準階位ベクトル空間について I

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(1)Title. 準階位ベクトル空間について I. Author(s). 櫻田, 邦範; 阿部, 裕; 石田, 敏之; 橋本, 功; 宮本, 裕. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 38(1) : 33-40. Issue Date. 1987-10. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6142. Rights. Hokkaido University of Education.

(2) WM^H^K^ (^2 §|5C) ^1^- Bgffi62^10^. Journal of Hokkaido University of Education (Section II C) Vol. 38, No. 1 October, 1987. On Quasi Ranked Vector Spaces, I. Kuninori SAKURADA, Yutaka ABE*, Toshiyuki ISHIDA**, Isao HASHIMOTO"* and Yutaka MIYAMOTO"** Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 002. * Shinoro Junior High School, Sapporo 002 ** Teine Junior High School, Sapporo 063 *** Teine-Higashi Junior High School, Sapporo 063 **** Miyanooka Junior High School, Sapporo 063. i^^^ bA^m<:-^T, i EB ^ $g-P"T ^ ^-5 EB @( ^** ^ ^**. ^ $ :^***. w>i.wi-^.^m^w^m tLffirtiA^^^^^. M^S.^&HHWS.. +LffirpA^®£4'^%. Abstract In her paper [3], Shizu Nakanishi introduced the concept of ranked union spaces, and in [6] she defined classes of ranked vector spaces called (UCs-N) spaces, (UN) spaces, etc. In [6] and [7], she also showed that certain important vector spaces are definable as complete (UCs-N) spaces, complete (UN) spaces, etc. In this paper, we make a study of quasi ranked vector spaces. And we propose to show that if such spaces satisfy certain conditions,. then they can be defined as the ranked union spaces of metrizable ranked spaces, in such a way that : for the sequence of points, the r-convergence of the sequence under the quasi ranked vector space coincides with the r-convergence under the ranked union space.. (33;.

(3) 34 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA, Isao HASHIMOTO, Yutaka MlYAMOTO. §1. The fundamental terminology and notations for ranked spaces that are used throughout this paper are referred to in [5] and [6], We denote by (& either the real number field or the complex number field, and denote by N the set of all non-negative integers.. Let (E, T(x), T „) be a ranked space of indicator Wo in which E is a vector space over <E>. We denote the origin of E by o. Needless to say, Ux, Vx, etc. denote fundamental sequences of center x (x-f. s. for short). For the fundamental sequences u={U,} and v={V,},. and for x £ E and A. E 3>, we denote the sequences of sets : {U,+ V,}, {U, + x} and U£/,} by u+v, u+x and \.u, respectively. For the sequences of sets {A,} and {-8,}, {A,} > {B{} (or {B,} < {A,}) means that, for every A,, there exists a B, such that A, ^> 5,, while {A,}~{5,} means that {A,} > {B,} and {B,} > {A,}. Definition 1. A ranked space (E, T(x), Tn) in which E is a vector space over $ will be called a quasi ranked vector space if it satisfies the following condition (I): (I) (Io) For every x £ E and every u^, there exists a Wg such that Wg + x > M^. Conversely, for every x £ E and every Uo, there exists a w;r such that Wx> u.o + x. (Ii) For every Uo and every Vo, there exists a Wo such that Wo > Uo + ^0. (Iz) For every Uo and every X. £ <E>, there exists a Wo such that Wo > ^.%o.. [5, Proposition 26] says that in a quasi ranked vector space condition (I) is equivalent to the following condition (II): (II) (Hi) For every Ux and every Vy, there exists a Wx+y such that w^+,, > Ux + Vy. (IIa) For every Ux and every X ^ d>, there exists a. w^, such that w^ > \Ux.. Definition 2. Let E be a quasi ranked vector space. Let us denote by 2 the set of all fundamental sequences of center o in E. For Uo and Vo in 2, Uo~Vo is defined to mean that z<o < Vg and Vo < u.o. It is easy to check that ~ is indeed an equivalence relation on 2. For each Uo ^ 2, let [%o] denote the set of all u £ 2 such that u^-Uo. Let us denote the collection of equivalence classes under the equivalence relation ~ by A. If a and ft are two elements of A, let Uo and Vo be elements of S representing these equivalence classes. Now we define a < ft. to mean that Uo < Vo. It is easy to check that this definition is independent of the choice of the representatives iio and Vo in the equivalence classes a and /3.. Then (A, <) becomes a directed set. Indeed, it is clear that (A, <) is an ordered set. For any a, /3 £ A, let Uo and ô be elements of S such that [%o1 =ra' and [ô] = /?. Since E is a quasi ranked vector space, by (Ii), there exists a Wg £ 2 such that Mo + Vg < Wo. Hence we have Uo < Wg and Vo < Wo. Let y = [Wg] ; clearly y £ A, and we have ot<y and /?<y.. (34).

(4) On Quasi Ranked Vector Spaces, I 35 Definition 3. Let £ be a quasi ranked vector space. For every Uy = {U,} f= 2, we. define E(uo) = {x eE | \fi(=N 3 ^,> 0 such that x £ Jl,C/,}, and £*f«J = Span £^J. ^ Lemma 1. If Uo, Vo <= 2 and Uo < Vg, then E(Uo) c E(Vo) and E*(u.o) c E*(Vo).. Proof. Put Uo = {Us} and fo = {V,-}, and let A: £ E(Uo). Since Mg < fg, for every i e TV, there exists a j(i) £ N such that C/,;,) C Y{. Then, since .K <= E(uo), there exists /l^,> > 0 such that x £ kjwUno. Thus we have x £ /l,(,)1/,-. Hence, x £ £'^y<,/). Therefore we have E(u.o) C. E(vJ andE*(u.o)Ê*(vo). Definition 4. For every a £ A, let Uo be an element of 2 representing a. We define. E ^ = E*(Uo). By Lemma 1, this definition is independent of the particular choice of the representative Uo in the equivalence class a.. Lemma 2. If a, /? e A awrf <»</?, ê% £^ c £ .. This follows from Lemma 1.. §2 Let E be a quasi ranked vector space. We take into account the following two conditions: (A. 1) (cf. [1] p. 272) For every 6>-f. s, Uo = { U,}, there exists an ;' £ N such that U, c £^(<,). (A. 1') For every (9-f.s. Uo.= {U,}, there exists an i (= N such that U, c E*(Ho). Lemma 3. Let E be a quasi ranked vector space,. (1) The condition (A. 1) implies the condition (A. 1'). (2) The condition (A. 1') is equivalent to the following condition (1 *): (1*) For every a £ A, there exists a 0-f. s. Ug = { £/,-} SMC/Z ^/ %o ^ a flwi^ Uo<^ E .. Proof. (1) This is clear. (2) (A. 1') implies (1*) : For every a <= A, let Vo = {V,} be a 0-f. s. in a. By (A. 1'), there exists an i ^ N such that Vi c E* (Vg). Let us put Uj = V.+j for each j £ N. Then Uo = {Uj} is a 0-f.s. such that %o ~ Vo. Therefore we have Uo £ a and Uo = V, c £* r^ = £.. d*) implies (A. 1'): For every 0-f.s. Uo = { U,}, let a = [ua\ . By (1*), there exists a 0-f.s. Vo = {Vs} such that Vo^ a and Vo c E _. Since Mo and ô are in ff, Uy ~ Vo. Hence, for Vo, there exists an ; such that Us c Vo. Thus we have U, c Yg c £_ = E* (Va) = E* (Ua).. Lemma4. Let E be a quasi ranked vector space. If, for every 0-f.s. Uo = {U,}, there. (35).

(5) 36 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA, Isao HASHIMOTO, Yutaka MlYAMOTO. exists a j such that [/,+i absorbs U, for all i -^ j, then the condition (A. 1) holds. Proof. Let Uo = { U,} be a 0-f.s., and let x £ y,-. For k with o ^ k ^j, put /I* = 1, then we have x £ Uj c U/, = A.* Ui,. For k= j + 1, since ^,-+1 absorbs L(,-, there exists a A./+I > 0 such that x £ ?7j c: Aj+i Uj+i. Assume that, for k ^ j + i, there exists a ij+i > 0 such that x £ A.J+! Ujî. Then, since Uj+i^-i absorbs Uj+,, there exists a // > 0 such that Uj+i c vUj+i-n for all v with | v \ -S//. Put /lj+,+i = /lj+,/<, we have -K E Aj+,+i C/,'+i+i. By. mathematical induction, for all k with k > j+1, there exists Xk > 0 such that x £ A.* C4. Therefore we have U, c E(uo).. We denote the family of all preneighborhoods of the origin of rank n by ff „. The system ff' n (n ^ N) will be called absorbent if it has the following property: Corresponding to each x e E, there exists a non-increasing sequence Uo ^> Ui 3 •••••• such that Us £ y mw and m(i)->co sis i -"co and such that each U. absorbs the point x, i.e., for each Ui, there exists /lo ^ ^ such that x £ ^.U, whenever | /l | ^ | A-o | .. Lemma 5. Let E be a quasi ranked vector space. If y „ (n^ N) is absorbent, then the following condition (E. 5) holds : (E. 5) (cf. [1] p. 270) For every x £ E, there is a 0-t.s. u<> £ 2 such that x <= E(Uo).. Moreover, suppose that each U £ y „ (n £ N) is circled, i.e., if x E U and | A | ^ l(/l £ S>), then A x £ U. Then, if (E. 5) holds, then y „ (n e N) is absorbent. Proof. Let x ^ E, and let ff „ (n £ N) be absorbent. Then, there exists a nonincreasing sequence {Ui} such that [/,£ y mw and wf^ -»oo as i ->oo and such that each U,. absorbs x. Let us define io = 0 and ^ = min {i e N | ;">4-i, w^^ > mfik-i)} for A ^ 1. Putting Vn = U, for each k ^ N, we have a 0-f.s. Vo = { V/J. Since V;, = (7; absorbs .v, for each k, there is a A/, > 0 such that x £ \i, V,,. Therefore x £ E (Vo). Next, suppose that each. U^ ff n (n<^ N) is circled, and that (E. 5) holds. For every x £ E, by (E. 5), there exists a Vo = { y/J £ 2 such that x £ E (Vo). Hence, for each k, there is a A/, > 0 such that x e /l;, V*.. Since V* is circled, we have x £ /l F* for all /l with | -I | > A/,. Thus ff „ (n <= N) is absorbent. Lemma 6. Let E be a quasl ranked vector space. The condition (E. 5) implies the following condition (E. 5'): (E. 5') (cf. [2] p. 106) For every x £ E, there exists an a £ A such that x £ E ^. Lemma 7. Let E be a quasi ranked vector space satisfying the following condition (14) and each U £ y „ (n £ N) is circled : (D For every 0-t.s. Uo, Uo > Uo + Uo.. Then, for every 0-t.s, Uo, we have E*(Uo) = E(uo).. Proof. To prove this Lemma, it is sufficient to show that E(u.o) is a vector subspace of. (36).

(6) On Quasi Ranked Vector Spaces, I 37. E. Put Uo = { U{}. Let x and 3' be any elements of E(uo), and let A and // be any elements of <E>. For any i £ N, by (14), there exists a k such that £/, ^ t4 + ^. Since A: and 3' are elements of E(uo), there exist an otk > 0 and a A > 0 such that x £ a,, U,, and y ^ /3,, U,,. Put. y, == | A. | ff* + | /< | /?A + 1 > 0. Then, since Uk is circled, (1/y,) {JLX + /xy) = (/l/y,) x + (p. / y,) y C a/y,,) ffA Ui, + (/.t/'y,) A U,C: U,,+ U.d U,. Hence A-t: + ,,3' £E y, £/,. Therefore. we have \x + p.y E £^0,. §3. Let {E, T (x), Tn) be a quasi ranked vector space satisfying the conditions (A. 1') and (E. 5'). Then, by (E. 5'), we have E = Ê_. Moreover, by Lemma 3, for each a e A, there exists a 0-f.s. Uo = {Umw} £ a such that m(0) < m(l) < •••••• and Um^ £ 7'm(o fi ^ N) and Umw c -E. • In this section, let us denote this Uy by Ug(a) = {U,n_w}. Moreover, if (E, T (x),. T,,) satisfies the condition (14), then we may assume that Uo (a) satisfies in addition to the above properties, the following: For every ;'e N, U,,^^ =3 Um^.+i) + Um^,+i)-. Definition 5. Let (E, T(x), Tn) be a quasi ranked vector space satisfying the conditions (A. 1') and (E, 5'), and let Uo (a) = { Un,^,,} be a 6>-f.s. such that u.o(a) e a and m^ (o) < m^ (1) < •••••• and Umî) ^ Tm.w (i e N) and y,n.(o) c £'^. Let us define. ^ (x) ={x+ U,n^ \i^N}(x(= EJ, ^ = {X + Umî) | A;£.£^} when n = m^fi) for some ;", and ^\ = </> otherwise.. Then the space E endowed with ^"(x) (x e E ) and ^" {n e N): (E ^"(x), ^°) becomes a ranked space satisfying the axioms (A) and (B) of Hausdorff. Definition 6. We will define on E a structure of ranked space as follows. Let us define ^(x) = u {y (x} | for some a <= A such that x e £,} (.c e £), ^n= U{^ | a<=A}(n(=N). Then the space (E, ^ (x), ^/n) becomes a ranked union space of the ranked spaces (E^, ^" (x),. T) (a e= A). Proposition 1. The ranked union space (E, ^ (x), ^,,) of the ranked spaces (E_, ^ "(x), ^/"J (a £ A) satisfies the condition (Ui): (Ui) For every x £ E and for every x-f.s, Ux in (E, ^(x), ^,,), there exists an x-f.s, v" in some component space (E_, ^ "(x), ^".), such that v". > %„.. Proof. For any x £ E, let %;c = [Umaa-i + A;} be an A;-/.S. in (E, ^(x), ^ „). We may assume that m,w < m,^ < ••••••. Then { L^;)} is a 0-f.s. in (E, T(x), T,,). Put a = [{ Un. (37).

(7) 38 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA, Isao HASHIMOTO, Yutaka MIYAMOTO „(,)}], then a £ A. IVIoreover, there exists a /? £ A such that x^. E since A; £ £ =au<i.£^.. Since A is a directed set, there exists a y E A such that a < y and /3 < y. Then, {C/m» (i) } < %o(y) since a < y, and x <= E c £ since /? < y. Hence, %o(y) + -v is an x-t.s. in (£' ., ^r(x), ^r) such that %<,(y) + x > {Un,^n + x} = u^.. Similarly, we have the following :. Proposition 2. Let (E, T(x),T „) be a quasl ranked vector space satisfying the conditions (A. 1') and (E. 57). Then the following condition holds : (U*) For every x £ E and for every x-f.s. Ux in (E, T(x), °f „), there exist an a e A and an x-f.s. v~ in (E , ^" (x), ^/"), such that v~ > Ux.. Proposition 3. Let (E, T(x), T n) be a quasi ranked vector space satisfying the conditions (A. 1') and (E. 5'). Then the following properties are mutually equivalent: (1) r-lim x, = x in (E, T{x), Tn)(2) There exists an a £ A such that r-lim x: = x in (E^, ^"{x), ^/°).. (3) r-lim x, = x in (E, ^(x), ^,,). Proof. (1) implies (2): Let r-lim K{ = x in (£', T(x), T „}• Then there exists an x-t.s.Ux in. (E, T(x), T,,) such that Ux > {x,} where x, = [Xk\ k ^ ;'} ^" e A^J. By Proposition 2, there exist an a £ A and an x-f.s. v in (E , ^ (x), ^ ) such that v > Ux. Hence we have v >. Ux > {x,}. Therefore r-lim Xi = x in (£' , ^^(^), ^/^)for some ft > a. (2) implies (1): Suppose that there exists an a £ A such that r-lim x, = x in (£',, ^"(x), % ). Let us put Uo(a) = {Um^w}, then we can assume that there exists a subsequence { u'n^t, },eN Of {U,nî)} SUCh that {U,,,^,.) + x},^n > {x,-}. Hence we have Uo(a) + x> { U,n^(i ) + X} > {xj}. Moreover, by (Io), there exists an x-f.s. Ux in (E, T(x), T „) such that Ux >Uo(a) + x. Hence we have Ux > Uo(a) + x > {xj}. Therefore r-lim x, = x in (E, T(x), nt'. (2) implies (3): Suppose that there exists an a £ A such that r-lim x, == x in (-£^, ^ (x), %' ). Then, there exists an x-f.s. u in (E , ^ (x), ^ ) such that u > {.?,}. Since M is also an x-t.s. in (E, ^(x), ^ „), we have r-lim x. = x in (.£, ^(x), ^ „).. (3) implies (2): If r'lim x, = x in (E, ^(x), % „), then there exists an x-t.s. Ux in (E, ^{x), ^,,) such that Ux > {x,} where x, == {^ | k > i} (i 1= N). By Proposition 1, there exist an a IE A and an x-f.s.v^ in (E^, ^ (x), ^,) such that y > Ux. Hence we have y > Ux > {x,}. Therefore we have r-lim x. = x in (E , ^/li(x), ^p)tor some /3>a.. Proposition 4. Let (E, T{x), T „) be a quasi ranked vector space satisfying the conditions (A. 1') and (E. 5'). Then the following properties are mutually equivalent: (1) (E, T(x), r,,) safes//es (r-T,) [resp. (r-T,)]. (2) (E^, ^"(x), ^[} (a (= A) satisfies (r-T,) [resp. (r-T,)].. (38).

(8) On Quasi Ranked Vector Spaces, I 39. (3) (E, ^(x), ^,,) satisfies (r-Tz) [resp. (r-Ti)]. Proof. (1) implies (2): Let a £ A, and let x and .y be two distinct points of E^. For every x-f.s. u\ and every y-f.s. v" in (E^, ^"(x), ^), since u^ and ^" are also f.s.'s in (E, T(x), T,,), by assumption we have u^ D ^ = ^.. (2) implies (1): Let x and y be two distinct points of (E, T(x), T,,). For every x-i.s. Ux and every y-f.s. Vy in (E, T(x), f „), by Proposition 2, there exist an a £ A and an ^-f.s. ^ in (£' ^. ^" (x), ^^) such that u^ > M^, and there exist a /3 ^ A and a y-f.s. vp, in £^ such that v^ > fy. By the definition of Uo(a) and HoW, we have Uo(a) + x > u and Uo(/3) + y> v . Since A is a directed set, there exists a y e A such that a < -y and /3 < y. Hence MoW + ^ and Uo(y) + y are f.s/s in (E , ^/7 (x), ^Y). By assumption, we have (Uo(y) + x) D (?ôfy) + ^ = <^. Since a < y and /? < y, Mofff^ < Mo6^ and %ofA) < Mo(y^. Hence we have Uo('y) + x > Uo(a) + x > u\ > Ux and Mo(y^ + y > Uo(p) + y > v",, > r.y. Therefore we have %x n ^' = <^.. Similarly, it follows that (2) ^ (3).. Proposition 5. Let (E, T(x), T „) be a quasi ranked vector space satisfying the conditions (A. 1'), (E. 5'), (14) and each U^ ffn (n e N) is circled. Then the following hold : (1) For each a £ A, (E ^ ^~(x), ^~) is a quasi ranked vector space. (2) (E, ^(x), ^,,) is a quasi ranked vector space.. Proof. (1): This is clear from [5, Proposition 28] . (2): (Io): Let x e E, and let Ux be any x-f.s. in (E, ^(x), ^/,,). By Proposition 1, there exist an a £ A and an x-t.s. v^ in (£'., ^ (A:),. ^°), such that v" > u^. Since (E ^°(x), ^°) satisfies (Io), there is a o-f.s. Wa'in {E ^'(x), % ) such that w + x > r. Then w is also a o-f.s. in (E, ^(x), ^/n) and we have w + x > v'^ > Ux. Similarly, for x £ E and for any o-f.s. Uo in (E, ^/(x}, ^/n), we have an x-t.s. Wx such that Wx > u.o + x.. (Ii): Let Uo and Vg be o-f.s. 's in (E, ^(x), ^,,). By Proposition 1, there exist an a £ A and a o-f.s. u_ in E _ such that M° > Ug, and there exist a /? £ A and a o-f.s. y in E. such that o. a. --•-•••. -•••••---. -. Q. --i//. -.-...—. -----. _„—_-. -.. ,„. _„. _..._-. _.. -. .._.. -^. ___-_. ^. v > Vg. Since A is a directed set, there is a y £ A such that a < y and /? < y. Since a < y, Uo < M°(y^, and since ft < y, v^< Uo(y). Since (-£'„, ^Y(x), ^Y) satisfies (Ii), there exists a o-f.s. wr in (-£'„, ^/r(.y), ^/r) such that M/ > Uo('y) + Mofy^- Then w7 is also a o-f.s. in (E, ^/(x), ^,,) and we have wr > Uo(y) + Uo('y) > u+ v > Uo + Vg. (12): Let u-o be any o-f.s. in (E, ^{x), ^ „), and let ^ £ €>. Then, by Proposition 1, there exist an a e A and a o-f.s. u in (E , ^~ (x), ^/"), such that u > %o. Since (E , ^/"(x), ?/") satisfies (Iz), there is a o-f.s. w~, in £'. such that w~ > /IM". Then w~^ is also a o-f.s. in (-£', ^/ (x), ^n), and we have w" > \Uo.. Theorem. Let (E, T(x), Tn) be a qnasi ranked vector space satisfying the conditions (A. 1'), (E. 5'), (D, (r-Ti) and each U ^ ^n (n (= N) is circled. Then the following hold : (1) For every a £ A, (E_ , ^/" (x), ^".) is a metrizable quasi ranked vector space.. (39).

(9) 40 Kuninori SAKURADA, Yutaka ABE, Toshiyuki ISHIDA, Isao HASHIMOTO, Yutaka MlYAMOTO. (2) (E, ^ (x), ^ „) is a ranked union space of metrizable qnasi ranked vector spaces (E ,,^"W.^:)(ff A). (3) r-lim x,,==x in (E. T (x), Tn)if and only if r-lim x,,=x in (E, ^(x), ^,,). Proof. By Proposition 3, 5 and [5, Proposition 14], we obtain this theorem.. Acknowledg'ement The authors would like to express their thanks to Professor S. Nakanishi for her valuable suggestions and warm encouragement.. References El] Y. Nagakura, Differential calculus in linear ranked spaces, Hiroshima Math. J., 8 (1978), 269-299. [ 2 ] Y. Nagakura, Differential calculus in a space with bi-convergences treatment by the method of ranked space, TRU Math., 21-1 (1985), 105-116. [ 3 ] S. Nakanishi, On the strict union of ranked metric spaces, Proc. Japan Acad., 50 (1974), 603-607. [ 4 ] S. Nakanishi, On ranked union spaces, Math. Japan., 23 (1978), 249-257. [ 5 ] S. Nakanishi, The method of ranked spaces proposed by Professor Kinjiro Kunugi, Math. Japan., 23 (1978), 291-323. [ 6 ] S. Nakanishi, On ranked union spaces and dual spaces, M:ath. Japan., 28 (1983), 353-370. [ 7 ] S. Nakanishi, On topological structures of spaces C ([0, I], X) and C([0,c»), X), I, II, HI, Math. Japan., 30 (1985) 485-493 ; 31 (1986)75-84 ; (to appear in Math. Japan., 32. (1987)). [ 8 ] M. Washihara, On ranked spaces and linearity, I, II, Proc. Japan Acad., 43 (1967), 584 -589:45(1969), 238-242.. (40).

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