PrefaceTheaimof thisbookistoreviewthetheoryof somebasictopicsinAnalysis and accompany the theory with problems and their solutions.Withtheproblemsthereadercantesthis/herunderstandingof thetheory and also discover extensions of the theory and additional resultswhich are not so standard in the literature. The topics coveredspan amoreorlessstandardadvancedundergraduateandgraduatecurricu-luminAnalysis. Moreprecisely,wefocusonthefollowingsubjects:1. MetricSpaces2. Topological Spaces(herethereisalsosomeintroductorymaterialonAlgebraicTopology)3. Measure,IntegrationandMartingales(includingLp-spaces)4. Measure andTopology(covering issues concerningthe interplaybetweenmeasuretheoryandtopology)5. Functional Analysis (with emphasis on basic Banach spacetheory).Each one of the above ve subjects corresponds to a dierent chap-ter. In the rst part of eachchapter,we present the basic theory,withall themaindenitionsandresults. Wealsoincludecommentsandremarksexpandingontheconceptsandresults, butnoproofs. Thisreviewmaterial will helpthereaderrefreshhis/herknowledgeofthetheory before tacklingthe problems. In eachchapterthe theory is fol-lowedbyproblemsandtheirdetailedsolution. Ineachchapter,thereareat least 170problems, markedwith, or accordingtotheirdiculty. Someofthoseproblemscomplementthetheory,whiletherestcheckthereadersunderstandingofthetheory. Westronglyencouragethereadertoputsomesubstantialpersonaleortintryingtosolvetheproblem,beforelookingatthesolution. Otherwise,theywill get no benet from reading the book. On the other hand, a seriousvvi Prefaceeort to solve the problem themselves and subsequently compare theirsolutionwiththeoneprovided(orcheckingtoseewheretheyfailedtocomeupwiththerightargumentstoproduceasolution)willhelpthemtoachieveasolidunderstandingofthetheory.Itisnoteasytoprovidetheoriginofeachoftheproblemsandoftheir solutions. They canbe tracedas problems included in the booksmentionedintheliterature(where they arestatedwithout proofs), orthey can be found in the problem books listed in the References, or theyare standard exercises in the public domain, or they were accumulatedthroughtheyearsfromteachingundergraduateandgraduatecoursesontheseorcloselyrelatedsubjects. Inany case,the books mentionedintheReferencescanbeavaluablesourceforadditional theoreticalmaterial andmore problems. Our bookis onlythe startingpoint(helpfulwehope).Theauthors express their gratitude toSpringer, NewYork, foritshighlyprofessionalassistanceandaboveallwewanttothankoureditor, Mrs. ElizabethLoew, forherstrongmoral support, patienceandkindunderstanding.Krakow,Poland LeszekGasi nskiAthens,Greece NikolaosS.Papageorgiouhttp://www.springer.com/978-3-319-06175-7