TRU Matheπ匿tics 15−2 〔1979）
COUNTABLE−CODIMENSIONAL SUBSPACES
OF TOPOLOGICAL VECTOR SPACES
Kazuo KERA， Naoki KITSUNEZAKI and Yutaka TERAO （Received November 16， 1979） N．Adasch， B． Emst and D． Ke迦［2］have proved that， in an ultrabarrelled space， a subspace of countable codjmension is also ultrabarrelled．〔ln［2］，the word”barrelled，’is used in stead of”ultrabarrelled°°in the sa皿e sense．） In this paper， we shall show that，皿der some conditions， the properties of l）eing uユtrabornologica1， countably ultrabarrelled or countably quasi二Ultra− barrelled are hereditary on pass血g to a subspace of countable cod迦ensi㎝． For locally convex spaces， the similar results have been obtained by J． H． Webb ［7］［8］． Throughout this paper， we assume that each vector topology is of Hausdorff t）？e． If a vector space E is e（luipped with a vector topologyτ， we denote this topological vector space by E［τ］． The following definitions are given by S．0． lyahen［4］． A topological vect・r・pace・E［・］i・callgd th・ ＊−inductiv・Z励・f・f・mily｛E。［・。］｝・f t・p・1・gical vect・r・paces by 1血ea・m・pPing・｛・α｝・血・・e・f・r each・・㌔卿・ 互。［τ。］int・E［・］・・d rh・u・i・n・f th・ve・t・r・ub・paces u。（E。）・P・n・E・if・i・ th・f血est vect・r t・p・1・9）「 m・king・v・ry u。、 c・nt血・・u・・In parti・己ar・E［・］i・ ・a11・d th・・t・ict ＊←indu・tive励鋤f｛En［・。］・炉ヱ・2・…｝・if E i・th・皿i・n ・fast・ictly血creas麺g se・lu・nce・f vect・r・ub・paces｛En｝・E［・］i・th・ ＊−induc−tiv・1皿t・f｛En［・。］｝by th・id・ntity卿P血9…nd・f・r each…。c・in・ides
with th・t。P・1・gy induced・by・n＋ヱ・ 』 Atopological vector space E［τ］is called uZtrαbornoZogicaZ if every bo皿d− ed linear mapping from E［T］ into any topological vector space is conti皿uous ［4］． F・rat・P・1・gical・ve・t・r・pace亦］，・・t・ub b。 th。 ultrab。rn。1。gica1・。p。1。gy associated withτ， which is the finest vector topology having the sante bounded subsets as τ ［1］． ㎜口・矧・］わθα閲Z励・抱・Z・θ鋤Z・p・…｛α｝あ・α輌W・f．V・・カ・・・・β・汐・・…”・励微けゐ・㈱伽・芦♂・E・α城・。カ・th・励…dt・ρ・Z・ev
㌶㌫嘘㌫d，欝・㌫ls㌶1；θ㌶㌘h ［τ］
N 4748 K．】くERA， N． KITSUNEZAKI AND Y． TERAO r PPoρτ・ Iet τ．be t｝ほvector topology on E such that E［τ．］ is the ★−induc− ・ive・皿・Of｛Ea［・。功】｝by・h・id・就i・y㎜卿g・． C・・訂・y・・i・f血er・㎞・．
1：：e㌶隠1エ轡Bぷ1↑1、蒜霊芸㌫：，：°C。e，9［2］二
bou【1φed linear maPPing， and it is c㎝tinuous， fbr亙IT］ is UltrabOrnologica1． Thus τ is filler thal1τ’． Hence we have T＝τ’． The fbllowing c（mditi㎝、 fbr a vector sdbspace P of a topological vector space E1［τ］is used by M．・Valdivia【6］and J． H．．Wel）b［7］【8］． ’ （☆）跡θα・h・b・taz・led subset B・f E［τ］。踊8・了拘鋤・・dinens伽玩伽・Zinear勧ZZ o了｛PUB｝． 』
In the fbllowing， fbr a vector sUl）space F of a topological vector space 亙【τ］・鳩den・te by i the induce’早@t。P・1・gy．・n Fl byτ・ 、 THEOREM 1・ 五θカE［τ］カθαηz己Z城ornolOθieaZ space3 and Fα〃θoカoヱ。8τψ一 Sρα0θof eoun励Zθ604伽θ随Z㎝劔亙．1アFsatisfies the eonditioh（ft）。 then牢担8αZ80 uZtubomologicaZ．
b・・ア・．L・t｛x。｝bg・b・・i・・f a・・呼1㎝・ntary鴨・t。・・Ub・pac？・f F・Put 亙・＝F皿d弘2＝1in・a・M1・f｛E。・￠。｝（n‘−1・2・…）・th・・Ei・th・皿i㎝・f E。 頑｛E。｝i・飢桓crea・i・g・明・en・6・f vect…mb・pace・。f E・L・t・。』th・ t°P°1°gy？f E。血duced．byτfb・eゆ・・Clea・1y・fぞ・B）・ tbe・・nditi。・（★）・ The ．followi皿g theorem enables us to obtain another resUlts about the here− dity under the condition of the closedness of sUbspaces． ㎜）肥M2・乙・品［・ロ・α抑・Z・φ・αZび・・t・r spa・θ。．Fカ・α・Z・sbd・v・・励 szthspaσe°f e拠鋤Z…di・・n・i・・碗【・L｛⑳。・n−1・2・’°°｝be aカ・・i・・f・6°ゆZ㎝θη剛〃・σ励』限・θG・アろ．写ヱ［・1】b・咽・雇㌦1［・n＋1】カ・輪
Zinea？．huZZ・ア｛与Xn｝Uith the i』・d鋤・Z・ev・n＋ヱ（炉1・2・…〕・1個・】 i・免★−i・・luctiv・z嚇・了｛En【τn】｝痴』w’n・apping・・th・n・G Z・・z…dCtnd eこそ］i・i・・m・rphi・加幼θ⑭琵θηカ・卿θ・了叩】勿σ・ ’
COUNTABLE−CODIMENSIONAL SUBSPACES 49 ke・・τ・By・the assufirption・Ei・th・mi㎝・f E。 ・nd｛En｝i・飢血crea・血9 sequence of vector sdbspaces of E． Consider the proj ectiQPπfrom E［τ］onto F［モ］p・・a11・1 t・G鋤d 1・t㌦b・th・・est・i・ti㎝・f・・n En［・。］・th・n㌦i・ continuous・fbr F is closed and of finite codimension i皿En［τn］・ Since E［τ］ i・the★−i・ductiv・limit・f｛En［・。］｝・・i・c㎝t迦・u・【4］・lt f・11㎝・th・t Gis closed and F［〒］ is isomorphic to the quotient space of E［τ］by C． F・r eaCh・P・・i・i鴨血・・g・・i，・・t ｛・in・炉・…2・…｝be a・equ・・ce・f…− sed balanced neighbourho〔xls of the origin in a topological vector space E［τ］ ・㏄h・ha・Ui”＋’・Ui”＋i・Ua” f・r 9・・炉・，ヱ．・。…・・f f・r・eaCh・n，♂・∩膓．、〃膓i・ 。b。。rb。nt（1，。皿iv。ru、），th㎝〔ρi， ca11。d飢。Z城＝Z〔。カ。？niv。・。u。・uZtra− bαr？eZ）of type（α）． A topological vector space E［τ］is called eountabZy uZt− ？ctbαr？eZZed（countabZy（luasi−uZtT，abctr）re ZZeのif every ultrabarre1（bomivorous ultrabarre1） of type （α） is a neighbouエhood of the origin． 00ROLLARY 1． Let E［τ］』αoo協加力Zy uZtuabαm，eZZed sραoθ． and FαoZosed 抄θ已加r8必spαθ・of・0朔鋤Zθ・・dimension in E［τ］． Then F［モ担SαZSO eOuntubZy
uZtrCtZ）Ctr・reZZed． 』
tW・・f・1・et｛En［・。］｝b・the sdb・paces・斑・］・・n・tru・t・d垣皿…em 2・ A・・E［・］i・c・皿t・bly・Ultrabarrell・d・E［・］i・th・ ＊−indu・tiv・limit・f｛En［・n］｝ ［2］． So， by Theorem 2， F［〒］ is isomo□phic to a quotient space of E［τ］by a closed vector subspace． Since the property of being co皿tably ultrabarコrelled is carried over to quotient spaces ［5］， F［〒］ is countably uユtrabarrelled． COROLLARY 2． 1｝θ力石’［τ］ Z）θα eozmtably quasi一祝Z加αわαヱn’e ZZed sραoθ， aηd F αoZosed veetor subspaee O王eountabZe eodemension in E［τ］．ヱアFSαtisfies the eondition（ht’）． then F閨asαZSO countUbZy quasi−uZtrαカαr！T・eZZed． f「・・f・L・t｛En［・。］｝b・th・ ・Ul）・pace・・珂・］・・n・t・ruct・d in th・・rem 2・ By・th・ ・・nditi・n（AA）・ea・h b・u・d・d・ub・et・f E［・］i・c・nt・in・d in・㎝・En・ As E［τ］ is countably quasi−uユtrabarrelled， E［τ］ is the k−inductive limit of ｛En［・n］｝［2］・S・・by血・…m2・P［i］i・i・・m・rphi・t・aq・・tient・pace・f E［τ］by a closed vector sdbspace． Since the property of being comtably quasi−ultrabarrelled is carried over to quotient spaces ［5］，we have the conclusion． ． REMARK． In this corollary， the word”countably quasi−ultrabarrelled・’can be replaced by tt（luasi−Ultrabarrelled”．50 K．KERA， N．．KITSUNEZAKI AND Y L TERAO Atopological vector space E，［τ］is an uZt？a−DR−spαoρ， if E［T］is comtably quasi−Ultrabarrelled ・and E［τ］has a f血dainental seqqence of bomded sets［3］． COROI工ARY 3． Let E［τ1ゐθan uZtra−DB−spασθand FαoZo8θ〈ヱveetor Stthspaρe of E［τ］．ヱアFsatisfies ’the●qndition（☆）．．then F［τ］is aLSOαn uZtl・CZ−DR−8Pα0θ・ llpoof． B）r the assu哩ti㎝， it follows that P is a closed vector sul）space of countal）ie cOdimension in the co皿｛lably quasi−uユtrabarrelled space E【τ］． By Corollary 2， R［そ】 is countably quasi−Ultrabarrelled． F［そ］has clearly a funda一 ㎜tal sequence。f．bO皿ded sets． Tlius F向is an Ultra−PF−space．’ REbvlARK． For a locally cQnvex pF・space孕［T】，M． the similar result without the closedness of F． Valdivia．［6］has obtained Here， we give an example of a pair of a top◎10gical vector space and its sUbspace砲ich satisfies the assurrption of Theorem l and corollaries of
