrentlytheworldleaderintermsofthelevelandvolumeofmathematical research, the level of mobility is very high, and for this reason there are no US mathematical schools in the Soviet or French sense of the term. One can talk not only about the Soviet school of mathematics but also, more specifically, of the Moscow, Leningrad, Kiev, Novosibirsk, Kharkov, and other schools. In all these places, there were constellationsof distinguished scientists with large numbersof students,conductingregularseminars.Thesedistinguishedscientistswereoftennot merelyadvisorsandleaders,butoftentheyeffectivelybecamespiritualleadersina verygeneralsense. AcharacteristicfeatureoftheMoscowmathematicalschoolisthatitstressesthe necessity for mathematicians to learn mathematics as broadly as they can, rather thanfocusingonanarrowfieldinordertogetimportantresultsassoonaspossible. The Moscow mathematical school is particularly strong in the areas of alge- bra/algebraicgeometry,analysis,geometryandtopology,probability,mathematical physics and dynamical systems. The scenarios in which these areas were able to developinMoscowhavepassedintohistory.However,itispossibletomaintainand develop the Moscow mathematical tradition in new formats, taking into account modern realities such as globalization and mobility of science. There are three recently created centers—the Independent University of Moscow, the Faculty of Mathematics at the National Research University Higher School of Economics (HSE) and the Center for Advanced Studies at Skolkovo Institute of Science and Technology(SkolTech)—whosemissionistostrengthentheMoscowmathematical traditioninnewways.HSEandSkolTechareuniversitiesofferingofficiallylicensed full-timeeducationalprograms.Mathematicalcurriculaattheseuniversitiesfollow not only the Russian and Moscow tradition but also new global developments in mathematics. Mathematical programs at the HSE are influenced by those of the Independent University of Moscow (IUM). The IUM is not a formal university; it is rather a place where mathematicsstudentsof differentuniversitiescan attend specialtopicscoursesaswellascourseselaboratingthecorecurriculum.TheIUM was the main initiator of the HSE Faculty of Mathematics. Nowadays, there is a closecollaborationbetweenthetwoinstitutions. While attempting to further elevate traditionally strong aspects of Moscow mathematics, we do not reproduce the former conditions. Instead of isolation and academic inbreeding, we foster global sharing of ideas and international cooperation. An important part of our mission is to make the Moscow tradition ofmathematicsatauniversitylevelapartofglobalcultureandknowledge. The“MoscowLectures”seriesservesthisgoal.Ourauthorsaremathematicians of different generations. All follow the Moscow mathematical tradition, and all teach or have taught university courses in Moscow. The authors may have taught mathematics at HSE, SkolTech, IUM, the Science and Education Center of the Steklov Institute, as well as traditionalschools like MechMath in MGU or MIPT. Teaching and writing styles may be very different. However, all lecture notes are PrefacetotheBookSeriesMoscowLectures vii supposedtoconveyalivedialogbetweentheinstructorandthestudents.Notonly personalitiesofthelecturersareimprintedinthesenotes,butalsothoseofstudents. We hope that expositions published within the “Moscow lectures” series will provideclearunderstandingofmathematicalsubjects,usefulintuition,andafeeling oflifeintheMoscowmathematicalschool. Moscow,Russia IgorM.Krichever VladlenA.Timorin MichaelA.Tsfasman VictorA.Vassiliev Preface Wherefore I perceive that there is nothing better, thanthatamanshouldrejoiceinhisownworks; for thatishisportion:forwhoshallbringhimtoseewhat shallbeafterhim? Ecclesiastes. Thisbookistheresultofthesubstantialreworkingandenlargingoflectureson realandfunctionalanalysisgivenovertheyearsbytheauthorsattheDepartment ofMechanicsandMathematicsofMoscowStateUniversityandalsoattheFaculty ofMathematicsoftheHigherSchoolofEconomicsinMoscow. A course on the theory of functions and functional analysis, with the title “Analysis-III”, was first introduced in the university curriculum (for all students ofthethirdyear,notasa8000-course)atMekhMatinthe1940sontheinitiativeof A.N. Kolmogorov, who became the first lecturer together with S.V. Fomin, who gavethiscourseattheDepartmentofPhysics. Severalyearslatertheclassicaltext- book[331]byKolmogorovandFominwaspublished,whichtothepresentdayre- mainsamongthebestuniversitycourses. LaterthiscoursewasgivenatMekhmat by I.M. Gelfand, G.E. Shilov and other eminent mathematicians. Analysis-III gathered several previously existing courses of the theory of functions of a real variable, integral equations, and also elements of variational calculus. At present atMekhmatandatothertopmathematicaldepartments,thecourseofrealanalysis (more precisely, the theory of the Lebesgue integral) and variational calculus (op- timal control) have again become separate courses (though, there exist university programmes of functional analysis that include the Lebesgue integral). On the other hand, advanced courses of functional analysis now include, unlike in Kol- morogov’s time, the spectral theory of selfadjoint operators and elements of the theoryofSobolevspaces. Severalexcellentmorerecentbooksonfunctionalanalysisaresuitableastext- books, such as Reed, Simon [502], Rudin [522], Werner [628], and many others mentioned in the bibliographic comments (which list several hundred titles that we looked through over several decades in the libraries of dozens of universities and mathematical institutes all over the world), as well as fundamental treatises ix x Preface of a reference nature, such as Dunford, Schwartz [164], Edwards [171], and Kan- torovich,Akilov[312]. However,mostofthemeitherdonotcoverallthematerial includedinadvancedcoursesofrealandfunctionalanalysis(inparticular,needed for PhD students) or, at the opposite extreme, they contain too much additional materialnotseparatedfrom thenecessaryminimumandbytheir organization and style are not textbooks, but rather advanced courses for researchers and readers already familiar with the subject. This book aims at giving a modern exposition of all the material from functional analysis traditionally presented in a one-year orone-and-a-halfyearcourse(includingrealanalysis)andnecessaryforthestudy of partial differential equations, mathematical physics, optimal control, and the theoryofstochasticprocesses. There are three levels of exposition in this book oriented towards different categories of readers: 1) a relatively standard course for mathematical depart- ments of universities; this part occupies less than one half of the whole book; the corresponding material is distributed over the main sections of Chapters 1–9 and includes approximately one half of all the exercises; 2) complementary mate- rial for advanced students and PhD students which includes the main sections of Chapters 10–12 and a small number of subsections from the additional sections (called“Complementsandexercises”)inotherchaptersaswellassomeexercises; 3) finally, more special information deserving the attention of more professional readers(thisinformationispresentedinthesections“Complementsandexercises” inallchapters). Weemphasizethatthecorematerialconnectedwithlevels1)and2)occupies less than 300 pages, i.e., somewhat less than half of the book (this material corre- spondstoapproximately100academiclecturinghours,andalecturehourenables onetocoveronaverage2–3pagesofbooktext). This structure makes the present text different from many existing books on the subject, although we have been obviously influenced by many of these books. Ourgoalistoofferamoderntextbookforabroadreadershipcoveringallnecessary materialinrealandfunctionalanalysisforadvancedstudentsandPhDstudents. Of course,asmentionedin3),wealsostepoutofanyformalcurriculumandpresent additionalusefulandinterestingfactswhicheitherdemonstratetheconnectionsof this area with other areas and applications, or belong to the classical foundations of functional analysis and admit a relatively simple presentation (for example, we give with complete proofs the Eberlein–Shmulian and Krein–Milman Theorems). However, all such facts are placed in complementary sections and can be omitted on the first reading. In addition, these complements are not used in the main part of the book and are independent of each other. Naturally, certain questions are presented in the aforementioned treatises in more detail and in greater depth, and there are interesting applications discussed in other textbooks but not mentioned at all here. As follows from what has been said, we have not aimed at creating a universal encyclopedia of functional analysis. The bibliographic comments give further references for reading in many classical and modern directions, which along with complementary sections can make this book useful also for a broader audienceofresearchersinthemostdiversefields.