ISSN sub h tt p : divided by hline to the power of Banach hline divided by B a n sub : slash 1 slash w from period m to 735 w\ begin w hyphen{ a l i 8787 g n ∗} open parenthesis a to the power of electronic sub t h hyphen a na ly s to the power of closing parenthesis i s periodISSN o ˆ rg{ toBanach the power{\ off r a c c{\ J hr sub u l e J{3em to the}{0.4 power pt }}{ of periodB Math a o n u}}} r n a{ periodh \ subf r a c l{ Analt t o f pperiod : sub}{\ Mr 1 ua l e open{3em}{0.4 pt }}}ˆ{ a parenthesisc J { th h} toˆ{ the. power} { ofJ 2007} Math sub e m{ ao t i to u the power r ofn closing a } parenthesis. { l comma} Anal no c{ a lo to the f power} . of{ periodM } 1 sub1 { a } ( { t h }ˆ{ 2007 }ˆ{ ), } { e m a t i } no { c a l }ˆ{ . } 1 { A } , n ˆ{ 1 }ˆ{ −− } { a }ˆ{ 1 } { l }ˆ{ 0 } { y } Banach ac J. Math . Anal . 1 (2007), no. 1 ,n1−−10 sis A comma n to the power of 1 subBa to n the powerhJ ofourn endash a l subof lM toa theth powere ma of ti 1 subca ly toA thea powerl y of 0 s i s s i s } { : { ISSN/ } 1ttp: { / } w ˆ{ . m } { 735 { w w } − 8787 } ( { a }ˆ{ e l e c t r o n i c }ˆ{ ) } { t ON .. STEFAN .. BANACHh .. AND:/1 ..w SOME.m .. OF(electronic .. HIS) .. RESULTSis.org / 735ww−8787 a t h−a nalys h KRZYSZTOF− a na CIESIELSKI l y s to} thei power s of 1 . o rg } \endDedicated{ aON l i g n ∗} to Professor STEFAN Themistocles BANACH M period Rassias AND SOME OF HIS RESULTS Submitted by P period Enflo \ centerlineAbstract period{ON ..\quad In theSTEFAN paper a short\quad biographyBANACHKRZYSZTOFCIESIELSKI of\ Stefanquad BanachAND \quad comma1 SOME .. a few\quad storiesOF \quad HIS \quad RESULTS } about Banach and the Scottish Caf acute-e and some results that nowadays are named \ [ KRZYSZTOF CIESIELSKI ˆDedicated{ 1 }\ ] to Professor Themistocles M . Rassias by Banach quoteright s surname are presentedSubmitted period by P . Enflo 1 period .. Introduction Abstract . In the paper a short biography of Stefan Banach , a few stories Banach Journal of Mathematical Analysis i s named after one of the most .... out hyphen \ centerline { Dedicatedabout Banach to Professor and the Scottish Themistocles Cafe ´ and some M .results Rassias that nowadays} are named standing mathematicians in the XXthby century Banach comma ’ s surname Stefan Banach are presented period . .. Thus it is natural to .. recall .. in .. the .. first .. i ssue .. of the j ournal .. some .. information .. about .. Banach period .. A \ centerline { Submitted by P . Enflo 1} . Introduction very shortBanach biography Journal comma of some Mathematical of his most eminent Analysis results i s named and some after stories one will of be the most out - presented in this article period \ centerlinestanding{ Abstract mathematicians . \quad In in the the paperXXth century a short , Stefan biography Banach of . Stefan Thus Banach it is natural , \quad a few stories } 2 periodto .. Banach recall and the in Scottish the .. Caf first e-acute i ssue of the j ournal some information Stefan Banach was born in the Polish city Krak acute-o w on 30 March comma .. 1 892 period .. Some \ centerlineabout{ about Banach Banach . and A the very Scottish short biography Caf $ ,\ someacute of{e his} $ most and eminent some results results and that nowadays are named } sources ..some give .. stories the .. date will .. be 20 presented .. March comma in this .. however article ... it .. was .. checked .. open parenthesis in particular comma .. in the \ centerline {by Banach ’2 s . surnameBanach are and presented the Scottish . } Caf e´ parish sourcesStefan by the Banach author .... was open born parenthesis in the open Polish square city bracket Krak 1 2o´ closingw on square 30 March bracket , closing 1 892 parenthesis . .... and by Roman Ka suppress-l u z-dotaccent a open parenthesis open square bracket 26 closing square bracket closing parenthesis closing \ centerlineSome{1 . sources\quad Introduction give the date} 20 March , however it was checked parenthesis ....( in that particular the date , in the 30 March is correct period \noindentparishBanach sources Journal by the of author Mathematical ( [ 1 2 ] Analysis ) and by i Roman s named Ka after lu z˙ a one ( [ 26 of ] the ) ) most that \ h f i l l out − Banach quoterightthe date s parents were not married period .. It is not much known about his mother comma Katarzyna30 Banach March comma is correct after . whom he had a surname period .. It was j ust recently discovered \noindentopen parenthesisstanding see open mathematicians square bracket 25 in closing the XXth square century bracket closing , Stefan parenthesis Banach that . she\quad was aThus maid itor servant is natural and to \quad r eBanach c a l l \quad ’ s parentsin \quad werethe not\ marriedquad f i . r s t It\quad is noti much ssue known\quad aboutof the his j mother ournal \quad some \quad information \quad about \quad Banach . \quad A Banach quoteright, s father comma Stefan Greczek comma who veryhline short biography , some of his most eminent results and some stories will be presentedKatarzyna in this Banach article , after . whom he had a surname . It was j ust recently discovered Date : Received( see [ 25: .. ] 1 ) 4 that June she 2007 was semicolon a maid Accepted or servant : 25 September and Banach 2007 ’ period s father , Stefan Greczek , 2000 Mathematicswho Subject Classification period .. Primary 0 1 A 60 semicolon Secondary 46 hyphen 3 comma 46 B 25 period \ centerlineKey words and{2 phrases. \quad periodBanach .. Banach and thecomma Scottish functional\ analysisquad Caf comma $ Scottish\acute{ Cafe} acute-e$ } sub comma Lvov School of mathe hyphen Stefanmatics Banach comma Banach was born space in period the Polish city Krak $ \acute{o} $ w on 30 March , \quad 1 892 . \quad Some s o1 u r c e s \quad g i v e \quad the \quad date \quad 20 \quad March , \quad however \quad i t \quad was \quad checked \quad ( in particular , \quad in the Date : Received : 1 4 June 2007 ; Accepted : 25 September 2007 . \noindent parish sources by the author \ h f i l l ( [ 1 2 ] ) \ h f i l l and by Roman Ka \ l u $ \dot{z} $ 2000 Mathematics Subject Classification . Primary 0 1 A 60 ; Secondary 46 - 3 , 46 B 25 . a ( [ 26 ] ) ) \ h f i l l that the date Key words and phrases . Banach , functional analysis , Scottish Cafe ´, Lvov School of mathe - matics , Banach space . \noindent 30 March is correct . 1 \ hspace ∗{\ f i l l }Banach ’ s parents were not married . \quad It is not much known about his mother ,

\noindent Katarzyna Banach , after whom he had a surname . \quad It was j ust recently discovered ( see [ 25 ] ) that she was a maid or servant and Banach ’ s father , Stefan Greczek , who

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline {Date : Received : \quad 1 4 June 2007 ; Accepted : 25 September 2007 . }

\ centerline {2000 Mathematics Subject Classification . \quad Primary 0 1 A 60 ; Secondary 46 − 3 , 46 B 25 . }

Key words and phrases . \quad Banach , functional analysis , Scottish Caf $ \acute{e} { , }$ Lvov School of mathe − matics , Banach space .

\ centerline {1 } 2 .. K period CIESIELSKI \noindentwas a soldier2 \ openquad parenthesisK.CIESIELSKI probably assigned orderly to the officer under whom Katarzyna was a servant closing parenthesis .. could not .. marry Katarzyna because of some military rules period .. Banach \noindent2was K .a CIESIELSKI soldier ( probably assigned orderly to the officer under whom Katarzyna was was grewwas up with a soldier the owner ( probably of the laundry assigned Franciszka orderly P suppress-lto the officer owa and under her niecewhom Maria Katarzyna period was Banach attended school in Krak o-acute w .. and took there final exams period .. He was .. in hyphen \noindentaa servant servant ) could) \quad notcould marry not Katarzyna\quad marry because Katarzyna of some military because rules of some. Banach military rules . \quad Banach t erestedwas in mathematics grew up with and he the was owner very good of the at laundryit period .. Franciszka However comma P lowa he thoughtand her that niece Maria . wasin this grew .. area up nothing with the .. much owner .. new of .. the can belaundry .. discovered Franciszka .. and .. he P ..\ l decidedowa and to .. studyher niece .. at Maria . Banach attendedBanach attended school school in Krak in Krak $ \acuteo´ w{o} and$ took w \quad thereand final took exams there . He final was exams in . \quad He was \quad in − the Technical- University at .. Lvov .. open parenthesis Politechnika Lwowska closing parenthesis period .. In those t imes both Krak o-acutet erested w and in Lvov mathematics were in the t and erritory he was governed very by good Austro at hyphen it . Hungary However period , he thought.. Therefore that \noindentBanach movedt erested to Lvov period in mathematics .. His studies andthere he were was interrupted very good by the at First it World . \quad However , he thought that in t h i s in\quad thisarea area nothing nothing\quad muchmuch new\quad cannew be\quad discoveredcan be \quad andd i she c o v e decided r e d \quad and \quad he \quad decided to \quad study \quad at War andto Banach study came back at to the Krak Technical o-acute w periodUniversity at Lvov ( Politechnika Lwowska ) theAs a Technical mathematician University comma Banach at \ wasquad selfLvov hyphen\quad taught( period Politechnika .. He did not Lwowska study mathematics ) . \quad commaIn those t imes both Krak $ .\acute In{ thoseo} $ t w imes and both Lvov Krakwere ino´ w the and t Lvov erritory were ingoverned the t erritory by Austro governed− Hungary by . \quad Therefore howeverAustro he attended - Hungary some lectures . at Therefore the Jagiellonian Banach University moved comma to Lvov especially . His de studieshyphen there were Banachl ivered moved by Stanis to suppress-l Lvov . aw\quad ZarembaHis period studies .. In 1 there 9 1 6 comma were ainterrupted very important by event the took First place World period .. Hugo War andinterrupted Banach came by back the First to Krak World $ War\acute and{o Banach} $ w came . back to Krak o´ w . Steinhaus commaAs a mathematicianan outstanding mathematician , Banach was comma self - then taught already . well He known did not comma study spent mathematics some t ime in Krak, however acute-o he w period attended .. Once some comma lectures during at his the evening Jagiellonian walk at the University Planty Park , especially in the centre de - Asof a Krak mathematician o-acute w comma , Banach .. Steinhaus was .. self heard− thetaught words .. . quotedblleft\quad He Lebesgue did not .. integral study quotedblrightmathematics period , .. In those howeverl he ivered attended by Stanis somelaw lectures Zaremba at . the In Jagiellonian 1 9 1 6 , a very University important event , especially took place de . − t imes .. it Hugo Steinhaus , an outstanding mathematician , then already well known , spent lwas ivered a very by modern Stanis mathematical\ l aw Zaremba t erm comma . \quad so SteinhausIn 1 9comma 1 6 a , l a itt very le surprised important comma eventstartedto took place . \quad Hugo some t ime in Krak o´ w . Once , during his evening walk at the Planty Park in Steinhaustalk with two , young an outstanding men who were mathematician speaking about the , Lebesgue then already measure period well known.. These , spent some t ime inthe Krak centre $ \ ofacute Krak{oo´} w$ , w Steinhaus . \quad Once heard , during the words his evening “ Lebesgue walk integralat the ”Planty . Park in the centre two menIn were those Banach t imes and Otto it Nikodym was a very period modern .. Steinhaus mathematical told them aboutt erm a , problem so Steinhaus , a l itt le o fhe Krak was currently $ \acute working{o} on$ comma w , \ andquad a fewSteinhaus days later Banach\quad visitedheard Steinhaus the words and \quad ‘ ‘ Lebesgue \quad integral ’’ . \quad In those t imes \quad i t was a verysurprised modern , started mathematical to talk with t erm two , young so Steinhaus men who were , a speaking l itt le about surprised the Lebesgue , started to presentedmeasure him a correct . so These lut ion two period men were Banach and Otto Nikodym . Steinhaus told talkSteinhaus with realised two young that Banach men who had an were incredible speaking mathematical about thetalent Lebesgue period .. Stein measure hyphen . \quad These two menthem were aboutBanach a and problem Otto he Nikodym was currently . \quad workingSteinhaus on , told and a them few days about later a problem Banach haus wasvisited j ust moving Steinhaus to Lvov and where presented he got a Chair him a period correct .. He so offered lut ion Banach . a position heat was the Technical currently University working period on .. , Thus and Banach a few started days his later academic Banach career visited and t each Steinhaus hyphen and presented himSteinhaus a correct realised so that lut Banach ion . had an incredible mathematical talent . Stein - ing studentshaus comma was j however ust moving he did to not Lvov graduate where period he got a Chair . He offered Banach a position Steinhaus .. used to .. say that the .. discovery of Banach was his greatest .. mathe hyphen \ hspace ∗{\atfthe i l l } TechnicalSteinhaus University realised . that Thus Banach Banach had started an incredible his academic mathematical career and t talent each . \quad S te i n − matical discovery- ing students period .. , It however must be he noted did here not that graduate .. many . outstanding mathematical results are due to Steinhaus period \noindent hausSteinhaus was j ust used moving to say to that Lvov the where discovery he got of a Banach Chair was . \ hisquad greatestHe offered mathe Banach a position There ..- i s matical .. a curious discovery .. story . how It.. Banachmust be got noted .. his here .. Ph that period D many period outstanding .. He .. was mathematical.. being forced to atwrite the .. Technical a .. Ph period University D period .. paper . \quad .. andThus .. take Banach .. the .. examinations started his comma academic .. as .. careerhe .. very and .. quickly t each .. obtained− ing studentsresults , are however due to he Steinhaus did not . graduate . many importantThere results i commas a curious but he kept story saying how that he Banach was not ready got and his perhaps Ph . D . He was he would invent .. something more interesting period .. At .. last the university authorities Steinhausbeing\quad forcedused to to write\quad asay Ph that . D the .\ paperquad discovery and take of Banach the examinations was his greatest , \quad mathe − became nervousas he period very .. Somebody quickly wrote downobtained Banach many quoteright important s remarks results .. on , some but problems he kept comma saying maticaland .. this discovery .. was .. accepted . \quad .. asIt .. an must .. excellent be noted .. Ph here period that D period\quad .. dissertationmany outstanding period .. But mathematical .. an .. exam .. was resultsthat are duehe was to not Steinhaus ready and . perhaps he would invent something more interesting also .. required. At period last .. One the .. university day .. Banach authorities was .. accosted became .. in nervous the .. corridor . Somebody .. and .. asked wrote to .. down go to .. a .. Dean quoteright s .. room comma .. as .. quotedblleft some .. people .. have .. come .. and .. they .. want .. to .. There \quadBanachi s ’\quad s remarksa c u r i o on u s some\quad problemss t o r y how , and\quad thisBanach was got accepted\quad h i s as\quad anPh . D . \quad He \quad was \quad being forced to know .. someexcellent Ph . D . dissertation . But an exam was also required . writmathematical e \quad detailsa \quad commaPh and . D you . \ willquad certainlypaper be\ ablequad toand answer\quad their questionstake \quad quotedblrightthe \quad periodexaminations , \quad as \quad he \quad very \quad q u i c k l y \quad obtained many importantOne day results Banach , but was he kept accosted saying in that the he wascorridor not ready and and asked perhaps to go Banach willinglyto a .. answered Dean ’ s the .. room questions , ascomma “ .. some not .. realising people that have .. he was come j ust .. being and they heexamined would by invent a special\quad commissionsomething that had more come interesting to Lvov for this . purpose\quad periodAt \quad last the university authorities became nervouswant to . \quad knowSomebody some wrote mathematical down Banach details ’ , sand remarks you will\quad certainlyon some be able problems to , In 1 922answer Banach comma their questions aged 30 comma ” . was Banach appointed willingly Professor at answered the Jan Kazimierz the questions Uni hyphen , not andversity\quad at ..t Lvov h i s period\quad ..was After\ thequad Firstaccepted World War\quad .. Polandas got\quad .. backan it\ squad .. independencee x c e l l e n t \quad Ph . D . \quad dissertation . \quad But \quad an \quad exam \quad was a l s o \quadrealisingr e q u i that r e d . \ hequad wasOne j ust\quad beingday \ examinedquad Banach by a was special\quad commissionaccosted that\quad hadin the \quad c o r r i d o r \quad and \quad asked to \quad go and Krakcome o-acute to w Lvov and Lvov for this were purpose again in Poland. period toIn\ factquad commaa \quad .. BanachDean was ’ interested s \quad inroom nothing , \ butquad .. mathematicsas \quad period‘ ‘ some .. He\quad wrote downpeople \quad have \quad come \quad and \quad they \quad want \quad to \quad know \quad some mathematicalIn 1 details 922 Banach , and , agedyou will 30 , was certainly appointed be Professorable to answer at the Jan their Kazimierz questions Uni ’’ . only a small- versity part of his at results Lvov period . .. He After was speaking the First about World mathematics War comma Poland introducing got back it s Banachnew ideas willingly comma .. solving\quad problemsanswered all the the t ime\quad periodq u .. e s Andrzej t i o n s .. , Turowicz\quad not comma\quad who knewrealising Banach that \quad he was j ust \quad being examinedindependence by a special and commission Krak o´ w thatand Lvov hadwere come again to Lvov in Poland for this . purpose . very well commaIn fact used , to say Banach that two was mathematicians interested in should nothing have but fo llowed mathematics Banach all . He wrote In 1 922down Banach only , aged a small 30 ,part was of appointed his results Professor . He was at speaking the Jan Kazimierzabout mathematics Uni − , v e r s i t yintroducing at \quad Lvov new . ideas\quad ,After solving the problems First World all the War t ime\quad .Poland Andrzej got Turowicz\quad back i t s \quad independence and Krak, who $ \acute knew{ Banacho} $ w very and well Lvov , used were to again say that in twoPoland mathematicians . should have fo llowed Banach all In f a c t , \quad Banach was interested in nothing but \quad mathematics . \quad He wrote down only a small part of his results . \quad He was speaking about mathematics , introducing new i d e a s , \quad solving problems all the t ime . \quad Andrzej \quad Turowicz , who knew Banach very well , used to say that two mathematicians should have fo llowed Banach all ON STEFAN BANACH AND SOME OF HIS RESULTS .. 3 \ hspacethe t ime∗{\ andf i l written l }ON STEFAN down everything BANACH he AND said SOME period OF .. ThenHIS RESULTS comma probably\quad the3 majority of what Banach did would have been saved period .. Nevertheless comma his results are incredible period ON STEFAN BANACH AND SOME OF HIS RESULTS 3 \noindentSome of themthe will t imebe recalled and written in the s equel down of this everything paper period he .. said For more . \ detailsquad Then comma , see probably the majority of what Banachthe t did ime would and written have downbeensaved everything . \quad he saidNevertheless . Then , probably , his results the majority are incredible of . open squarewhat bracket Banach 1 1 closing did would square have bracket been comma saved .. . open Nevertheless square bracket , 1 his 3 closing results square are incrediblebracket period SomeAn important of them role will in mathematicians be recalled quoteright in the s l ife equel in Lvov of was this played paper by the . Scottish\quad For more details , see [ 1 1 ]. , \ Somequad of[ them 1 3 ] will . be recalled in the s equel of this paper . For more details , Caf acute-esee open [ 1 parenthesis1 ] , [ 1 see 3 ] open . square bracket 1 0 closing square bracket closing parenthesis period .. It was a place of their meetings commaAn important where they role were in eating mathematicians comma drinking ’ l comma ife in Lvov was played by the Scottish Anspeaking important .. about role .. mathematics in mathematicians comma .. stating ’ l ..ife problems in Lvov .. and was .. so played lving them by periodthe Scottish .. They .. used Caf $ \Cafacutee´({esee} [ 1($ 0 ] ) see[10]). . It was a place\quad of theirIt was meetings a place , where of their they were meetings eating , , where they were eating , drinking , to write sodrinking lut ions , on speaking marble tables about in the caf mathematics acute-e sub period , stating .. However problems comma .. after and each such so lving visit speakingthe tables\ werequad carefullyabout cleaned\quad bymathematics the staff period , ..\ Probablyquad s t some a t i n difficultg \quad proofsproblems of \quad and \quad so lving them . \quad They \quad used them . They used to write so lut ions on marble tables in the caf e´ However toimportant write so theorems lut ions disappeared on marble in this way tables period in .. theTherefore caf after $ \ someacute t ime{e} Banach{ . } quoteright$ . \quad s However , \quad after each such visit the tables, were after each carefully such visit cleaned the tables by the were staff carefully . \quad cleanedProbably by the some staff . difficult Probably proofs of wife commasome L-suppress difficult sub proofs ucj a ofcomma important .. bought theorems .. a .. special disappeared .. book .. open in this parenthesis way . called Therefore .. later .. the .. Scottish .. Bookimportant closing parenthesis theorems .. thatdisappeared .. was in this way . \quad Therefore after some t ime Banach ’ s w i f e $after , \ someL { tucj ime Banach a , ’} s$ wife\quad, L boughtucja, bought\quad a a\quad specials p e c i a book l \quad (book called\quad ( c a l l e d \quad l a t e r \quad the \quad S c o t t i s h \quad Book ) \quad that \quad was always keptlater .. by the waiters Scottish .. and given Book to .. ) a mathematician that was when always ordered kept period by .. the The waiters and alwaysproblems kept comma\quad .. solutionsby the comma waiters .. and\ rewardsquad and were given written to down\quad in thea book mathematician period .. One reward when ordered . \quad The problemsgiven , \quad tos a o lmathematician u t i o n s , \quad whenand ordered rewards . were The written problems down , in solutions the book , and . \quad One reward became particularlyrewards were famous written period down .. On in 6 November the book .. . 1 936 One .. Stanis reward l-suppress became aw particularly Mazur stated famous the becamefollowing particularly problem open parenthesis famous . in\ thequad S cottishOn 6 Book November the problem\quad had1 the 936 number\quad 1 53S t closing a n i s \ parenthesisl aw Mazur period stated the following. problem On 6 November ( in the S 1 cottish 936 Stanis Book law the Mazur problem stated had the the following number 1 problem 53 ) . ( in Problemthe .. 2 S period cottish 1 period Book .. the Assume problem that a had continuous the number function 1 53 on )the . square .. open square bracket 0 comma 1 closing square bracket to the power of 2 and the Problem 2 . 1 . Assume that a continuous function on the square [0, 1]2 \noindentnumber epsilonProblem greater\quad 0 are2 given . 1 period . \quad .. DoAssume there exist that numbers a continuous a sub 1 comma function period periodon the period square comma\quad a sub$ n [ 0 , 1and ] the ˆ{ number2 }$ andε > 0 theare given . Do there exist numbers a1, ..., an, b1, ..., bn, c1, ..., cn, comma b subsuch 1 comma that period period period comma b sub n comma c sub 1 comma period period period comma c sub n comma numbersuch that $ \ varepsilon > 0 $ are given . \quad Do there exist numbers $ a { 1 } ,. .vextendsingle-vextendsingle-vextendsingle-vextendsingle . , a { n } , b { 1 } , . f . open . parenthesis , b x{ comman } y, closing c parenthesis{ 1 } ,... minus sum from k , c { n } , $ = 1 to n c sub k f open parenthesis a sub k comma yk=1 closing parenthesis f open parenthesis x comma b sub k closing parenthesis such that X vextendsingle-vextendsingle-vextendsingle-vextendsingle|f(x, y) − c lesskf(a ork, equal y)f(x, epsilon bk)|for ≤ ε any x comma y in open square bracket 0 comma 1 closing square bracket ? n \ beginNow{ ita i l si g often n ∗} said that the problem was about the existenceforany ofx, Schauder y ∈ [0, 1]? basis \arrowvertin arbitrary separablef ( Banach x space , period y ) .. However− \ commasum ˆ{ thatk was = not 1 known} { atn that} c t ime{ periodk } f ( a { k } ,It y was only )Now in f 1 955 it ( i that s x often Alexandre , said b that Grothendieck{ k the} problem) showed\arrowvert was open about parenthesis the\ leq existence open\ varepsilon square of Schauder bracket\\ 1 9 basisf closing o r any square bracket x , closingy \ in parenthesisin[ arbitrary 0that the , separable existence 1 ] Banach ? space . However , that was not known at that t \endof{ sucha l i g numbers nime∗} . iIt s wasequivalent only toin so 1 955called that .. quotedblleft Alexandre the Grothendieck approximation showed problem ( quotedblright [ 1 9 ] ) that comma the i period e period .. the existence Nowproblem it iof i s f everysuchoften compactnumbers said linearthat i s equivalent operator the problem T from to so wasa Banachcalled about space “ the the X into existence approximation a Banach of Schauder problem ” basis , i . e inspace arbitrary Y is. a limit the separable in problem norm of operatorsBanach i f every space offinite compact rank . \ linearquad periodHowever operator .. The problem ,T thatfrom was was aespecially Banach not known space atX thatinto t ime . Itattractive was onlya as Banach .. in Mazur 1 955 space offered that aY prize Alexandreis a : limit.. a live in Grothendieckgoose norm period of operators .. The showed .. approximation offinite ( [ rank 1 9 problem ]. ) that Thethe problem existence open parenthesiswas especially and comma attractive .... consequently as comma Mazur .... offered the .... original a prize .... : Mazur a quoteright live goose s .... . problem The closing parenthesis ....\noindent was .... soapproximationof lved such .... only numbers in problem .... 1 972i s .... equivalent by to so called \quad ‘‘ the approximation problem ’’ , i . e . \quad the problema Swedish( i mathematician and f every , consequently compact Per Enflo linear , .... the open original operator parenthesis Mazur then $ T 28 ’ $ s years fromproblem old a closing Banach ) was parenthesis space so lved .... $ onlyopen X $ parenthesis in into 1 a open Banach square bracketspace 1 7 $ closing972 Y $ square is a bracket limit closing in norm parenthesis of operators who shortly offinite after rank . \quad The problemby was especially attractivegiving a solutiona Swedish as \ camequad mathematician toMazur Warsaw offered and got Per from Enflo a prize Mazur ( the : then\ prizequad 28 period yearsa live old goose ) ( [ . 1\ 7quad ] ) whoThe shortly\quad approximation problem The formerafter Scottish Caf acute-e in 2006 open parenthesis now a bank closing parenthesis \noindentgiving( and a solution, \ h f i l l cameconsequently to Warsaw ,and\ h got f i l l fromthe Mazur\ h f i l l theo r prize i g i n a . l \ h f i l l Mazur ’ s \ h f i l l problem ) \ h f i l l was \ h f i l l so lved \ h f i l l only in \ h f i l l 1 972 \ h f i l l by The former Scottish Caf e´ in 2006 ( now a bank ) \noindent a Swedish mathematician Per Enflo \ h f i l l ( then 28 years old ) \ h f i l l ( [ 1 7 ] ) who shortly after

\noindent giving a solution came to Warsaw and got from Mazur the prize .

\ centerline {The former Scottish Caf $ \acute{e} $ in 2006 ( now a bank ) } 4 .. K period CIESIELSKI \noindentThere are4 plenty\quad of storiesK.CIESIELSKI about mathematicians in the Scottish Caf acute-e sub period .. Here comma let me recall two of them period 4 K . CIESIELSKI ThereOnce are Mazur plenty stated a of problem stories and Hermanabout mathematicians Auerbach started thinking in the it over Scottish period Caf $ \acute{e} { . }$ \quad Here , let me recallThere two are of plenty them of . stories about mathematicians in the Scottish Caf e´. Here , After a whilelet me Mazur recall said two that of comma them to . make a puzzle more interesting comma .. he offered a bottle of wineOnce as a Mazurreward period stated .. Thena problem Auerbach and said Herman : .. quotedblleft Auerbach Ah comma started so thinking I give up period it over .. .Wine does Oncenot Mazur agree with stated me quotedblright a problem period and Herman Auerbach started thinking it over . After aAfter while a Mazur while Mazur said that said ,that to , make to make a puzzle a puzzle more more interesting interesting , \quad he offeredhe o f fa e r e d a The secondbottle story of i swine connected as a reward with Lebesgue . Then quoteright Auerbach s visit said to Lvov : in “ 1 Ah 938 , period so I give .. Lebesgue up . Wine bottlecame to of Lvov wine comma as delivered a reward a lecture . \quad and afterThen that Auerbach was invited said to the : Scottish\quad Caf‘‘ Ah acute-e , so sub I period give up . \quad Wine does not agreedoes with not me agree ’’ .with me ” . A waiter gaveThe Lebesgue second the story menu i s period connected .. Lebesgue with Lebesgue comma who ’ s didn visit quoteright to Lvov int know 1 938 Polish . commaLebesgue .. studied the menu for a while comma .. gave it back and said : .. quotedblleft I eat .. only dishes which are well The secondcame story to Lvov i s , connected delivered a with lecture Lebesgue and after ’ that s visit was invited to Lvov to thein 1 Scottish 938 . Caf\quade´. ALebesgue Case 1 quotedblrightwaiter gave Case Lebesgue 2 period the menu . Lebesgue , who didn ’ t know Polish , studied cameThe period to Lvov s ince , the delivered end of the a First lecture World War and up after to .. 1 that 939 was was the invited Golden to the Scottish Caf $ \acute{e} { . }$ A waiterthe gave menu Lebesgue for a while the , menu gave . \ itquad backLebesgue and said : , who “ I didn eat ’ only t know dishes Polish which , are\quad studied age for Polishwell mathematics period .. In particular comma in those t imes t ime Banach obtained a theseries menu of remarkable for a while results , period\quad .. Banachgave it can back be regarded and said as the : creator\quad of functional‘ ‘ I eat \quad only dishes which are well analysis comma which at that period could be named .. quotedblleft the Polish branch of mathemat hyphen \ beginics quotedblright{ a l i g n ∗} period .. In Lvov comma together with Banach00 comma there worked many outstanding mathemati hyphen \ l e f t . de f i ned\ begin { a l i g n e d } &’’ \\ defined cians comma for example Steinhaus comma Mazur comma .. Juliusz. Schauder comma W suppress-l adys suppress-l aw Orlicz comma&. .. Stan\end{ a l i g n e d }\ right . \end{ a l i g n ∗} Ulam and MarkThe Kac period .. open s ince parenthesis the end the of last the two First .. moved World to theWar USA up before to the1 939 war was closing the parenthesis Golden period .. Polish mathematicians worked actively also in other centers comma .. especially in Warsaw semicolon .. for The periodage s for ince Polish the mathematics end of the . First In particular World War , in up those to \quad t imes1 t 939ime Banachwas the obtained Golden examplea comma series Wac of remarkable to the power results of suppress-l . Banach aw .. S canierpi be acute-n regarded ski comma as the .. creator Karol .. of Borsuk functional comma .. Kazimierz .. Kuratowskiage for .. Polish and .. Stefan mathematics . \quad In particular , in those t imes t ime Banach obtained a series ofanalysis remarkable , which results at that period . \quad couldBanach be named can be “ regarded the Polish as branch the creator of mathemat of functional - Mazurkiewiczics ” should. In be Lvov mentioned , together period with Banach , there worked many outstanding mathemati analysisBanach and , Steinhaus which at init that iated period in 1 928 a could new j ournal be named comma\quad .. quotedblleft‘‘ the Studia Polish Mathematica branch of quotedblright mathemat − i c s ’ ’ .-\ ciansquad ,In for Lvov example , together Steinhaus with , Mazur Banach , Juliusz, there Schauder worked many , W ladys outstanding law Orlicz mathemati , − which publishedStan Ulam papers and j ust Mark on functional Kac analysis ( the last period two .. It was moved one of to the the first USA j ournals before the war ) . ciansin history , for that example was specialized Steinhaus in some particular, Mazur areas , \quad of mathematicsJuliusz openSchauder parenthesis , W the\ l veryadys \ l aw O r l i c z , \quad Stan Ulam andPolish Mark Kacmathematicians\quad ( the worked last actively two \quad alsomoved in other to centers the USA , before especially the in war Warsaw ) . \quad P o l i s h first one was .. quotedblleft Fundamental Mathematicae quotedblright closing parenthesis period mathematicians; for worked example actively, Wac aw also in S ierpi othern´ centersski , Karol , \quad Borsukespecially , Kazimierz in Warsaw ; \quad f o r In .. 1 93Kuratowski 1 .. the .. fundamental and .. Stefan monograph Mazurkiewicz .. on .. functional should .. analysisbe mentioned .. by .. Banach . .. was examplepublished period$ , .. Wac It .. ˆwas{\ ..l quotedblleft}$ aw \quad OperacjS e i .. e r l p iniowe i $ quotedblright\acute{n} comma$ s k i.. in , ..\quad 1 932 ..Karol published\quad .. inBorsuk .. French , \quad Kazimierz \quad Kuratowski \quad and \quad Stefan MazurkiewiczBanach should and be Steinhaus mentioned init .iated in 1 928 a new j ournal , “ Studia Mathematica .. quotedblleft” Th which acute-e published orie papers j ust on functional analysis . It was one of the first j des op to the power of acute-e rations l in e-acute aires quotedblright .. as the first volume in the s eries .. quotedblleft Banach andournals Steinhaus in history init that iated was in specialized 1 928 a in new some j ournal particular , \ areasquad of‘‘ mathematics Studia Mathematica ( the ’’ Mathematicalvery Mono hyphen whichgraphs published quotedblright papers period .. j For ust many on years functional it was the analysis most basic book . \quad on functionalIt was analysis one of comma theup first j ournals in historyfirst that one was was specialized “ Fundamenta in Mathematicae some particular ” ) . areas of mathematics ( the very to the momentIn when 1 93 the 1 famous the monograph fundamental .. open square monograph bracket 1 6 on closing functional square bracket analysis .. was published .. open parenthesis see also .. open square bracket 30 closing square bracket closing parenthesis period \noindentbyfirst Banach one was was\quad published‘‘ Fundamenta . It Mathematicae was “ Operacj ’’ e ) . l iniowe ” , in 1 It .. should932 .. be published .. noted .. that in .. only French .. in 40“ s .. Th thee´ ..orie t erm des .. quotedblleftope´ rations functional l in e´ aires .. analysis ” as quotedblright the .. was .. being In \quad first1 93 volume 1 \quad in thethe s\ eriesquad fundamental “ Mathematical\quad Monomonograph - graphs\ ”quad .on For\ manyquad yearsf u n c t it i o n a l \quad a n a l y s i s \quad by \quad Banach \quad was used periodwas .. the In Banach most basic t imes book other on names functional were of use analysis comma , especially up to the .. quotedblleftmoment when the theory the famous of l inear publishedoperators quotedblright . \quad I tperiod\quad was \quad ‘ ‘ Operacj e \quad l iniowe ’ ’ , \quad in \quad 1 932 \quad published \quad in \quad French \quad ‘ ‘ Th $ \acute{monographe} $ o r i e [ 1 6 ] was published ( see also [ 30 ] ) . It should be In 1 939noted Lvov was captured that by only the Soviet in 40 Union s comma the in t 1 erm 941 Hitler “ functional quoteright s soldiers analysis took ” was desLvov $ for op 4 years ˆ{\ periodacute ..{e Banach}}$ spent rations .. the l .. inS econd $ \ Worldacute War{e} ..$ in .. a Lvovi r e s comma ’ ’ \quad .. l ivingas underthe first volume in the s eries \quad ‘‘ Mathematical Mono − graphs ’being ’ . \quad used .For In many Banach years t imes it was other the names most were basic of use book , especially on functional “ the analysis theory , up extremelyof difficult l inear conditions operators period ” . .. After the war Lvov was taken by the Soviet Union toagain the and moment Banach when planned the to famous go to Krak monograph acute-o w where\quad he[ would 1 6 have ] \quad taken awas Chair published \quad ( see a l s o \quad [ 30 ] ) . I t \quad shouldIn 1 939\quad Lvovbe was\quad capturednoted by\ thequad Sovietthat Union\quad , inonly 1 941\quad Hitlerin ’ s40 soldiers s \quad tookthe \quad t erm \quad ‘‘ functional \quad a n a l y s i s ’ ’ \quad was \quad being at the JagiellonianLvov for University4 years . period Banach .. He died spent j ust a the few days S econdbefore the World move War period .. in He is Lvov , l usedburied . in\quad LychakovIn Banach Cemetery t open imes parenthesis other names Cmentarz were L-suppress of use yczakowski , especially closing\ parenthesisquad ‘‘ thein Lvov theory period of .. Now l inear operatorsiving ’’ under . extremely difficult conditions . After the war Lvov was taken by the comma in frontSoviet of Union again and Banach planned to go to Krak o´ w where he would have the buildingtaken of the a Chair Mathematics at the Institute Jagiellonian of the JagiellonianUniversity University . He died there j is ust a few days before the Ina 1 monument 939 Lvov of Banachwas captured period by the Soviet Union , in 1 941 Hitler ’ s soldiers took Lvov formove 4 years . . He\quad is buriedBanach in Lychakovspent \quad Cemeterythe \quad ( CmentarzS econdLyczakowski World War \ )quad in Lvovin \quad Lvov , \quad l iving under extremely. difficult Now , in front conditions of the building. \quad ofAfter the Mathematics the war Lvov Institute was taken of the by Jagiellonian the Soviet Union again andUniversity Banach plannedthere is a to monument go to Krak of Banach $ \acute . {o} $ w where he would have taken a Chair at the Jagiellonian University . \quad He died j ust a few days before the move . \quad He i s buried in Lychakov Cemetery ( Cmentarz \L yczakowski ) in Lvov . \quad Now , in front of the building of the Mathematics Institute of the Jagiellonian University there is a monument of Banach . ON STEFAN BANACH AND SOME OF HIS RESULTS .. 5 \ hspaceThe monument∗{\ f i l l } ofON Banach STEFAN in Krak BANACH acute-o AND w SOME OF HIS RESULTS \quad 5 The Scottish Book survived the war period .. It was taken to Poland by L-suppress sub ucj a Banach ON STEFAN BANACH AND SOME OF HIS RESULTS 5 \ centerlineand later on{ translatedThe monument to English of by Banach Steinhaus in period Krak .. $ Ulam\acute let the{o} problems$ w } from the The monument of Banach in Krak o´ w book circulate in the United States period .. In 1 98 1 the book was published in English The Scottish Book survived the war . It was taken to Poland by L Banach Thein Scottish the version preparedBook survived by Dan Mauldin the war .. open . \quad parenthesisIt was open taken square bracket to Poland 27 closing by square $ ucja\L bracket{ ucj closing a parenthesis}$ Banach and laterand on later translated on translated to English to English by Steinhaus by Steinhaus . \ .quad UlamUlam let let the the problems problems from from the period .. Thisthe translation book circulate is remarkably in the United States . In 1 98 1 the book was published in bookvaluable circulate as besides in the the problems United and so States lut ions . ....\quad open parenthesisIn 1 98 1 if there the arebook closing was parenthesis published .... in it .... English includes several in the versionEnglish prepared in the version by Dan prepared Mauldin by Dan\quad Mauldin( [ 27 ] ( )[ 27 . ]\quad ) .This This translation translation is is remarkably commentsremarkably and remarks about the continuation of the investigations inspired by the problems from the Scottish Book period .. It is a large and important piece of math hyphen \noindentvaluablevaluable as besides as besides the problems the problems and so and lut ions so lut ( if ions there\ areh f i ) l l it( includes if there several are ) \ h f i l l i t \ h f i l l includes several ematics periodcomments and remarks about the continuation of the investigations inspired by the The .. international .. mathematical .. centre .. in .. Poland comma .. created in .. 1 972 .. i s .. named \noindentproblemscomments from and the remarks Scottish about Book the . continuation It is a large of and the important investigations piece of math inspired - by after Banachematics period . .. The Banach Center i s a part of the Institute of Mathematics of the thePolish problems Academy from of Sciences the andScottish has it s Book main office . \quad in WarsawIt is period a large .. Conferences and important took piece of math − ematics . The international mathematical centre in Poland , created in 1 place in Warsaw972 i comma s named recently after they haveBanach been . organizedThe Banach also in B Center cedilla-e dlewoi s a partperiod of .. the There Institute are Banach .. Center Publications that .. publishes proceedings .. of selected .. conferences The \quadofinternational Mathematics of\ thequad Polishmathematical Academy of\quad Sciencesc e n tand r e has\quad it sin main\quad officePoland in Warsaw , \quad created in \quad 1 972 \quad i s \quad named and semesters. Conferencesheld at the International took place Stefan in WarsawBanach Mathematical , recently they Center have period been .. Up organized also in afterto now Banach comma 77 . volumes\quad The wereBanach published Center period i s a part of the Institute of Mathematics of the Polish AcademyB cedilla of− e Sciencesdlewo . and There has are itBanach s main office Center in Publications Warsaw . \quadthatConferences publishes took 3 periodproceedings .. Some results named of selected by Banach quoteright conferences s surname and semesters held at the International placeNow we in turn Warsaw to some , mathematical recently they results have of Banach been period organized .. As mentioned also in above B $ comma cedilla −e $ dlewo . \quad There are Banach \Stefanquad Center Banach Publications Mathematical thatCenter\quad . Uppublishes to now , proceedings 77 volumes were\quad publishedo f s e l e . c t e d \quad conferences only some of his discoveries3 . were published semicolon nevertheless comma they present themselves andan enormous semesters collection held period at theSome .. Here International commaresults we presentnamed Stefan only by Banachsome Banach of the Mathematical most ’ s surname important Center results . \quad Up to now , 77Now volumes we turn were to some published mathematical . results of Banach . As mentioned above , which areonly nowadays some named of his after discoveries him semicolon were .. published the results ; will nevertheless be recalled and , they it will present be themselves said where they were published period \ centerlinean{ enormous3 . \quad collectionSome results . Here named , we present by Banach only ’ some s surname of the most} important results Before ..which that comma are nowadays .. an .. important named .. after fact .. him should ; .. thebe .. results mentioned will period be recalled .. It .. was and .. checked it will ..be what names appear most frequently in the t it les of mathematical and physical papers Now we turnsaid to where some they mathematical were published results . of Banach . \quad As mentioned above , published ..Before in the .. 20 that th .. century , an period important .. It .. turned fact .. out .. should that .. it .. be was .. mentioned Banach quoteright . sIt .. name that onlygot the some first of place his period discoveries .. The second were place published was obtained ; by nevertheless Sophus Lie comma , they the third present by themselves an enormouswas collection checked what . \quad namesHere appear , we most present frequently only some in the of t it the les most of mathematical important results Bernhardand Riemann physical period papers published in the 20 th century . It turned out whichCertainly are comma nowadays Banach named deserves after such a him position ; \quad mainlythe because results of Banach will spaces be commarecalled and it will be said wherethat they it were was published Banach .’ s name that got the first place . The second place nowadayswas one obtained of the most by important Sophus mathematical Lie , the third notions by periodBernhard .. A BanachRiemann space . is a normed complete vector space period .. It was .. formally defined in the paper .. open square bracket 5 closing square bracket Before \quadCertainlythat , \ ,quad Banachan deserves\quad important such a position\quad mainlyf a c t \quad becauseshould of Banach\quad spacesbe \quad , mentioned . \quad I t \quad was \quad checked \quad what semicolon ..nowadays for one of the most important mathematical notions . A Banach space is a namessome t appear ime mathematicians most frequently in Lvov called in the those t spaces it les .. quotedblleft of mathematical a space of typeand B physical quotedblright papers period .. The publishednormed\quad completein the \ vectorquad 20 space th \.quad Itcentury was formally . \quad definedI t \quad in theturned paper\quad [ 5 ]out ; \quad that \quad i t \quad was \quad Banach ’ s \quad name that got thefor first some place t ime . mathematicians\quad The second in Lvov place called was those obtained spaces by Sophus“ a space Lie of type , the B ”third . by BernhardThe Riemann .

Certainly , Banach deserves such a position mainly because of Banach spaces , nowadays one of the most important mathematical notions . \quad A Banach space is a normed complete vector space . \quad I t was \quad formally defined in the paper \quad [ 5 ] ; \quad f o r some t ime mathematicians in Lvov called those spaces \quad ‘‘ a space of type B ’’ . \quad The 6 .. K period CIESIELSKI \noindentname quotedblleft6 \quad BanachK.CIESIELSKI space quotedblright was probably used for the first t ime by Maurice Fr acute-e chet comma in 1 928 period .. Note that independently such spaces were introduced by Norbert Wiener comma 6 K . CIESIELSKI \noindenthowever Wienername thought ‘‘ Banach that the space spaces ’’ would was not probably be of importance used for and the gave first t ime by Maurice Fr $ \acute{e} $ name “ Banach space ” was probably used for the first t ime by Maurice Fr e´ chet , in chetup period , in .. A long t ime later Wiener wrote in his memoirs that the spaces quite j ustly 1 928 . 1\quad 928 .Note Note that that independently independently such such spaces spaces were were introduced introduced by Norbert by Norbert Wiener Wiener , , should behowever named after Wiener .. Banach thought alone that comma the .. spaces as .. sometimes would not they be were of called importance .. quotedblleft and gave Banach up hyphen . howeverWiener spaces Wiener quotedblright thought period that the.. For spaces more details would comma not see be open of importance square bracket and 1 4 closing gave square bracket and open up . \quadA longA long t ime t later ime Wiener later Wiener wrote in wrote his memoirs in his that memoirs the spaces that quitethe spaces j ustly quite should j ustly square bracketbe 1 named 5 closing after square bracket Banach period alone , as sometimes they were called “ Banach - shouldThere .. be are named .. some after .. points\quad .. whichBanach .. show alone .. why .. , the\quad .. introductionas \quad .. ofsometimes Banach .. spaces they were called \quad ‘ ‘ Banach − Wiener spacesWiener ’’ spaces . \quad ” .For For more more details details , see , see [ 1 4 [ ] 1 and 4 ] [ 1 and 5 ] .[ 1 5 ] . was .. so .. importantThere period are .. some For .. a points variety .. of which reasons .. show function why .. spaces the .. are ..introduction very useful .. in of many investigationsBanach and spaces applications was periodso important .. To a large . extent For comma a variety modern mathematics of reasons is function Thereconcerned\quad withare the\ studyquad ofsome general\quad structuresp o i n period t s \quad .. Thewhich essential\quad thingshow i s finding\quad the why \quad the \quad introduction \quad o f Banach \quad spaces was \quadspacesso \quad areimportant very useful . \quad in manyFor \ investigationsquad a v a r i e and t y \ applicationsquad o f reasons . To\ aquad largef u n c t i o n \quad spaces \quad are \quad very u s e f u l \quad in r ight generalizationextent , modern period .. mathematics Insufficient generality is concerned can be too with restrictive the study and a great of general deal structures . manyof generality investigations may result .. and in a applicationss ituation where .. . litt\quad le canTo be aproved large and extent applied period , modern mathematics is concernedThe with essential the study thing i of s finding general the structures r ight generalization . \quad .The Insufficient essential generality thing i scan finding the The spacebe introduced too restrictive by Banach and comma a great especially deal of pointing generality out completeness may result comma in attestsa s ituation to where rhis ight genius generalization semicolon he hit the . \ traditionalquad Insufficient nail on the head generality period can be too restrictive and a great deal of generalitylitt le can may be result proved\quad and appliedin a s . ituation The space where introduced\quad litt by Banach le can , be especially proved and applied . Banach quoterightpointing sout .. great completeness .. merit .. was , attests .. that to comma his genius.. in .. principle ; he hit comma the traditional .. it .. was nail.. thanks on the .. to .. him .. that .. theThe space introduced by Banach , especially pointing out completeness , attests to his geniushead ; . he hit the traditional nail on the head . quotedblleftBanach geometric ’ squotedblright great waymerit of lo oking was at that spaces , was in init principle iated period , .. itThe elements was thanks of some general spaces might be functions or number sequences comma but when fitted into the structure Banach ’to s \quad himgre that at \quad the “merit geometric\quad ” waywas of\quad lo okingthat at ,spaces\quad wasin init\quad iatedp . r i n c The i p l e , \quad i t \quad was \quad thanks \quad to \quad him \quad that \quad the of a Banachelements space they of some were regarded general as spaces .. quotedblleft might be points functions quotedblright or number comma sequences as the elements , but of when a .. quotedblleft space quotedblright‘‘ geometric period ’’ way of lo oking at spaces was init iated . \quad The elements of some general spaces mightfitted beinto functions the structure or number of a Banach sequences space they , but were when regarded fitted as into “ the points structure ” , as At t imesthe this elements resulted in of remarkable a “ space s implifications ” . At t imes period this resulted in remarkable s implifications ofToday a Banach comma spacealmost ninety they years were after regarded it s introduction as \quad comma‘‘ thepoints notion ’’ of a , Banach as the space elements of a \quad ‘ ‘ space ’ ’ . At t imes. this resulted in remarkable s implifications . remains fundamentalToday , in almost many areasninety ofyears mathematics after it period s introduction .. The theory , theof Banach notion spaces of a Banach space i s being developed to this day comma and new comma interesting comma and o ccasionally surprising Today , almostremains ninety fundamental years in after many it areas s introduction of mathematics , . the The notion theory of of a Banach Banach spaces space results ..i are s being .. obtained developed be many to researches this day period , and .. new In particular , interesting comma , and .. some o ccasionally really important surprising remainsresults were fundamental obtained in the in end many of the areas 20 th of century mathematics by William Timothy. \quad GowersThe theory period of Banach spaces i s beingresults developed are to obtained this day be , manyand new researches , interesting . In , particular and o ccasionally , some really surprising Some problemsimportant he solved results waited were for obtainedthe so lut ion in thes ince end Banach of the quoteright 20 th century s t imes by period William .. For Timothy his r eresearch s u l t s comma\quad Gowersare \quad was awardedobtained in 1 998 be with many the researches Fields medal period. \quad In particular , \quad some really important resultsGowers were obtained . Someproblems in the end he solved of the waited 20 th for century the so lut by ion William s ince Timothy Banach ’ Gowers s t imes . In the same. paper For his open research square bracket , Gowers 5 closing was awarded square bracket in 1 998there with i s proved the Fields the famous medal Banach . Fixed Point Theorem periodSome problems he solved waited for the so lut ion s ince Banach ’ s t imes . \quad For h i s research ,In Gowers the same was paper awarded [ 5 ] in there 1 998 i s proved with the the famousFields Banach medal . Fixed Point Theorem It says . It says Theorem 3 period 1 period .... Let open parenthesis X comma d closing parenthesis .... be a complete metric space and a Theorem 3 . 1 . Let (X, d) be a complete metric space and a function functionIn the f same : X right paper arrow [ X 5 ] there i s proved the famous Banach Fixed Point Theorem . f : X → X I tbe says a contracting operation comma .... i period e period .... there exists a lambda in open parenthesis 0 comma 1 closing be a contracting operation , i . e . there exists a λ ∈ (0, 1) such that parenthesis .... such that d open parenthesis f open parenthesis x closing parenthesis comma f open parenthesis y closing parenthesis d(f(x), f(y)) ≤ closing\noindent parenthesisTheorem less or 3 equal . 1 . \ h f i l l Let $ ( X , d ) $ \ h f i l l be a complete metric space and a function $ f :λd X(x, y)\rightarrowfor any x, y ∈ XX. $ Then there exists a unique p ∈ X such that f(p) = p. lambda d openThe parenthesis theorem xwas comma the use y closing of the parenthesis method of for successive any x comma approximations y in X period .... and Then a general there exists a unique p in X such thatversion f open parenthesis of the property p closing that parenthesis was known = p period earlier in some concrete cases . \noindentThe theorembe was a contracting the use of the method operation of successive , \ h fapproximations i l l i . e . and\ h f a i l l there exists a $ \lambda \ in ( 0 , 1Now ) $ we\ h turn f i l l tosuchthat$d some fundamental (theorems f ( on functional x ) , analysis f . ( y ) ) \ leq $ general version ofOne the propertyof the most that importantwas known earlier of them in some i s the concrete Hahn cases - Banach period Theorem . Now we turn to some fundamental theorems on functional analysis period Theorem 3 . 2 . Let X be a real normed vector space , p : X → a \noindentOne of the most$ \lambda important ofd them ( i s xthe Hahn , hyphen y )$forany$x Banach Theorem period , y \ in XR . $ \ h f i l l Then there exists a unique function such that p(αx + (1 − α)y) ≤ αp(x) + (1 − α)p(y) for each x, y ∈ X and $ pTheorem\ in 3 periodX$ 2 suchthat period .. Let X $f .. be a ( real normed p ) vector = space p comma .$ p : X right arrow R a function such α ∈ [0, 1]. that p open parenthesis alpha x plus open parenthesis 1 minus alpha closing parenthesis y closing parenthesis less or equal Assume that ϕ : Y → is a lin ear functional defined on a alphaThe theorem p open parenthesis was the x closing use of parenthesis the method plusR open of parenthesis successive 1 minus approximations alpha closing parenthesis and a p open parenthesis y closing generalvector version subspace of theY propertyof X thatwith was the known property earlierϕ(x) in≤ somep(x) concretefor al l casesx ∈ .Y. parenthesis forThen each x comma there y in exists X .. and alpha a in linear open square bracket 0 comma 1 closing square bracket period Assume .. that phi : Y right arrow R .. is .. a .. lin ear functional defined .. on .. a vector subspace Y functional ψ : X → su ch that ψ(x) = ϕ(x) for al l x ∈ Y and ψ(x) ≤ p(x) for \ centerlineof X .. with{ ..Now the weproperty turn phi to openR some parenthesis fundamental x closing theorems parenthesis on less functional or equal p open analysis parenthesis . x} closing parenthesis .. for al l x in Y period .. Then .. there .. exists .. a .. linear \ centerline {One of the most important of them i s the Hahn − Banach Theorem . } functional psi : X right arrow R su ch that psi openallx parenthesis∈ x. x closing parenthesis = phi open parenthesis x closing parenthesis for al l x in Y .... and psi open parenthesis x closing parenthesis less or equal p open parenthesis x closing parenthesis for\noindent Theorem 3 . 2 . \quad Let $ X $ \quad be a real normed vector space $ , p : X \rightarrow R $The a function theorem has such several versions ( compare [ 29 ] ) . al l x in x periodIt was published by Banach in [ 3 ] . Independently , it was ( in a s thatThe theorem $ p has ( several\alpha versionsx open + parenthesis ( 1 compare− \ openalpha square) bracket y 29 ) closing\ leq square\ bracketalpha closingp parenthesis ( x ) + (imper 1 case− ) \alpha s l ightly) earlier p ( discovered y )$ foreach$x by Hans Hahn , y (\ [in 20 ]X ) $ \quad and period which was a generalization of $ \Italpha was published\ in by Banach[ 0 in .. , open 1 square ] bracket . $ 3 closing square bracket period .. Independently comma .. it was .. open parenthesis in a s imper case closing parenthesis \noindents l ightly ..Assume earlier ..\ discoveredquad that .. by $ ..\ Hansvarphi .. Hahn:Y .. open parenthesis\rightarrow open squareR bracket $ \quad 20 closingi s \quad squarea bracket\quad closinglin ear functional defined \quad on \quad a vector subspace parenthesis$ Y $ .. which .. was .. a .. generalization .. of o f $ X $ \quad with \quad the property $ \varphi ( x ) \ leq p ( x ) $ \quad f o r a l l $ x \ in Y . $ \quad Then \quad the re \quad e x i s t s \quad a \quad l i n e a r

\noindent functional $ \ psi :X \rightarrow R$ su ch that $ \ psi ( x ) = \varphi ( x )$ forall $x \ in Y $ \ h f i l l and $ \ psi ( x ) \ leq p ( x ) $ f o r

\ begin { a l i g n ∗} a l l x \ in x . \end{ a l i g n ∗}

\ centerline {The theorem has several versions ( compare [ 29 ] ) . }

It was published by Banach in \quad [ 3 ] . \quad Independently , \quad i t was \quad ( in a s imper case ) s l i g h t l y \quad e a r l i e r \quad d i s c o v e r e d \quad by \quad Hans \quad Hahn \quad ( [ 20 ] ) \quad which \quad was \quad a \quad generalization \quad o f ON STEFAN BANACH AND SOME OF HIS RESULTS .. 7 \ hspacehis result∗{\ fromf i l l ..}ON 1 922 STEFAN period BANACH .. Banach AND did not SOME know OF about HIS Hahn RESULTS quoteright\quad s paper7 semicolon .. neverthelesss comma Banach quoteright s version was stronger as Hahn proved a theorem in the case where X ON STEFAN BANACH AND SOME OF HIS RESULTS 7 \noindenti s a Banachhis space result period from .... It\ shouldquad be1 noted 922 .that\quad a s implerBanach version did ofnot the theorem know about open parenthesis Hahn ’s in paper ; \quad neverthelesss , Banach ’his s result version from was 1 stronger 922 . Banach as Hahn did proved not know a theorem about Hahn in the ’ s paper case ;where neverthe- $ X $ the caselesss where , X Banach i s the space ’ s version of continuous was strongerreal functions as Hahn on a compact proved interval a theorem closing in parenthesis the case where was published much earlier by Eduard Helly open parenthesis open square bracket 2 1 closing square bracket semicolon see \noindentXi s a Banach space . \ h f i l l It should be noted that a s impler version of the theorem ( in open squarei bracket s a Banach 22 closing space square . It bracket should closing be noted parenthesis that period a s impler .... Neither version Banach of the nor theorem ( in Hahn knew about Helly quoteright s theorem period \noindentthethe case case where whereX i s $ the X $ space i s of the continuous space of real continuous functions on real a compact functions interval on a ) compact interval ) The complexwas versionpublished of the much theorem earlier was provedby Eduard later comma Helly (.. [ in 2 .. 1 1 ] 938 ; see independently [ 22 ] ) . Neither Banach by G period A period Soukhomlinov and by H period F period Bohnenblust open parenthesis open square bracket 8 closing \noindentnorwas published much earlier by Eduard Helly ( [ 2 1 ] ; see [ 22 ] ) . \ h f i l l Neither Banach nor square bracketHahn closing knew parenthesis about Helly and A ’ period s theorem Sobczyk . open parenthesis open square bracket 3 1 closing square bracket closing parenthesis period \noindent HahnThe knew complex about version Helly of the ’ s theorem theorem was . proved later , in 1 938 independently The Hahnby hyphen G . A Banach . Soukhomlinov Theorem i s and regarded by H by . F many . Bohnenblust authorities as ( one [ 8 of ] ) three and A . Sobczyk ( [ 3 1 basic principles]). of functional analysis period .. The two others are the Banach hyphen Steinhaus TheTheorem complex and version the Banach of Closed the Graphtheorem Theorem was provedperiod later , \quad in \quad 1 938 independently byG . A .The Soukhomlinov Hahn - Banach and byH Theorem . F i. s Bohnenblust regarded by ( many [ 8 authorities ] ) andA as . Sobczyk one of three ( [ 3 1 ] ) . The Banachbasic hyphen principles Steinhaus of functional Theorem was analysis proved in. open The square two bracket others 7 are closing the Banachsquare bracket - Steinhaus period .. It says : Theorem 3 period 3 period .. Let X .. be a Banach space comma Y be a normed vector space period .. Consider The Hahn −TheoremBanach and Theorem the Banach i s regarded Closed Graph by many Theorem authorities . as one of three the family F of al lThe lin ear Banach bounded - Steinhaus functions from Theorem X .. to wasY period proved .. If in for [ any 7 ] x . in X It .. says the : basics et open principles brace bar bar of T functional open parenthesis analysis x closing . parenthesis\quad The bar two bar : others T in F closing are the brace Banach .. is bounded− Steinhaus comma .. then Theorem 3 . 3 . Let X be a Banach space ,Y be a normed vector space . theTheorem s et open and brace the bar Banach bar T bar Closed bar : T Graphin F closing Theorem brace .. . is bounded period Consider the family F of al l lin ear bounded functions from X to Y. If for Now recall the Banach Closed Graph Theorem period \ centerlineany{Thex ∈ BanachX the− sSteinhaus et {|| T (x) Theorem||: T ∈ F} wasis bounded proved ,in [ then 7 ] the . \ squad et {||I tT || says: T ∈ F} : } Theoremis 3 periodbounded 4 period . .. Let X .. and Y .. be Banach spaces comma .. and T be a lin ear operator from X .. to Y period .. Then T isNow bounded recall if and the only Banach if the Closed graph of Graph T is clos Theorem ed in X times . Y period \noindentThis theoremTheorem i s closely 3 .related 3 . to\quad the veryLet important $ X $ Banach\quad Openbe Mapping a Banach space $ , Y $ be a normed vector space . \quad Consider Theorem 3 . 4 . Let X and Y be Banach spaces , and T be a lin thePrinciple family period $ F $ of al l lin ear bounded functions from $ X $ \quad to $ Y . $ \quad I f f o r any ear operator from X to Y. Then T is bounded if and only if the graph of T $ xTheorem\ in 3 periodX $ 5\ periodquad ..the Let X .. and Y .. be Banach spaces comma .. and T be a lin ear operator from is clos ed in X × Y. sX et .. onto $ \{\ Y openmid parenthesis\mid we assumeT that( T x is surjective ) \mid closing\mid parenthesis:T period ..\ in Then forF any\} open$ subset\quad U ofi s X bounded the , \quad then the s et $ \{\mid This\mid theoremT i\ smid closely\mid related:T to the very\ in importantF \} $ Banach\quad Openi s bounded Mapping . s et T openPrinciple parenthesis . U closing parenthesis .. is open in Y period Both theorems were published in open square bracket 6 closing square bracket period Theorem 3 . 5 . Let X and Y be Banach spaces , and T be a lin ear \ centerlineLet us mention{Now also recall another the theorem Banach on functional Closed analysis Graph comma Theorem .. nowadays . } called operator from X onto Y ( we assume that T is surjective ) . Then for any frequently the Banach hyphen Alaoglu Theorem period .. It says open subset U of X the \noindentTheorem 3Theorem period 6 period 3 . 4 .. .Let\quad X to theLet power $ Xof * $ .. be\quad the dualand space $ Y of a$ Banach\quad spacebe X Banach period .. spaces Then the , clos\quad ed and $ T $ bes a et linT (U ear) is operator open in fromY. unit bal l in X to the power of * ..Both is compact theorems in X to were the powerpublished of * .. in with [ 6 the ] . weak * endash .. topology period $A X proof $ of\quad this theoremto $ was Y given . $ in 1\ 940quad byThen Leonidas $ Alaoglu T $ openis bounded parenthesis if open and square only bracket if the 1 closing graph square of bracket $ T $ is clos ed inLetus $X mention\times also anotherY . $ theorem on functional analysis , nowadays called closing parenthesisfrequently and in the the Banach - Alaoglu Theorem . It says case of separable normed vector spaces was published in 1 929 by Banach open parenthesis open square bracket 4 closing square This theoremTheorem i s closely 3 . 6 . relatedLet X to∗ thebe the very dual important space of a Banach Banach Open space MappingX. Then the bracket closing parenthesis period ∗ ∗ P r i n c i p lclos e . ed unit bal l in X is compact in X with the weak ∗−− topology . Banach did notA work proof only of on this functional theorem analysis was given period in .. For 1 940 example by Leonidas comma today Alaoglu his name ( [ 1 ] ) and in i s connectedthe with famous Banach hyphen Tarski Theorem on paradoxical decomposition \noindentof the ballTheorem period .. The 3 . theorem 5 . \quad may beLet formulated $ X $ in the\quad followingand way $ periodY $ \quad be Banach spaces , \quad and $ T $ becase a lin of separable ear operator normed from vector spaces was published in 1 929 by Banach ( [ 4 ] ) . Theorem .. 3Banach period 7 did period not .. work If B onlysubset on R functional to the power analysis of 3 .. is . a three For hyphen example dimensional , today hisbal lname comma .. then there .. exist$ Xpairwise $ \quad onto $Y ( $ we assume that $T$ is surjective ) . \quad Then for any open subset $ U $ o fi X s connected the with famous Banach - Tarski Theorem on paradoxical decomposition disjoint sof ets the A subball 1 . comma The period theorem period may period be comma formulated A sub in n ..the and following is ometric way transformations . I sub 1 comma period period period comma I sub n .. such that B = Theorem 3 . 7 . If B ⊂ 3 is a three - dimensional bal l , then \noindentA sub 1 cups period et $ period T period ( U cup A)sub $ n\quad commais ..R and openin for s ome $Y k in open .$ parenthesis 1 comma n closing parenthesis : I there exist pairwise disjoint s ets A , ..., A and is ometric transformations sub 1 open parenthesis A sub 1 closing parenthesis comma period1 periodn period comma I sub k open parenthesis A sub k closing \ centerlineI1, ...,{Both In such theorems that wereB = A published1 ∪ ... ∪ An, inand [ 6 for ] s . ome} k ∈ (1, n): I1(A1), ..., Ik(Ak) parenthesis ..are are pairwise pairwise disjoint disjoint comma , B = I sub 1 open parenthesis A sub 1 closing parenthesis cup period period period cup I sub k open parenthesis A sub k closing B = I (A ) ∪ ... ∪ I (A ),I (A ), ..., I (A ) are pairwise disjoint ,B = parenthesisLet us mention comma I also sub1 k1 plusanother 1 openk theoremk parenthesisk+1 on Ak+1 functional sub k plusn n 1 closing analysis parenthesis , \quad commanowadays period period called period comma I sub nfrequently open parenthesis the A subBanach n closing− Alaoglu parenthesis Theorem .... are pairwise . \quad .... disjointI t says comma B = I sub k plus 1 open parenthesis A sub k plus 1 closing parenthesis cup period period period cup I sub n open parenthesis A Ik+1(Ak+1) ∪ ... ∪ In(An) sub\noindent n closing parenthesisTheorem 3 . 6 . \quad Let $ X ˆ{ ∗ }$ \quad be the dual space of a Banach space $ X . $The\ theoremquad ThenThe .. lo theorem oks the very clos strange edlo oks comma very .. strange as .. it comma , as .. in it fact , comma in fact .. says , thatsays .. that we can we.. double can the unitvolume bal !double .. l The in point the $Xˆ i volume s{ that ∗ the }$ ! ball\quad The is split pointis into compact piecesi s that that in the are $ ball non X hyphenˆis{ split ∗ } measurable$ into\quad pieces periodwith that the are weaknon $ ∗ { −− }$ \quadThe ..topology proof- measurable relays . .. on ... the The .. axiom proof .. of relays choice period on .. It the .. was axiom.. published of .. choice in .. open . square It bracket was 9 closing square bracket periodpublished .. For .. more in [ 9 ] . For more information , see [ 1 8 ] . \ hspaceinformation∗{\ f i comma l l }A proofsee open of square this bracket theorem 1 8 closing was given square bracket in 1 940 period by Leonidas Alaoglu ( [ 1 ] ) and in the \noindent case of separable normed vector spaces was published in 1 929 by Banach ( [ 4 ] ) .

Banach did not work only on functional analysis . \quad For example , today his name i s connected with famous Banach − Tarski Theorem on paradoxical decomposition o f the b a l l . \quad The theorem may be formulated in the following way .

\noindent Theorem \quad 3 . 7 . \quad I f $ B \subset R ˆ{ 3 }$ \quad i s a thre e − dimensional bal l , \quad then there \quad exist pairwise disjoint s ets $A { 1 } ,...,A { n }$ \quad and is ometric transformations $ I { 1 } ,...,I { n }$ \quad such that $B =$ $ A { 1 }\cup ... \cup A { n } , $ \quad and for s ome $ k \ in ( 1 , n ) : I { 1 } (A { 1 } ),...,I { k } (A { k } ) $ \quad are pairwise disjoint ,

\noindent $ B = I { 1 } (A { 1 } ) \cup ... \cup I { k } (A { k } ),I { k + 1 } (A { k + 1 } ),...,I { n } (A { n } ) $ \ h f i l l are pairwise \ h f i l l disjoint $ , B =$

\ begin { a l i g n ∗} I { k + 1 } (A { k + 1 } ) \cup ... \cup I { n } (A { n } ) \end{ a l i g n ∗}

The theorem \quad lo oks very strange , \quad as \quad i t , \quad in f a c t , \quad says that \quad we can \quad double the volume ! \quad The point i s that the ball is split into pieces that are non − measurable . The \quad proof relays \quad on \quad the \quad axiom \quad o f c h o i c e . \quad I t \quad was \quad published \quad in \quad [ 9 ] . \quad For \quad more information , see [ 1 8 ] . 8 .. K period CIESIELSKI \noindentNot all the8 mathematical\quad K.CIESIELSKI theorems and notions which are now frequently called by Banach quoteright s name were decribed above period .. Let us mention here comma for instance comma Banach Not all the8 mathematical K . CIESIELSKI theorems and notions which are now frequently called integral commaNot Banach all the generalized mathematical limit open theorems parenthesis and introduced notions which in open are square now bracket frequently 2 closing called square bracket closing parenthesisby Banach and ’ Banach s name algebra were period decribed .. Banach above . \quad Let us mention here , for instance , Banach integralby , Banach Banach ’ generalized s name were limit decribed ( introduced above . in Let [ us 2 ]mention ) and Banachhere , for algebra instance . \quad Banach algebras, wereBanach a kind of integral restructurization , Banach of Banach generalized spaces limit .... open( parenthesis introduced instead in [ 2 of ] a ) vector and Banach space there is taken a r ing and in addition a multiplication of elements closing parenthesis period .. Banach \noindentalgebraalgebras. were Banach a kind of restructurization of Banach spaces \ h f i l l ( instead of a vector algebrasalgebras were introduced were a in kind 1 94 1 of by restructurization a Russian mathematician of Banach Israil spaces M period Gelfand ( instead period of a vector One .. shouldspace .. there mention is taken .. here a .. r also ing ..and the in .. addition Banach hyphen a multiplication Mazur distance of elements .. open parenthesis ) . Banach introduced .. in .. open square\noindent bracketspace 6 closing there square is bracket taken closing a r parenthesis ing and incomma addition a multiplication of elements ) . \quad Banach algebrasalgebras were introduced were introduced in 1 94in 1 1 94 by 1 a by Russian a Russian mathematician mathematician Israil Israil M M . . Gelfand Gelfand . which is. a suitable One defined should distance mention between two here i somorphic also Banach the spacesBanach period - Mazur distance ( OneThe\ firstquad volumeshould of the\quad Banachmention Journal of\quad Mathematicalhere \quad Analysisa l si os dedicated\quad the \quad Banach − Mazur distance \quad ( introduced \quad in \quad [ 6 ] ) , which isintroduced a suitable in defined [ 6 ] )distance , which is abetween suitable two defined i somorphic distance between Banach two spaces i somorphic . to ThemistoclesBanach M spaces period .Rassias period .. It is nice to notice that some of the achievements of Th period MThe period first .. volumeRassias have of the a particular Banach connection Journal of with Mathematical Banach and the Analysis mathematics i s dedicated to Thefrom first the Scottish volume Caf of acute-e the Banach sub period Journal of Mathematical Analysis i s dedicated to ThemistoclesThemistocles M . M Rassias . Rassias . \ .quad It isIt nice is tonice notice to noticethat some that of the some achievements of the achievements of of One of theTh most . M important . Rassias mathematicians have a particular of the Lvov connection group was withStan Ulam Banach comma and the mathematics who was very young in his Lvov days period .. In .. 1 936 he moved to the USA where he from the Scottish Caf e´ \noindentlater on becameTh . aM very . \ famousquad Rassias scientist. period have .. a The particular reader i s referred connection to the wonderful with Banach and the mathematics from the ScottishOne of the Caf most $ \ importantacute{e} mathematicians{ . }$ of the Lvov group was Stan Ulam , volume ..who open was square very bracket young 32 in closing his Lvov square days bracket . period In .. 1 As936 was he mentioned moved to above the USA comma where .. Ulam he played a great ro le in circulating One of thelater most on becameimportant a very mathematicians famous scientist of . the The Lvov reader group i s was referred Stan to Ulam the wonderful , the mathematicsvolume from [ the 32 Scottish ] . Caf As acute-e was mentioned after the S econd above World , War Ulam period played a great ro le in whoWith was the very names young of Ulam in and his Rassias Lvov there days i s connected . \quad aIn mathematical\quad 1 936 t erm he comma moved to the USA where he later oncirculating became athe very mathematics famous scientist from the . Scottish\quad The Caf readere´ after the i s S referredecond World to War the . wonderful now widely knownWith theas Ulam names endash of Ulam Hyers endash and Rassias Rassias there stability i s period connected .. Let X a andmathematical Y be real Banach t erm , volumespaces period\quad ..[ The 32 stability ] . \quad of UlamAs endash was mentioned Hyers endash above Rassias ,approximate\quad Ulam is ometries played on a restricted great ro le in circulating the mathematicsnow widely from known the as Scottish Ulam – Caf Hyers $ –\ Rassiasacute{e stability} $ after . the Let SX econdand WorldY be real War . domainsBanach .. S open parenthesis spaces . bounded The stability .. or .. unbounded of Ulam – closing Hyers parenthesis – Rassias .. approximate for .. into .. mapping is ometries .. f : S right arrow Y .. satisfying on restricted domains S ( bounded or unbounded ) for into mapping Withbar the bar f names open parenthesis of Ulam x and closing Rassias parenthesis there minus i f s open connected parenthesis a y mathematical closing parenthesis t bar erm bar , minus bar bar x minus f : S → Y satisfying ynow bar bar widely less or known equal epsilon as Ulam especially−− Hyers where−− Y i sRassias a Banach spacestability period . \quad Let $X$ and $Y$ be real Banach spaces .|| \fquad(x) − fThe(y) || stability − || x − y ||≤ ofε especially Ulam −− whereHyers −−Y i sRassias a Banach approximate space . is ometries on restricted In .. 1 940 StanIn Ulam 1 940 stated Stan the Ulam problem stated concerning the problem the stability concerning of homomor the hyphen stability of homomor - domainsphisms :\ ..quad Let G sub$S 1 .. be ( $a group bounded and let\ Gquad sub 2or be\ aquad metricunbounded group with a ) metric\quad d andf o r let\quad i n t o \quad mapping \quad phisms : Let G be a group and let G be a metric group with a metric d $ fepsilon : greater S 0\ ..rightarrow be given period1 Y .. Does$ \ therequad exists a t a i s delta f y i n greater g2 0 such that if a function h : G sub 1 right arrow G sub 2 and let ε > 0 be given . Does there exist a δ > 0 such that if a function satisfies the inequality d open parenthesis h open parenthesis xy closing parenthesis comma h open parenthesis x closing \noindenth : $ G\mid1 → G\2midsatisfiesf the ( inequality x ) −d(h(xyf), h( (x)h(y y)) < )δ for\ allmid x, y\mid∈ G1, −then \mid \mid parenthesis hthere open exists parenthesis y closing parenthesis closing parenthesis less delta for all x comma y in G sub 1 comma .. then therex − existsy \mid \mid \ leq \ varepsilon $ especially where $ Y $ i s a Banach space . a homomorphism H : G1 → G2 with d(h(x),H(x)) < ε for all x, ∈ G1? Roughly a homomorphismspeaking H : : G subWhen 1 right does arrow a G linear sub 2 with mapping d open near parenthesis an h open “ approximately parenthesis x closing linear parenthesis ” comma H openIn \quad parenthesis1 940 x closing Stan parenthesis Ulam stated closing the parenthesis problem less concerning epsilon for all the x comma stability in G sub of 1 ? homomor .. Roughly− mapping exists ? T . M . Rassias gave a so lut ion in [ 28 ] , introducing some phismsspeaking : :\ ..quad WhenLet does a $ linear G { mapping1 }$ near\quad an ..be quotedblleft a group approximately and let $G linear{ quotedblright2 }$ be a mapping metric group with a metric $ d $ andcondition l e t for map - pings between Banach spaces . A particular case of exists ? ..Rassias T period ’ s M theorem period Rassias was the gave result a so lut of ion Donald in open H square .Hyers bracket 28 ( closing [ 23 ] square ) . bracket Now itcomma introducing some$ \ conditionvarepsilon for map hyphen> 0 $ \quad be given . \quad Does there exist a $ \ delta > 0 $ such that if a function $ h :can G be{ 1 said}\ thatrightarrow the study ofG Ulam{ 2 –}$ Hyers – Rassias stability in it s present pings betweenform Banach was spaces started period by .. the A particular paper .. of case Th .. of . Rassias M . Rassias quoteright [ s 28 theorem ] . was For the more satisfiestheinequalityresult of Donald H period .. Hyers $d .. open ( parenthesis h ( open xy square ) bracket , 23 h closing ( square x bracket ) h closing ( parenthesis y ) period ) < \ delta $information forall $x of such , kind y of stability\ in G and{ the1 } basic, $ papers\quad onthen the subjthere ect exists , see [ 24 ] .. Now it .. can. be said that the study of Ulam endash Hyers endash Rassias .. stability in it s .. present .. form was .. started by the paper .. of Th period M period \noindent a homomorphism $HLet us : end G with{ 1 some}\ morerightarrow anecdotesG . { 2 }$ with $ d ( h ( Rassias openThere square bracket is 28 a closing story square to bracket the period effect .. For more that information , upon of such publication kind of stability and the basic papersx ) , H ( x ) ) < \ varepsilon $ f o r a l l $ x , \ in G { 1 } ? $ \quad Roughly speaking, : \ Banachquad When ’ s does monograph a linear “ mapping Theory of near operations an \quad .‘‘ Linear approximately operations linear ” was ’’ mapping on the subj ect comma see open square bracket 24 closing square bracket period 00 e x i s t s $ ? $ \quad T . M . Rassias gave a so lut ion in [ 28 ] , introducing some condition for map − Let us enddisplayed with some in more some anecdotes Lvov bookshops period on shelves labelled “ Medical Books pingsThere between .. is .. a .. Banach story .. to spaces .. the .. . effect\quad .. thatA commaparticular .. upon\ ..quad publicationcase \ commaquad ..of Banach Rassias quoteright. ’ s theorem s .. monograph was the resultquotedblleft of DonaldIn Theory some of H Polish operations . \quad “ periodHyers cit ies ..\ Linear ,quad including operations( [ 23 Krak ] quotedblright )o´ . w\quad and was WarsawNow displayed i t ,\quad there in somecan are Lvov streets be bookshops said that the study of Ulam −− Hyerson shelves−−namedRassias labelled “ .. Banach\ quotedblleftquad streetstability Medical ” . Case in In Warsaw it 1 quotedblright s \quad , nowpresent the Case Mathematical 2 period\quad form Institute was \quad of Warsawstarted by the paper \quad o f Th . M . RassiasIn some PolishUniver [ 28 .. ] - quotedblleft . s ity\quad hasFor it cit s ies house more comma at information Banach including street Krak of acute-o . such The w kind and 1 Warsaw of 983 stability International comma there and Congress are the streets basic named papers onquotedblleft the subjof Mathe Banach ect - ,street maticians see quotedblright [ 24 took ] . placeperiod in .. In Warsaw Warsaw . comma A fewnow theforeign Mathematical mathematicians Institute of found Warsaw Univer hyphen s ity hasout it s house that at there Banach i s street a street period in .. Warsaw The .. 1called 983 International Banach Street Congress , of Matheand this hyphen is the last \ centerlinematiciansstop took{ Let on place aus in certain end Warsaw with trolley period some ..line moreA few . foreign anecdotes Curious mathematicians about . } Banach found outStreet that , they got on the there i strolley a street ,in got Warsaw off called Banach Street comma .. and this is the last stop on a Therecertain\quad trolleyi s line\quad perioda ..\ Curiousquad s about t o r y Banach\quad Streetto \quad commathe they\ gotquad on thee f f trolley e c t \ commaquad that got off , \quad upon \quad publication , \quad Banach ’ s \quad monograph ‘‘ Theory of operations . \quad Linear operations ’’ was displayed in some Lvov bookshops on shelves labelled \quad ‘ ‘ Medical $\ l e f t . Books\ begin { a l i g n e d } &’’ \\ &. \end{ a l i g n e d }\ right . $

In some Polish \quad ‘‘ cit ies , including Krak $ \acute{o} $ w and Warsaw , there are streets named ‘‘ Banach street ’’ . \quad In Warsaw , now the Mathematical Institute of Warsaw Univer − s ity has it s house at Banach street . \quad The \quad 1 983 International Congress of Mathe − maticians took place in Warsaw . \quad A few foreign mathematicians found out that there i s a street in Warsaw called Banach Street , \quad and this is the last stop on a certain trolley line . \quad Curious about Banach Street , they got on the trolley , got off ON STEFAN BANACH AND SOME OF HIS RESULTS .. 9 \ hspaceat the∗{\ lastf stop i l l } commaON STEFAN .. and BANACHwere confronted AND SOME by a s OF izable HIS empty RESULTS area period\quad ..9 They arrived at the unanimous conclusion that what they were facing was not .. quotedblleft Banach street quotedblright \noindent at the last stop , \quad andON were STEFAN confronted BANACH by AND a SOME s izable OF HIS empty RESULTS area 9 . \quad They arrived at but ratherat a the quotedblleft last stop Banach , and space were quotedblright confronted period by a s izable empty area . They arrived theThe unanimous li st of references conclusion is far from that complete what comma they as were thereare facing enormous was numbers not \quad of ‘‘ Banach street ’’ but ratherat the a ‘‘ unanimous Banach space conclusion ’’ . that what they were facing was not “ Banach street papers connected” but rather with Banach a “ Banach period .. space Here ”comma . except the original papers mentioned in the article commaThe are li given st of only references some papers is farin English from completewhere the reader , as there can find are several enormous numbers of Theadditional li st ofinformation references period is far from complete , as there are enormous numbers of papers connectedpapers connected with Banach with Banach . \quad .Here Here , except, except the the original papers papers mentioned mentioned in in the Acknowledgementsthe article : .. , The are photographsgiven only somewere taken papers by Danuta in English C iesielska where and the reader can find several articleKrzysztof , Ciesielski are given period only some papers in English where the reader can find several additionaladditional information information . . References Acknowledgements : The photographs were taken by Danuta C iesielska 1 periodand .. L period Krzysztof Alaoglu Ciesielski comma .. . Weak topologies of normed linear spaces comma Ann period Math period 4 1 open parenthesisAcknowledgements 1 940 closing : parenthesis\quad The comma photographs 252 endash were 267 period taken by Danuta C iesielska and Krzysztof2 period .. S Ciesielski period Banach . comma Sur le probl acute-eReferences me de la mesure comma Fund period Math period 4 open parenthesis 1 . L . Alaoglu , Weak topologies of normed linear spaces , Ann . Math . 4 1 ( 1 940 ) , 252 – 267 . 1 924 closing parenthesis comma 7 endash 33 period 2 . S . Banach , Sur le probl e´ me de la mesure , Fund . Math . 4 ( 1 924 ) , 7 – 33 . \ centerline3 period .. S{ periodReferences Banach} comma Sur les fonctionelles lin acute-e aires comma Studia Math period 1 open parenthesis 1 929 3 . S . Banach , Sur les fonctionelles lin e´ aires , Studia Math . 1 ( 1 929 ) , 2 1 1 – 2 1 6 . closing parenthesis comma 2 1 1 endash 2 1 6 period 4 . S . Banach , Sur les fonctionelles lin e´ aires II , Studia Math . 1 ( 1 929 ) , 223 – 239 . \ centerline4 period .. S{1 period . \quad BanachL comma . Alaoglu Sur les , fonctionelles\quad Weak lin acute-e topologies aires II ofcomma normed Studia linear Math period spaces 1 open , Ann parenthesis . Math 1 . 4 1 ( 1 940 ) , 252 −− 267 . } 5 . S . Banach , Sur les ope´ rations dans les ensembles abstraits e t leur application aux e´ 929 closing parenthesis comma 223 endash 239 period quations int e´ grales , Fund . Math . 3 ( 1 922 ) , 1 33 – 1 8 1 . \ centerline5 period .. S{2 period . \quad BanachS comma . Banach Sur les , op Sur to the le power probl of acute-e $ \acute rations{e} dans$ les me ensembles de la mesure abstraits e, t Fund leur application . Math . 4 ( 1 924 ) , 7 −− 33 . } 6 . S . Banach , Th e´ o rie des ope´ rations lin e´ aires , Monografie Matematyczne 1 , aux e-acute quations Warszawa 1 932 . 7 . S . Banach , H . Steinhaus , Sur le principle de la condensation de s ingularit \ centerlineint acute-e grales{3 . comma\quad FundS . Banachperiod Math , Sur period les 3 open fonctionelles parenthesis 1 lin922 closing $ \acute parenthesis{e} $ comma aires 1 33 , endash Studia 1 8 Math 1 . 1 ( 1 929 ) , 2 1 1 −− 2 1 6 . } e´ s , Fund . Math . 9 ( 1 927 ) , 50 – 6 1 . period 8 . H . F . Bohnenblust and A . Sobczyk , Extensions of fuctionals on complete linear spaces \ centerline6 period .. S{ period4 . \quad BanachS comma . Banach Th acute-e , Sur o rie les des fonctionelles op to the power of e-acutelin $ rations\acute lin{ acute-ee} $ aires aires comma II Monografie , Studia Math . 1 ( 1 929 ) , 223 −− 239 . } , Bull . Amer . Math . Soc . Fund . Math . 44 ( 1 938 ) , 9 1 – 93 . Matematyczne 1 comma Warszawa 1 932 period 9 . S . Banach and A . Tarski , Sur la d e´ composition des ensembles de points en parties 5 .7 period\quad ..S S . period Banach Banach , Sur comma les H period $ op ˆSteinhaus{\acute comma{e}} Sur$ le rations principle dans de la condensation les ensembles de s ingularit abstraits acute-e e t s leur application aux respec - tivement congruentes , Fund . Math . 6 ( 1 924 ) , 244 – 277 . comma$ \acute Fund{e period} $ quationsMath period 1 0 . K . Ciesielski , Lost Legends of Lvov 1 : The Scottish Caf e´ Math . Intelligencer 9 ( 1 i n9 t open $ parenthesis\acute{e} 1 927$ closing grales parenthesis , Fund comma. Math 50 . endash 3 ( 1 6 1922 period ) , 1 33, −− 1 8 1 . 987 ) no . 4 , 36 – 37 . 8 period .. H period F period Bohnenblust and A period Sobczyk comma Extensions of fuctionals on complete linear spaces 1 1 . K . Ciesielski , Lost Legends of Lvov 2 : Banach ’ s Grave , Math . Intelligencer 10 ( 1 comma6 . \quad Bull periodS . Banach , Th $ \acute{e} $ o rie des $opˆ{\acute{e}}$ rations lin $ \acute{e} $ 988 ) no . 1 , 50 – 5 1 . airesAmer , period Monografie Math period Matematyczne Soc period Fund 1 , period Warszawa Math 1 period 932 44 . open parenthesis 1 938 closing parenthesis comma 9 1 1 2 . K . Ciesielski , On some details of Stefan Banach ’ s life , Opuscula Math . 13 ( 1 993 ) endash7 . \ 93quad periodS . Banach , H . Steinhaus , Sur le principle de la condensation de s ingularit $ \acute{e} $ , 7 1 – 74 . 1 3 . K . Ciesielski and Z . Pogoda , Conversation with Andrzej Turowicz s ,9 period Fund .. . S Math period . Banach and A period Tarski comma Sur la d acute-e composition des ensembles de points en parties respec , Math . Intelligencer 10 ( 1 988 ) no . 4 , 1 3 – 20 . 1 4 . K . Ciesielski and Z . Pogoda , hyphen9 ( 1 927 ) , 50 −− 6 1 . Mathematical diamonds , manuscript , the translation from the book published in Polish “ Diamenty tivement congruentes comma Fund period Math period 6 open parenthesis 1 924 closing parenthesis comma 244 endash 277 matematyki ” , Pro ´szyn´ ski i S - ka , 1 997 . 1 5 . R . Duda , The dis covery of Banach spaces period8 . \quad H . F . Bohnenblust and A . Sobczyk , Extensions of fuctionals on complete linear spaces , Bull . , in : European mathematics in the last centuries ( W . Wi e − cedilla slaw , ed . ) , Wroclaw 2005 Amer1 0 period . Math .. K. Socperiod . Ciesielski Fund . Mathcomma . Lost 44 Legends ( 1 938 of ) Lvov , 9 1 1 :−− .. The93 Scottish . Caf acute-e sub comma Math period , 37 – 46 . 1 6 . N . Dunford and J . T . Schwartz , Linear Operators , Interscience Publishers , Intelligencer 9 open parenthesis 1 987 closing parenthesis no period 4 comma New York , vol . I , 1 958 . 1 7 . P . Enflo , A counterexample to the approximation property in 9 .36\ endashquad S 37 . period Banach andA . Tarski , Sur la d $ \acute{e} $ composition des ensembles de points en parties respec − Banach spaces , Acta Math . 130 ( 1 973 ) , 309 – 3 1 7 . tivement1 1 period congruentes.. K period Ciesielski , Fund comma . Math Lost Legends . 6 ( of 1 Lvov 924 2 ) : Banach , 244 −− quoteright277 . s Grave comma Math period Intelligencer 1 8 . R . M . French , The Banach - Tarski theorem , Math . Intelligencer 10 ( 1 988 ) no . 4 , 10 open parenthesis 1 988 closing parenthesis no period 1 comma 2 1 – 28 . 1 9 . A . Grothendieck , Produits t ensoriels topologiques e t espaces nucleaires , Mem \noindent50 endash1 5 1 0 period . \quad K . Ciesielski , Lost Legends of Lvov 1 : \quad The Scottish Caf $ \acute{e} { , }$ . Amer . Math . Soc . 16 ( 1 955 ) . 20 . H . Hahn , U¨ ber lineare Gleichungssysteme Math1 2 .period Intelligencer .. K period Ciesielski 9 ( 1 comma 987 ) On no some . 4 details , of Stefan Banach quoteright s life comma Opuscula Math period 13 in linearen R a¨ umen , J . Reine Angew . Math . 157 ( 1 927 ) 2 1 4 – 229 . 2 1 . E . open36 −− parenthesis37 . 1 993 closing parenthesis comma 7 1 endash 74 period Helly , U¨ ber lineare operationenen , Sitzgsber . Akad . Wiss . Wien . Math - Nat . 12 1 ( 1 9 1 3 period .. K period .. Ciesielski and Z period Pogoda comma .. Conversation .. with Andrzej .. Turowicz comma .. Math 1 2 ) , 265 – 297 . period\noindent .. Intelligencer1 1 . \quad K . Ciesielski , Lost Legends of Lvov 2 : Banach ’ s Grave , Math . Intelligencer 10 ( 1 988 ) no . 1 , 22 . H . Hochstadt and E . Helly , Father of the Hahn - Banach Theorem , 5010−− open5 parenthesis 1 . 1 988 closing parenthesis no period 4 comma 1 3 endash 20 period Math . Intelligencer 2 ( 1 980 ) , no . 3 , 1 23 – 1 25 . 23 . D . Hyers , On the s ta b ility of 1 4 period .. K period Ciesielski and Z period Pogoda comma Mathematical diamonds comma manuscript comma the translation the linear functional equation , Proc . Nat . Acad . Sci . USA 12 1 ( 1 941 ) , 222 - 224 . from\noindent the 1 2 . \quad K . Ciesielski , On some details of Stefan Banach ’ s life , Opuscula Math . 13 ( 1 993 ) , 7 1 −− 74 . 1book 3 . published\quad K. in Polish\quad quotedblleftCiesielski Diamenty and Zmatematyki . Pogoda quotedblright , \quad Conversation comma Pr o-acute\quad szy towith the power Andrzej of n-acute\quad ski Turowicz , \quad Math . \quad Intelligencer i S10(1988)no hyphen ka comma 1 . 997 4 period ,13 −− 20 . 11 4 5 period. \quad .. RK period . Ciesielski Duda comma and .. The Z dis . Pogodacovery of Banach , Mathematical spaces comma diamonds in : European , manuscript mathematics in , the the last translation centuries from the bookopen published parenthesis W in period Polish Wi e-cedilla ‘‘ Diamenty s l-suppress matematyki aw comma ed ’’ period , Pr closing $ \acute parenthesis{o} commaszy ˆWroc{\acute l-suppress{n}} aw$ 2005 s k i i S − ka , 1 997 . comma1 5 . 37\ endashquad R 46 . period Duda , \quad The dis covery of Banach spaces , in : European mathematics in the last centuries (1 W 6 . period Wi ..$ eN− periodc e d i l l Dunford a $ s and\ l Jaw period , ed T . period ) , SchwartzWroc \ l commaaw 2005 Linear , 37 Operators−− 46 comma . Interscience Publishers comma1 6 . New\quad YorkN comma . Dunford vol period and J . T . Schwartz , Linear Operators , Interscience Publishers , New York , vol . II , comma 1 958 1 958 . period 11 7 7 . period\quad .. PP period . Enflo Enflo , comma A counterexample A counterexample to to the the approximation approximation property property in Banach in spacesBanach comma spaces Acta , Math Acta Math . period130 ( 1 973 ) , 309 −− 3 1 7 . 130 open parenthesis 1 973 closing parenthesis comma 309 endash 3 1 7 period \noindent1 8 period1 .. R8 period . \quad M periodR . M French . French comma The , The Banach Banach hyphen− Tarski theorem theorem comma , Math Math period . Intelligencer Intelligencer 10 10 open ( 1 988 ) no . 4 , 2 1 −− 28 . parenthesis1 9 . \quad 1 988A closing . Grothendieck parenthesis no period , Produits 4 comma t 2 ensoriels1 endash 28 period topologiques e t espaces nucleaires , Mem . Amer . Math . Soc1 9 period. 16 ( .. A1 period 955 ) Grothendieck . comma Produits t ensoriels topologiques e t espaces nucleaires comma Mem period Amer period20 . Math\quad periodH. \quad Hahn $ , \ddot{U} $ ber lineare \quad Gleichungssysteme \quad in linearen R $ \Socddot period{a} 16$ open umen parenthesis , \quad 1J. 955 closing\quad parenthesisReine Angew period . \quad Math . 15720 period ( 1 927 .. H ) period 214 .. Hahn−− 229 comma . U-dieresis ber lineare .. Gleichungssysteme .. in linearen R dieresis-a umen comma .. J period2 1 ... Reine\quad AngewE . period Helly .. Math $ , period\ddot{U} $ ber lineare operationenen , Sitzgsber . Akad . Wiss . Wien . Math − Nat. 121(1912) , 265157−− open297 parenthesis . 1 927 closing parenthesis 2 1 4 endash 229 period 2 1 period .. E period Helly comma U-dieresis ber lineare operationenen comma Sitzgsber period Akad period Wiss period Wien\noindent period Math22 . hyphen\quad NatH. period\quad 12 1Hochstadt open parenthesis\quad 1 9and 1 2 closing\quad parenthesisE. \quad commaHelly , \quad Father \quad o f the \quad Hahn − Banach \quad Theorem , \quad Math . \quad Intelligencer 2(1980)265 endash 297 ,no.3 period ,123 −− 1 25 . 2322 . period\quad .. HD period . Hyers .. Hochstadt , \quad ..On and the .. E periods ta b .. Helly ility comma of the .. Father linear .. of functional the .. Hahn hyphen equation Banach , .. Proc Theorem . Nat . \quad Acad . Sci . USA comma12 1 .. ( Math 1 941 period ) , .. 222 Intelligencer− 224 . 2 open parenthesis 1 980 closing parenthesis comma no period 3 comma 1 23 endash 1 25 period 23 period .. D period Hyers comma .. On the s ta b ility of the linear functional equation comma Proc period Nat period .. Acad period Sci period USA 12 1 open parenthesis 1 941 closing parenthesis comma 222 hyphen 224 period 1 0 .. K period CIESIELSKI \noindent24 period ..1 D 0 period\quad H periodK.CIESIELSKI Hyers and T period M period Rassias comma Approximate homomorphisms comma Aequationes Math period comma 44 open parenthesis 1 992 closing parenthesis comma 1 0 K . CIESIELSKI \noindent1 25 endash24 1 53 . \ periodquad D . H . Hyers and T . M . Rassias , Approximate homomorphisms , Aequationes Math . , 44 ( 1 992 ) , 24 . D . H . Hyers and T . M . Rassias , Approximate homomorphisms , Aequationes Math . , 125 25 period−− 1 .. 53 E period . Jakimowicz and A period Miranowicz open parenthesis eds period closing parenthesis comma Stefan 44 ( 1 992 ) , 1 25 – 1 53 . Banach endash remarkable life comma .. bril liant math hyphen 25 . E . Jakimowicz and A . Miranowicz ( eds . ) , Stefan Banach – remarkable life , bril liant \noindentematics comma25 . Gda\quad n-acuteE . sk Jakimowicz University Press and comma A . AdamMiranowicz Mickiewicz ( eds University . ), Press Stefan comma Banach 2007 period−− remarkable life , \quad bril liant math − math - ematics , Gdan ´ sk University Press , Adam Mickiewicz University Press , 2007 . 26 . R ematics26 period , .. Gda R period $ \ Kaacute suppress-l{n} $ u z-dotaccent sk University a comma Press Through , Adam a reporter Mickiewicz quoteright Universitys eyes period .. Press The Life , of 2007 Stefan . . Kalu z ˙ a , Through a reporter ’ s eyes . The Life of Stefan Banach , Birkha ¨ user 1 996 . 27 Banach26 . comma\quad BirkhR . Ka dieresis-a\ l u user $ \ 1dot 996{ periodz} $ a , Through a reporter ’ s eyes . \quad The Life of Stefan Banach , Birkh . R . D . Mauldin ( ed . ) , The Scottish Book . Mathematics from the Scottish Cafe ´ Birkha ¨ $ \27ddot period{a} ..$ R userperiod 1 D 996 period . Mauldin open parenthesis ed period closing parenthesis comma, The Scottish Book period user 1 98 1 . Mathematics27 . \quad fromR the . D Scottish . Mauldin Caf acute-e ( ed sub . comma) , The Birkh Scottish a-dieresis Book user . Mathematics from the Scottish Caf $ \acute{e} { , }$ 28 . T . M . Rassias , On the stability of the linear mapping in Banach spaces , Proc . Amer . Birkh1 98 1 $ period\ddot{a} $ user Math . Soc . 72 ( 1 978 ) , 297 – 300 . 29 . W . Rudin , Functional analysis , second edition , 128 98 period 1 . .. T period M period Rassias comma On the stability of the linear mapping in Banach spaces comma Proc period McGraw - Hill , 1 99 1 . 30 . M . Reed , B . Simon , Methods of modern mathematical physics , I Amer period Math period . Functional analysis , Aca - demic Press , 1 972 . 3 1 . G . A . Soukhomlinov , U¨ ber Fortsetzung \noindentSoc period28 72 open . \quad parenthesisT . M 1 978. Rassias closing parenthesis , On the comma stability 297 endash of 300 the period linear mapping in Banach spaces , Proc . Amer . Math . von linearen Funktionalen in linearen komplexen Ra ¨ umen und linearen Quaternionenra ¨ umen , Soc29 period . 72 ..( W 1 978period ) Rudin , 297 comma−− 300 Functional . analysis comma second edition comma McGraw hyphen Hill comma 1 99 1 Mat . Sb . 3 ( 1 938 ) , 353 – 358 . period29 . \quad W . Rudin , Functional analysis , second edition , McGraw − H i l l , 1 99 1 . 32 . S . Ulam 1 909 – 1 984 . Los Alamos Sci . No . 1 5 , Special Issue ( 1 987 ) , Los Alamos 3030 . period\quad ..M M . period Reed Reed , B comma . Simon B period , Methods Simon of comma modern Methods mathematical of modern mathematical physics , physicsI . Functional comma I period analysis , Aca − National Laboratory , Los Alamos , NM , 1 987 . pp . 1 – 3 1 8 . Functionaldemic Press analysis , comma 1 972 Aca . hyphen 1 Mathematics Insitute , Jagiellonian University , Reymonta 4 , 3 0 - 0 59 Krak 3demic 1 . \ Pressquad commaG . A 1 972 . Soukhomlinov period $ , \ddot{U} $ ber Fortsetzung von linearen Funktionalen in linearen komplexen o´ w , Poland . R3 $1 period\ddot{ ..a} G$ period umen A undperiod linearen Soukhomlinov Quaternionenr comma U-dieresis $ \ berddot Fortsetzung{a} $ umen von linearen , Mat Funktionalen . Sb . 3 ( in 1 linearen 938 ) , 353 −− 358 . E - mail address : K rzy sz t o f . Cie s i el ski @ i m . u j . edu . pl komplexen \noindentR a-dieresis32 umen . \quad und linearenS . Ulam Quaternionenr 1 909 −− a-dieresis1 984 umen . Los comma Alamos Mat period Sci . Sb No period . \quad 3 open1 parenthesis 5 , Special 1 938 closingIssue ( 1 987 ) , Los Alamos National parenthesisLaboratory comma , Los 353 endash Alamos 358 , period NM , 1 987 . pp . 1 −− 3 1 8 . 32 period .. S period Ulam 1 909 endash 1 984 period Los Alamos Sci period No period .. 1 5 comma Special Issue open parenthesis1 Mathematics 1 987 closing Insitute parenthesis , Jagiellonian comma Los Alamos University National , Reymonta 4 , 3 0 − 0 59 Krak $ \acute{o} $ w ,Laboratory comma Los Alamos comma NM comma 1 987 period pp period 1 endash 3 1 8 period Poland1 Mathematics . Insitute comma Jagiellonian University comma Reymonta 4 comma 3 0 hyphen 0 59 Krak o-acute w comma Poland period \ centerlineE hyphen mail{E address− mail : K address rzy sz t o :Krzy f period Cie sz s t i el o ski f at . i Cie m period s i u el j period ski edu $@$ period im. pl u j . edu . pl }