Iterat. J. Math. & Math. Sci. Vol. i0 No. 2 (1987) 227-232 227.-SIMILAR BASES MANJUL GUPTA ad P.K. KAMTHAN Departmet of Mathematics Idia Istitute of Techology Kapur, 208016 INDIA (Received May I, 1986) ABSTRACT. Correspodig to a arbitrary sequece space, a sequece {x i a locally covex space (l.c.s.) (X,T) is said to be %-similar to a sequece {y i aother l.c.s. (Y,S) if for a arbitrary sequece {a of scalars, {a P(X)} E % for all P DT <---> {a q(y )} g % for all q D S, where D T ad D s are respectively the family of all T ad S cotiuous semiorms geeratig T ad S. I.this ote we ivestigate coditios o ad the spaces (X,T) ad (Y,S) which ultimately help to characterize % similarity betwee two Schauder bases. We also determie relatioship of this cocept with %-bases. KEY WORDS AND PHRASES. (K)-property, ormal topology, semi--base, -base, fully -base, similar sequeces, -similar sequeces, ormal ad symmetric sequece spaces. gbo AMS SUBJECT CLASSIFICATION CODES. 46A35, 46A45, 46A05. i. MOTIVATION. Similar bases i the literature have bee studied with a view to fid the compariso betwee a arbitrary base ad a ow base of apparetly simple looig elemets. However, for certai eutral ad obvious reasos, the similarity of bases is defied i terms of the covergece of the uderlyig ifiite series. Sice there are differet modes of covergece, oe may tal of similarity of bases depedig upo the particular id of covergece or a particular type of a Schauder base, e.g. similar bases, absolutely similar bases etc. (see [I]). I the literature of Schauder bases, we have recetly itroduced differet types of %-bases" which geeralize several cocepts of absolute bases i a locally covex space (cf. [2]) ad hece it becomes imperative to exploit this otio ad itroduce a id of similarity, that is, %-similarity which should have i a atural way some relatioship with %-bases ad at the same time should exted the results o absolute similarity. This is what we have doe i this ote ad would discuss the same i subsequet pages. 2. TERMINOLOGY. Uless specified otherwise, we write throughout (X,T) ad (Y,S) for two arbitrary Hausdorff locally covex spaces (l.c. TVS) cotaiig Schauder bases (S.b.)
228 M. GUPTA ad P.K. KAMTHAN {X E {X ;f ad {y} E {y;g} respectively (cf. [3] for details ad uexplaied terms o basis theory hereafter). We write D T to mea te family of all T- cotiuous semiorms geeratig the topology T of (X,T) ad attach a similar meaig for D S. The symbol (X,T) (Y,S) is used to mea the existece of a topological isomorphism R from X oto Y ad coversely. Let 6 x {{f(x)}: x e X }; 6y {{g(y)}: y e Y} ad A x Pe D T -Wp Ay P DS q, where p* E {P(X)}; q* {q(y)} ad A is a arbitrary sequece space (s.s) For a Kthe set P, we write A(P) for its KSthe space. For discussio ad uexplaied terms o s.s., we refer to [4]; cf. also [5] whereas for the theory of l.c. TVS our refereces are [6] ad [7]. that A s.s. A is said to satisfy the (K)-propert if there exists 8 i A x such K E K 8 if 18l > 0 (2.1) Without further otice, wheever we cosider a s.s. A as a l.c. TVS, it would be assumed that A is equipped with its ormal topology (A,Ax) Fially, followig [8], we recall DEFINITION 2.2: A s.b. {x ;f for a l.c. TVS (X,T) is called (i) a semi-a-base if 6X AX, (ii) a A-base if 6 X A X ad (iii) a fully A-base if X AX ad for each p i D T the map (X,T) (A (A Ix)), (x) {f(x)p(x)} tluous. is cop p NOTE: If {X;f is a semi-l-base, the for each p i D T ad i Ix, the semiorm Qp,y o X is well defied, where Qp,y(x) l If(X) lltlp(x) for all x e X. (2.2) If l satisfies (2.1) ad {x "f is a fully l-base, the T is also geerated by the family {Qp,y: p e DT,Y e Ix} of semiorms o x; i particular, for each p i D T, there exists r i D T so that for all x i X, 3. I-SIMILAR BASES. p(x) _< l_ If(X) IP(X) -< K8 p,8(x) Let us recall from [I] the followig: _< r(x). (2.3) DEFINITION 3.1. A sequece {X cx is said to be similar to a sequece {y cy, writte as {X {Y } if for e i ilixi coverges i X> i eiy coverges i Y. i (3.2) The cocept of absolute similarity itroduced i [I], is abstracted to A- similarity as follows:
A-SIMILAR BASES 229 DEFINITION 3.3. For a arbitrary s.s. l, a sequece {x X is said to be l-slmilar to a sequece {y }c Y, to be writte as {X {Y } provided for {ep(x)} e l, for all p e D T <---> {eq(y )} e l, for all q e D S. The followig are straightforward. PROPOSITION 3.5. {X {Y <-----> PROPOSITION 3.6. {X {Y <=> AX Ay. PROPOSITION 3.7. If l is ormal, the (3.4) is equivalet to for e i {p(ex )} e l, for all p e D T <=> {q(y)} e l, for all q e D s. (3.8) (3.4) NOTE. I particular, if I Z1 we have }l-{y } <---> x is absolutely similar to {y as discussed i {x []. <=> A(P) A(Q), where A(P) ad A(Q) are Kthe spaces correspodig to P {{P(X)}: p e D T) ad the KSthe sets Q {{q(y)}: q e D S} respectively. PROPOSITION 3 8 Let {x ad {y be l-bases. The {x {y <---> {x {y }. PROOF. It suffices to observe that 6 x ad 3.6. 4. FURTHER CONDITIONS. A x, 6y ay ad ow apply Propositios 3.5 I order to obtai more characterizatios of l-slmilarlty of bases, let us brigforth the followig coditios for two arbitrary sequeces {x X ad {Y Y- for all p e D T, q e D S such that p(x) q(y ), for all Z I. (4.1) for all q e D S, p e D T such that q(y) P(X), for all Z I. (4.2) PROPOSITION 4.3. If l is ormal ad coditios (4.1) ad (4.2) are satisfied, the {X! {y} For obtaiig the coverse of the precedig result, we restrict ourselves to metrlzable l.c. TVS. If (X,T) is a metrlzable l.c. TVS, the T is also geerated by DX {Pi: i I}, where Pl P2 Similarly, whe (Y,S) is metrlzable, S is geerated by Dy {ql q2! " }" With this bacgroud, (4.1) ad (4.2) tae the followig forms: for all i i, j ad M i > 0 such that pi(x) Miqj(Y), for all 1.(4.4) for all j > I, i > ad K > 0 such that qj(y < Kjpi(x) for all > 1.(4.5) Hereafter, i all subsequet pages, we will let be a ormal s.s., satisfyig (2.1). Also, we cosider aother s.s. which will throughout be assumed to be ormal ad symmetric alog with I Ix. LEMM 4.6. Let (X,T) ad (Y,S) be metrlzable. Suppose that l is also symmetric ad let l cotai a elemet with >> 0, that is, a > 0, for each I. Assume
230 M. GUPTA ad P.K. KAMTHAN the truth of the followig statemet for a i m: for all i i, {pi(x)} E A > {qj(=y)} E, for all Jl. (4.7) The (4.5) holds. PROOF. Sice A x is also symmetric, we may fid a elemet y i A x \ with Y >> 0. Let ow (4.5) be ot satisfied. The there exists j so that for each il, oe fids a positive iteger i( i < i+ I, i I) with qj(yi) > i pi(xl), for all i I. Defie yi/qj (Yi) i d 0 otherwise Now qj(iyi) Yi ad 0 otherwise ad so {qj(y, for otherwise its close up beig y woulb be i, a cotradictio. O the other had, let m I; the oe fids i > m so that Pm(iXi! Pi(iXi < ui for all i Z m. Hece {pm(x)} E A for each m I. This cotradicts (4.7) ad the lemma is proved. Symmetry cosideratios lead to THEOREM 4.8. Let (X,T) ad (Y,S) be metrizable, alog with A, satisfyig the coditios laid dow i Lemma 4.6. The {x A {y} <.> (4.4) ad (4.5) hold. 5. A-BASES AND SIMILARITY. First, I this sectio, we ivestigate coditios whe two fully A-bases are similar. we eed LEMMA 5.1. Let {x be a fully A-base for (X,T) ad {y a arbitrary sequece i (Y,S). The (4.2) is equivalet to the followig statemet: for all fiite sequeces {l }" for all q E D s, p E D T such that PROOF. Let (4.2) be satisfied. The lot some r e D T q(il iyi! P(il ixl (5.2) q(il iyi!bb il libilp(xi) ibb r(il =ixi ad this proves (5.2). The other part is obvious. Symmetry cosideratios i Lemma 5.1 ow easily lead to THEOREM 5.3. Let {x} ad {y be fully A-bases for m-complete l.c. TVS s (X,T) ad (Y,S) respectively. The the truth of (4.1) ad (4.2) yields that {x {y }.
A-SIMILAR BASES 231 Coversely, if (X,T) ad (Y,S) are Frchet spaces with l, satisfyig the coditios of Lemma 4.6, the {X {Y --> (4.4) ad (4.5) hold. PROOF. First part follows from Lemma 5.1 ad its symmetrizatlo. For the coverse, observe that {X ad {y are h-bases by Propositios 3.5 ad 3.6 of [8]. Now mae use of Propositio 3.8 ad Theorem 4.8. REMARKS. Observe that Lemma 5.1 ad the first part of Theorem 5.3 ad valid for a arbitrary sequece space l satisfyig the K-property. Sice h-bases ad fully h-bases are the same i a Frchet space (cf. [8], p.82), we ca rephrase Theorem 5.3 for Frchet spaces as follows: THEOREM 5.4. Let (X,T) ad (Y,S) be Frchet spaces ad {X } {Y be h-bases alog with l, satisfyig the coditios of Lemma 4.6. The {X (4.4) ad (4.5) hold. NOTE. The proof of the above theorem is also immediate from Propositio 3.8 ad Theorem 4.8. 6. THE ISOMORPHISM THEOREM. two I order to obtai the mai result of this sectio, let us itroduce the followig coditios: {p(ex )} e l, for all p e D T ==> {q(ey)} e l, for all q DS; (6.1) {p(=x )} e l, for all p D T ==> {q(ey)}, for all q e DS, (6.2) where e, {X } {Y are arbitrary sequeces i X, Y respectively ad l, are the s.s. as specified i Lemma 4.6. NOTE. Whe (X,T) ad (Y,S) are metrizable, the coditios (6.1) ad (6.2) will be cosidered with D ad D T S beig replaced by D ad X Dy (cf. remars followig Propositio 4.3). Thus we have PROPOSITION 6.3. If the hypothesis of Lemma 4.6 holds, the (4.5) --> (6.1) --> (6.2) ---> (4.5). Next, we have PROPOSITION 6.4. Let (X,T), (Y,S) be a metrizable l.c. TVS ad R:X Y a cotiuous liear map. For a sequece {X X, let Y RX. The either of the equivalet coditios (4.5), (6.1) ad (6.2) holds. PROOF. For each j > i there exist i such that qj(rx) KjPi(X) for all x i X ad so (4.5) holds. The result ow follows from the precedig propositio. Coversely, we have PROPOSITION 6.5. Let (X,T) be a metrlzable l.c. TVS cotaiig a fully h-base {X } (Y S) a Frchet space cotaiig a sequece {y }. If either of the equivalet coditios (4.5), (6.1) or (6.2) holds, the there exists a cotiuous liear map R:(X,T) (Y,S) with RX Y _> I. PROOF. Defie R(m emxm m mym the by (4.5) ad (2.3), R:sp {x Y is cotiuous ad so its uique extesio is
232 M. GUPTA ad P.K. KAMTHAN the required cotiuous liear map. This completes the proof. PROPOSITION 6.6. Let (X,T) ad (Y,S) be m-complete such that (X,T) = (Y,S). Let {x be a semi -base for (X,T) ad Y RX I. The {y is a semi -base for (Y,S) ad {x {y }. PROOF. {y is clearly a S.b. ad let {g be the s.a.c.f, correspodig to {y}; ideed g f R-I I. The for each y i Y, ther exists a uique x i X with g(y) f(x) for all Z i. Sice (4.2) holds ad is ormal, {y;g is a semi -base for (Y,S). Cosequetly, both of these bases are -bases, cf. Propositio 3.5 of [8]. Thus X AX ad y Ay. But X Y "-> AX Ay ad ow apply Propositio 3.6. Fially, we have PROPOSITION 6.7. Let (X,T) ad (Y,S) be u-complete barrelled spaces havig semi -bases {x ad {y respectively such that {x {y with Rx Y > I. }. The (X,T) (Y,S) PROOF. By Propositio 3.4 {x {y ad so by a theorem of [9], (X,T) (Y,S) with Rx Y > i. REFERENCES i. ARSOVE, M.G. Similar Bases ad Isomorphisms i Frchet Spaces, Math. A. 135 (1958), 283-293. 2. KALTON, N.J. O Absolute Basis, Math. A. 200(1973), 209-225. 3. KAMTHAN, P.K. ad GUPTA, M. Theory of Bases ad Cocs, Advaced Publishig Program, Pitma, Lodo, U.K., 1985. 4. KAMTHAN, P.K. ad GUPTA, M. Sequece Spaces ad Series; Marcel Deer, New Yor, N.Y., USA, 1981. 5. RUCKLE, W.H. Sequece Spaces, Advaced Publishig Program, Pitma, Lodo, U.K., 1981. 6. HORVATH, J. Topological Vector Spaces ad Distributios I; Addiso-Wesly, Readig, Mass., USA, 1966. 7. KOTHE, G. Topological Vector Spaces I, Spriger-Verlag, Berli, W. Germay, 1969. 8. KAMTHAN, P.K., GUPTA, M. ad SOFI, M.A. -Bases ad Their Applicatios, J. Math. Aal. Appl. 88(I) (1982), 76-99. 9. JONES, O.T. ad RETHERFORD, J.R. O Similar Bases i Barrelled Spaces, Proc. Amer. Math. Soc. 18(4) (1967), 677-680.