片断：Finite-DifferenceEquations1.1AMYTHICALFIELDImaginethatagraduatestudentgoestoameadowonthefirstdayofMay，walksthroughthemeadowwavingaflynet，andcountsthenumberoffliescaughtinthenet.Sherepeatsthisritualforseveralyears，followingupontheworkofpreviousgraduatestudents.TheresultingmeasurementsmightlooklikethegraphshowninFigure1.1.Thegraduatestudentnotesthevariabilityinhermeasurementsandwantstofindoutiftheycontainanyimportantbiologicalinformation.Severaldifferentapproachescouldbetakentostudythedata.Thestudentcoulddostatisticalanalysesofthedatatocalculatethemeanvalueortodetectlong-termtrends.Shecouldalsotrytodevelopadetailedandrealisticmodeloftheecosystem，takingintoaccountsuchfactorsasweather，predators，andtheflypopulationsinpreviousyears.Orshecouldconstructasimplifiedtheoreticalmodelforflypopulationdensity.Stickingtowhatsheknows，thestudentdecidestomodelthepopulationvariabilityintermsofactualmeasurements.Thenumberoffliesinonesummerdependsonthenumberofeggslaidthepreviousyear.Thenumberofeggslaiddependsonthenurnberoffliesaliveduringthatsummer.Thus，thenumberoffliesinonesummerdependsonthenumberoffliesintheprevioussummer.Innaathematicaltenns，thisisarelationship，orfunction，Thisequationsayssimplythatthenumberoffliesinthet 1summerisde-terminedby（orisafunctionof）thenumberoffliesinsummert，whichistheprevioussummer.Equationsofthisform，whichrelatevaluesatdiscretetimes（e.g.，eachMay），arecalledfinite-differenceequations.N，iscalledthestateofthesystemattimet.Weareinterestedinhowthestatechangesintime：thedynamicsofthesystem.Sincethereal-worldecosystemiscomplicatedandsincethemeasurementsareimperfect，wedonotexpectamodellikeEq.1.1tobeabletoduplicateexactlytheactualflypopulationmeasurements.Forexample，birdseatflies，sothepopulationoffliesisinfluencedbythebirdpopulation，whichitselfdependsonacomplicatedarrayoffactors.TheassumptionbehindEq.1.1isthatthenumberoffliesinyeart 1dependssolelyonthenumberoffliesinyeart.Whilethisisnotstrictlytrue，itmayserveasaworkingapproximation.Theproblemnowistofigureoutanappropriatefonnforthisdependencethatisconsistentwiththedataandthatencapsulatestheimportantaspectsofflypopulationbiology.

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This book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding， for instance， my Undergraduate Analysis. I assume that the reader is acquainted with notions of uniform convergence and the like.　In this third edition， I have reorganized the book by covering integration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises， as well as some material about integration on the real line （e.g. on Dirac sequence approximation and on Fourier analysis）， and some material on functional analysis （e.g. the theory of the Gelfand transform in Chapter XVI）. These upgrade previous exercises to sections in the text.

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全国哲学社会科学“八五”规划中华社科基金研究课题。

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