（CUC_s-N）空間の双対空間についての注意

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(1)Title. （CUC_s-N）空間の双対空間についての注意. Author(s). 櫻田, 邦範; 阿部, 裕; 石田, 敏之; 橋本, 功. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 37(1) : 23-27. Issue Date. 1986-10. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6123. Rights. Hokkaido University of Education.

(2) (Ui2g[SA) î^- (maemo^. Journal of Hokkaido University of Education (Section U A) Vol. 37, No. I October, 1986. A Remark on the Dual Spaces of (CUCs-N) Spaces. Kuninori SAKURADA, Yutaka ABE*, Toshiyuki ISHIDA** and Isao HASHIMOTO*** Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 064 Shinoro Junior High School, Sapporo 002 * * Higashi Shiroishi Junior High School, Sapporo 003 * * * Teine Higashi Junior High School, Sapporo 063. (GUCs-N) ^©m-^^-^r®^:! ifflî® • Ngpr^* • ss^^** • W!h***. ^m^n±^w^^^ * w^±mt&^w.. ** ^??1^54^^. ***+L^rt)A^1^4'^-K. Abstract In [I], Professor S. Nakanishi introduced the concept of ranked union spaces, in connection with the treatment of the space Q) as a complete ranked vector space having a countable base. In [4], she succeeded in showing that the space S>' of distributions can also be treated as such a space. Subsequently, in [6], she showed more generally that the dual space X' of (CUCs-N) space X can be treated as a complete (UCs-N) space. But her treatment remains rather complicated. The purpose of this paper is to simplify the treatment. 1. Introduction For all the terminologies and notations concerning ranked spaces that are used throughout this paper we refer to [3] and [6]. N denotes the set of all non-negative integers.. First, let us recall the definitions of (Cs-N) spaces, (UCs-N) spaces and (CUCs-N) spaces according to Nakanishi [6]. Let X be a vector space, and suppose that, on X , there is defined a countable system of semi-norms pm(meN) to be non-decreasing, i. e., pv(x) ^ pi(x) < •••••• (x^. X). Put. (23).

(3) 24 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA and Isao HASHIMOTO. S(n)=lycX;pn(y)<l/2"i (ne/V), nx)=\x+S(n) ;neM (a;eX), t'n =|2;+S(n) \x^X\ (n^N). The ranked space (X, 1/{x), Y/n) is then called a ranked countably semi^normed space, or simply (Cs-N) space.. Let X be a vector space and let Xa (a£ Z) , where S is a directed set with the ordering 52, be a family of vector subspaces of X such that. (1.1) U tXo;ff£2;i=Z, (1.2) if a</3, then Xa^X,,.. Suppose that (1.3) in each Xa, there is defined a countable system of semi-norms pS (me N) which are non-decreasing, i.e., p?(x) ^> p°(x) ^ •••••• (x^Xa), in such a way that. (1.4) if ff^ ft, then pS (x) ^pS (x) for every me A? whenever rceZa. For such a collection as Xa and pS consider the (Cs-N) space Xa defined by the countable system of semi-norms pS (m e N) for every " . A ranked space X defined as the ranked union space of. such (Cs-N) spaces Xa (a'e2) is then called a (UCs-N) space, if it satisfies the following conditions (Ui) and (Uz): (Ui) for every x^X and for every fundamental sequence Ux of center x in X, there exists a fundamental sequence Vx of center x in some component space Xa, such that Vx>Ux; (Uz) for every fundamental sequence u in X , there exists a fundamental sequence v in some component space Xa, such that v>u. In particular, when 2= N and ^ is the usual ordering in N , such a (UCs-N) space is called a ranked countable union space of countably semi-normed spaces, or simply (CUCs-N) space.. Let Xa and pS (m^N), (a^Z), be a collection of vector subspaces and semi-norms satisfying (1.1)—(1.4) for a vector space X. Let X be the ranked union space of (Cs-N) spaces Xa. defined by p^(meN), (a^Z). If Xa{a<=Z) satisfies the following condition (1.5), then the ranked space X satisfies the conditions (Ui) and (Uz); hence X becomes a (UCs-N) space: (1.5) It XaCXfl, then a<^.. Moreover, if Xa(a^Z) satisfies (1.5) and the following condition (1.6), then X satisfies Hausdorff's axiom (B): (1.6) For any Xa, Xg, there exists an Xy such that Xr=XaC\X/,. (see [6] Proposition 4 (1), (3)).. 2. Dual spaces of (CUCs-N) spaces Let X be a vector space and Xi(leN) a sequence of vector subspaces of X such that U\X,;l<=N[=X and A'oC^C ••••••. Let X be a (CUCs-N) space determined as the ranked union space of (Cs-N) spaces Xi(leN}, each of which is defined by a countable system of semi-norms pm(m^N). As usual, the set of all r-continuous linear functionals on a ranked vector space X is called the. (24).

(4) A Remark on the Dual Spaces of (CUCs-N) Spaces 25 dual space of X , and denoted by X . For f^X' , let us put. pL^(f)=sup\\f(x)\ ;x^X,, pi,(x)^l\(l, mc=N). Then, by [6], for each I GN, Pl-m(f) <co for some m depending on I. Put p(/, /)=inf ImCA?; ?'-»,(/) <co>. Obviously, these pim(/) and p(l, f). are non-decreasing with respect to /,. Let us put £=\\={m(l)\ ;m(0)^m(l)< •••••-, where m(l)^N for each l^N\.. For A=|m(/)f£2: and A/=tm/(/)t e 2: , define AÂ/ to mean that m{l)^m'(l} for every / £ N. Then, 2; , with this ordering ^, is a directed set. For each Ae.S, A=!m(0!, we define. X^=\f^X';p(l, f)<m(l) for every / G N} . Then \X^; A<= £1 is a family of vector subspaces of X' such that U \X'^.; Ae Z\:=X' and X^ C^/, where A^ A/ (see [6] Proposition 10).. Now we will show that X can be defined as a (UCs-N) space. For A£ Z, A= lm(/)| and /e XA', let us put fc;(/)=max!p^,,,(/);/=0, 1, ••••••, J\U^N). Then, these k} are semi-norms on X^ and have the following properties.. Lemma 1. (1) If J^J', then k^. (/) S ft,A (/) for f^Xi (2) If A< A/, then k} (/) > /CJ'' (/) for f £ Z; and 7 e N. (3) For some A=|m(/)l, if k^(f)=0 for every j^N, then /=0. Proof. (1) and (2) are clear. To prove (3) suppose that /+0. Then, there exists a;oe Xi,. such that /(.x;o)+0. Put ,r,=.ro/(pml;/.i(.ro)+l), then |/(a;i)|>0. Hence k^(f)>0. This is a contradiction. By Lemma 1 (1), for each \e£, A=|m(/)i, we can define the space X'^ as a (Cs-N) space determined by the countable system of semi-norms k} (j £ N). In fact, let us put. S(A, j)=\feX^,k^f)<l/2J\ O'eAQ, and we define. Â(/)=t/+S(A, j);j^N\ (/ex;), ^ =|/+S(A, ^);/eX.(l O'eN). Then, the space X^ endowed with W^(f) (feX^) and W^j^N) : (X'^ 'W^(f), W}) becomes a (Cs-N) space.. Hence, by [6, Proposition 1 (1) and (2) J, the following proposition is induced more directly. Proposition 1. (cf. [6, Proposition 11]) Let .fi, f^X'^(i^N), A=im(/)|eZ'. (1) r-lim fi=f holds in the (Cs-fi) space (X^,'9^ (f),'%'}) if and only if fi^ f in each of semi-norms k^ (j <= N).. (25).

(5) 26 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA and Isao HASHIMOTO. (2) {ft} is a Cauchy sequence in the (Cs-N) space (X^lWA(;f),-'W^) if and only if it is a Cauchy sequence in each of semi-norms k} (j £ N}.. Moreover, by [6, Proposition 1 (3) and (4)] and Lemma 1 (3), we have the following. Proposition 2. The (Cs-N) sâce (X^., '%<(/), (?<''jA) i's fl ranked vector space satisfying. Hausdorffs axioms (1S) and (C) as well as (r-Ti) and having the properties (Mi), (M2) and (M,). Furthermore, by [6, Proposition 1 (5)], the following holds. Proposition 3. (cf. [6, Proposition 12]) The (Cs-N) space (XH, W^(f), ^) ('s complete. Proof. Let \fi^ be a Cauchy sequence in the (Cs-N) space X^ and let A=lm«)|e Z. Then, by Proposition 1 (2), i/,1 is a Cauchy sequence in each of semi-norms k^{je N). Hence, for each J ^ N and each positive e, there exists i(j, e) e N such that for {', i'~^ iU, e), k^f.-ffXe. For every x^X, since U \X,; I e N\=X, there exists j»<= N such that x^X». Then,. if l, r>l0'o, e), then \ft(y)-f,' (y) | ^ ^ (/<•-/,.)<£ for all </with y£ X,» and pA»,(y) sSL Therefore, for all i, i'î(j», e), l/,(3;)-/r(a;)| < e(rio,.i(a;)+l). This shows that l/.(a;)t is a Cauchy sequence in the scalar field. Hence )/; (3;) I converges. Let /(3;)=lim fi(x) for x^X. It is clear that/is linear. For each I e N, if i, i'~^. ('(/, e), then P-'mid (/.—/.•)</C(A (/(—/,.)< e , and consequently, for all ;>{'(;, e), P'-min (fi-f)<e. Now choose (' so that iî(l, e). Then, since /,C X^ pl-mu^f)<Pl-mni (f-fi) +P'-^(fi)^£+Pl-^,Afi}<(x> and/-/..ex'. Thus fGX' and p(l, f)^m(l) for all / GN. Therefore feXH. For each / with 0^l< j, p'-^^ft-f ,')^k^f .-/,.)< e for a.\\ i, i'î(j, e), and consequently, for all i î{j, e), P'-îAf.-f)^ e. Hence k^{ft-f)=ma.x\pl_^(f,-f) ; 0;a. i.^yiê for all iî(j, e). Thus, by Proposition 1 (1), l/d is r-convergent to / in X'f. . Therefore, by [6, Proposition 1 (5)], the (Cs-N) space X'n is complete.. Let AT? A if and only if X^=X^ . Then R is an equivalence relation on S. Denote the classification of I, with respect to R by Za, a e A and express \G S in tm(A, /)|. Let us put mB(/)=minim(A, /) ; AC Za\. Then, lm<,(/)|C.S, denoted by A(a) . We define the ranked space (X , W(f), Wj) as the ranked union space of the (Cs-N) spaces (X^,ai, ,'WW(f), euya}'} (ffCA), i.e., as the ranked space X' provided with the family of the. preneighborhoods of /:'%'(/)(/£ X ) and the family of the preneighborhoods of rank j: ett, (j £ N), which are defined by w{f} = U Wla (/); a e A for which ^ 3 /1 , att, =U Wal;a£A|.. (26).

(6) A Remark on the Dual Spaces of (CUCs-N) Spaces 27. Then, since the (Cs-N) spaces (X^, Ww(f), Ww) (a £ A) satisfy (1.5) and (1.6) by [6, Proposition 15], by Proposition 2, 3 and [6, Proposition 2, 4], we have: Theorem. (cf. [6, Theorem 2]) (1) The ranked space (X' ', 'W(f), <SUj) satisfies the conditions(Ui) and (U2) as the ranked union space of (Cs-N) spaces [X'â, Âlo'(/), W}M} (a (= A), ^MCC ?e ranked space (X/, W[f), Wj) is a (UCs-N) space. (2) The ranked space [X', W(f), W,} is a complete ranked vector space satisfying Hausdorffs axioms (B)- and (C) ffs well as (r— Ti) and having properties (Mi), (Mz) and (Ma). (3) r-lim fi=f holds in the (UCs-N) space X' if and only if r-lim /;=/ holds in some (Cs-N) space X'^.. (4) t/il is a Cauchy sequence in the (UCs-N) space X' if and only if it is a Cauchy sequence in some (Cs-N) space Xua .. References. [1] S. Nakanishi, On the strict union of ranked metric spaces, Proc. Japan Acad., 50(1974), 603-607. [2] S. Nakanishi, On ranked union spaces, Math. Japan., 23(1978), 249-257. [3] S. Nakanishi, The method of ranked spaces proposed by Professor Kinjiro Kunugi, Math. Japan. , 23(1978), 291-323.. [4] S. Nakanishi, The space of distributions treated as a ranked space, Proc. Japan Acad., 55(1979), 395-398; Math. Japon., 25(1980), 87-100. [5] S. Nakanishi, Main spaces in distribution theory treated as ranked spaces and Borel sets, Math. Japan., 26(1981), 179-201. [6] S. Nakanishi, On ranked union spaces and dual spaces, Math. Japan., 28(1983), 353-370.. (27).

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