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a n centerlineRow 1 6 Rowfa 2g i-dieresis-u . sub e y-d to the power of m-e E-a n s-O-dieresis-t sub n a-T e-l h-a n-n i-d T-a sub o-l U-s u i-n-n T-e sub r-e s-r z-i t-i sub o-y g-breve lu n [MDepartment−Sn begin f ofarray Mathematicsgf cg 6 nn i−d i e r e s i s −u nendf array g l−d f e y−d ^f m−e E−a g n g s−O−d i e r e s i s −t f n a−T g e−l h−a n−n i−d T−a f o−l Ua−s u i−n−n g T−e f r−e s−r z−i g t−i f o−y nbrevefgg g lu n ] 6 M−S l−dey−dm−eE−ans − O − dieresis − tna−Te − lh − an − ni − dT − ao−lU−sui−n−nT − er−es−rz−it − io−y˘glu nnoindent iDepartment− dieresis − u of Mathematics Department of Mathematics CONTENTS n centerlineIntroductionfCONTENTS .... period .... period ....g period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... period .... periodnnoindent .... periodIntroduction .... period ....n periodh f i l l ..... periodn h f i ....l l period. n h f .... i l l period. n h .... f i lperiod l . n ....h f period i l l . ....n h 5 f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l . n h f i l l 5 1 period .. Almost bounded sets and operators ..CONTENTS period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..nnoindent periodIntroduction .. period1 . ..n .quad period . .Almost .. period . bounded .. period . .. sets period . and ... period . operators . .. period . .n ..quad . period . .. ..nquad period . .. .. .n periodquad . .. .. period .nquad . .. 56. nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 6 22 . period1nquad . .. Almost EidelheitEidelheit bounded quoteright ' sets s and theorem s theorem operatorsn ..quad period .. ..nquad period .. ...n periodquad . .. periodnquad . .. .. periodnquad . .. period. n .quad .. period.. n .quad .. period . n ..quad period. ..nquad period. .. nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 1 3 period. .. period . .. period . .. period . .. period . 6 .. 2period . Eidelheit .. period ' s.. theorem period .. period . .. period . .. period . .. period . .. period . .. period .. period .. periodnnoindent....................13 .. period3 .. periodnquad ..Nuclear period .. 1 K 3 $ nddotfog $ the quotients nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 20 43 . period3nquad . .. Nuclear NuclearNuclear K Ko¨ o-dieresisKthe quotients $ nddot the quotientsfo .g $ . ..the period. subspaces . .. period . ..and period . completing . .. period . .. sequences period . .. period .nquad ... period .nquad .. period. n ..quad period. ..n periodquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 22 .. period. .. period . .. period . .. period . .. period . .. period . .. period . 20 .. period 4 . .. Nuclear period K ..o¨ periodthe subspaces .. period and .. period completing .. period .. period .. period .. periodnnoindentsequences .. period5 .. 20n .quad .Applications . .nquad . .. nquad . .. n .quad .. .nquad .. .nquad 22 . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 25 64 . period5nquad . .. Applications NuclearSpaces K of o-dieresis . continuous . the . subspaces . functions . and . completingn .quad . sequences. .nquad . ... periodnquad . .. period. nquad . .. period. n .quad .. .period. .nquad .. period. ..nquad period. ..n periodquad ... nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 30 periodReferences. .. period . ...nquad period .. ...n periodquad . .. .. periodnquad . .. .. periodn .quad .. period. .nquad .. period. 25nquad ..6 . 22 Spaces. nquad of continuous. nquad functions. nquad .. n .quad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 35 5 period......................30 .. Applications .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. periodn hspaceReferences .. period∗fn f i l.. l periodg .1 9 . 9 .. period1 . Mathematics . .. period . .. periodSubject . .. period . Classification . .. period . .. period . : Primary. .. period . 46 .. period .A 3 . ,.. 46period . A . 4 .. period, . 46 A .. 45period ; Secondary .. period .. 46 A 1 1 , period. .. period . ... period . ... period . .. period . .. 25 . 35 nnoindent6 period ..46A13 Spaces1 of9 9 continuous 1 ,46E10Mathematics functions Subject . .. Classification period .. period: Primary .. period 46 A .. 3 period , 46 A 4.. , period 46 A 45 .. ;period Secondary .. period 46 A 1 .. 1 ,period .. period .. period .. period46 .. periodA 1 3 , ..46 period E 1 0 . .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 30 n centerlineReferences ..f Received period .. period 1 0Received .. period6 . 1 1 0 .. 99 6period . 1 1 99 ; .. 1 revised ; period revised .. version versionperiod 1 ..2 . period 1 1 2 . 1 . .. 99 1 period 1 1 . .. 1 period 99 1 .. periodg .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 35 1 9 9 1 Mathematics Subject Classification : Primary 46 A 3 comma 46 A 4 comma 46 A 45 semicolon Secondary 46 A 1 1 comma 46 A 1 3 comma 46 E 1 0 period Received 1 0 period 6 period 1 99 1 semicolon revised version 1 2 period 1 1 period 1 99 1 period Introduction n centerlineIn this articlef Introduction our purpose willg be t o examine the conditions under which a locally convex space has a subspace or a quotient space which i s isomorphic to Ina this Fr acute-e article chet space our period purpose .. The will concrete be t Fr o e-acute examine chet spacesthe conditions we have in mind under are the which nuclear a locallyK o-dieresis convex the spaces space or the has space a subspace omega simeq orIntroduction K a to quotient the power of space N period which .. By i anuclear s isomorphic K o-dieresis to the space we mean a nuclear aFr Fr acute-e $Inn thisacute chet article spacefeg ourwhich$ purpose chet has a space basis will be and . t aonquadcontinuous examineThe the norm concrete conditions period Fr under $ whichnacute a locallyfeg $ convex chet space spaces has a we have in mind are the nuclear In opensubspace square or bracket a quotient 8 closing space squarewhich i bracket s isomorphic Eidelheit to a has Fr provede ´ chet spacethat every . The proper concrete Fr acute-e Fre ´ chet space spaces has we omega as a quotient nnoindentspacehave period inK mind .. $ For arenddot another thef nuclearog proof$ theof this spaces and results or about the space kernels of $ surjectionsnomega ontonsimeq omega commaK ^f N g . $ nquad By a nuclear K N $ nweddotK refero ¨fothe tg o$ spaces Vogt the .. or openspace the space square we! bracket mean' K : a 19 nuclearBy closing a nuclear square K bracketo ¨ the space period we .. mean The asearch nuclear for Fr nucleare ´ chet K space o-dieresis which the quotients of Fr acute-e chetFr spaceshas $ n aacute basisf andeg a$ continuous chet space norm . which has a basis and a continuous norm . was ..In initiated [ 8 ] Eidelheit by .. Bellenot has proved .. and that .. Dubinsky every proper .. open Fre ´ squarechet space bracket has 1! closingas a quotient square space bracket . period For another .. The present .. authors .. in .. openIn [ squareproof 8 ] ofEidelheitbracket this and 1 3 results closing has about proved square kernels bracket that of surjections.. every open parenthesis proper onto !; Frwe cf referperiod $ n tacute o Vogtfeg [ 19$ ] . chet The space search for has nuclear $ nomega $ as a quotient spacealsoK openo ¨ . then squarequad quotients bracketFor of another Fr 1 2e ´ closingchet proofspaces square was of bracket this initiated closing and by parenthesisresults Bellenot about showed and kernels that Dubinsky the assumption of [surjections 1 ] . of Theseparability present onto in the $ n theoremomega of Bellenot, $ weand referauthors Dubinsky t o in i Vogts redundant [ 1 3nquad ] ( period cf[ . 19 .. ]More . n preciselyquad The comma search they have for proved nuclear that K either $ n everyddotfog $ the quotients of Fr $ nacutefeg $ chetcontinuousalso spaces [ 1 2 operator ] ) showed from that a given the assumption Fr acute-e of chet separability space E into in the any theorem nuclear of Fr Bellenot e-acute andchet Dubinsky space comma i s redundant waswhich.nquad admits Moreinitiated precisely a continuous , they by normn havequad comma provedB e il l s that e boundedn o t eithernquad or every Eand has continuous an nuclearquad Dubinsky operator K dieresis-o fromn thequad aquotient given[ Fr1 periode ]´ chet .