Periodica Mathematica Hungarica Vol. 15 (3), (1984), pp. 219--231

DOMINATING SEQUENCES AND FUNCTIONAL EQUATIONS

M. GUPTA (Kanpur) and P. K. KAMTHAN (Kanpur)

1. Introduction

The notion of domination of sequences has been used by a number of mathematicians in the past with slight variations of terminology. Indeed, the domination between pair of sequences was considered by Banach ([4], p. 1638; [5], p. 11) around the year 1925. Later, Arsove ([2], [3]) used these notions in the s tudy of Schauder basis in Fr~chet spaces and constructed some examples of such sequences (for domination of infinite matrices, one is referred to [9], [10]). However, these concepts gained independent importance in the work of Singer [23] who also employed these notions to reformulate certain results on sequence spaces. He dealt with all the problems in Banach spaces.

In this note, we give various equivalent conditions for strict domination in terms of functional equations in topological vector spaces, characterize a base through the notions of domination and strict domination, and apply them to obtain a result on weakly ~-unconditionally Cauchy ser~es -- a con- cept introduced in [12] (cf. also Chapter 3 of [13]).

2. Terminological excerpts

Let X and Y be two topological vector spaces (TVS) over the same field K of reals or complex numbers. For a set A c X, we use the symbol sp {A} to denote the subspace of X generated by A and write [A] ~ s-p {A} the closure of sp {A} in X. For sequences {Xn} C X and {Yn} c Y, we introduce

A M S (MOS) subject classi/ica$ions (1980). Primary 46A35; Secondary 39B70. Key words and phrases. Domination of sequences, minimal sequences, AK-spaces,

Sehauder base, functional equation, weak unconditional Cauehy series, barrelled and bornological spaces.

220 GUI~YA, K A M T H A N : DOMINATING SEQUENCES

DEFINITIONS 2 .1 .

(i) The sequence {xn} dominates {Yn}, to be expressed as {x.} > {yn}, if for {~i} c K,

(2.2) convergence of ~ a i x i =~ convergence of ~ i Y i - i~ l i~ l

The sequences {Xn} and {yn} are said to be equivalent, to be written as {x~}

(ii) The sequence {x,} strictly dominates {Yn}, to be denoted by {xn} >~ >~ {Yn}, provided there exists a continuous linear operator T: [x,] ~ [y~] such that

(2.3) T(xn) : Yn, n -- 1, 2 . . . . .

I f {x,} >~ {Yn} >~ {Xn}, we say {x~} is strictly equivalent to {y~} and we write it as {x~} ~ {y~}.

We also need the notions contained in

DEFINITION 2.4. A sequence {xn} in a TVS X is said to be (i) min imal if xn ~ [x 1 . . . . , Xn_l, x~+ 1, . . .], n > 1 ;

(ii) complete if X : [x~] and (iii) a base if each x E X can be uniquely expressed as

(2 .5) x = fi(x)xi, fi(x) K, i 1. i~ l

Clearly, in the above representation the / / s are linear functionals on X such that fi(xj) -~ 5ij, where ~j is the Kronecker delta. In case these linear functionals are continuous, we say tha t {xn} is a Schauder base. We shall use the symbol {xn; fn} for a base {xn} provided we are interested to attach importance of the associated sequence {fn} of the co-ordinate functionals.

For various terms and results in locally convex spaces used in the sequel, we refer to [7] and [15]. However, we reserve the symbols X and :Y to denote t w o Hausdorff locally convex spaces (abbreviated hereafter as 1.c. TVS) and ~)x, @r for the families of all semi-norms generating the topologies of X and Y respectively. The notation X* is used for the topological dual of X, whereas X ' is the algebraic dual of X.

Concerning the theory of sequence spaces, we follow [13] (cf. also [15]). For the sake of completeness, let us recall only a few relevant terms. So, let co dencte the vector space of all scal~r-valued sequences, with the usual pointwise addition and scalar multiplication. As usual we write e n (n ~ 1) for the n- th unit vector in w and O for the subspace of o~, generated by the en's, n ~ 1; i.e., �9 = sp {e": n ~ 1}. A subspace 2 of o9 with �9 c 2, is called a sequence space. For x : {xn} in 2, the sequence x (n) : { x l , . . . : X n , . . . } :is

GUPTA, KAMTHAN: DOMINATING SEQUENCES 221

t e rmed as the n- th section o f x . A sequence space 2 equipped with a vector (linear) topo logy ~ is called a K-space if the co-ordinate project ions

P~: ~, -~ K, Pi(x) = x;, x = {xi} C

are continuous; and a K-space is called an AK-space if x (n) -~ x in ~, for each x ~ ~. An FK-space is a complete, metr izable locally convex K-sequence space. The fl-dual of a sequence space ~, is the space ~ def ined as

) ~ = {Y = {Yi} C ~o: , ~ x i y i converges for each {xi} C ~}- i~ l

3. Main results

In this section we give the main results of this paper. We assume through- ou t t ha t each subspace of a given 1.c. TVS is equipped wi th the subspace topology. We begin wi th

T ~ E o ~ 3.1. Let X be an 1.c. T V S and Y be a complete l.c. T V S such that @ x - {P~) and ~ r = {q~). For sequences {x~} c X and {y , ) c Y , the following conditions are equivalent.

(1) (i) {x.} >> {yn}, (ii) For every q~ E ~)y, there exists p~~ E @x and a constant C > 0 such

that

(3.2) q~ atyi ~ CP~o ~txi ,

for any finite sequence ~1 . . . . ' :r of scalars. (2) Assume that P = sp {x~} is bornological. Then the conditions (i) and

(ii) of (1) are equivalent to (iii) For each g E [Yn]*, the system of equations

(3.3) f(xn) : g(Yn) (n ~- l, 2 . . . . )

has a unique solution f E [x,]*. (3) I f the sequence {x,} is also minimal , the conditions (i), (ii) and (iii)

are equivalent to (iv) There exists a positive integer n o such that

{xn}%, >> {y.};..

(4) I f X and Y are metrizable and (xn; fn}, {y~, gn} are biorthogonal sys- tems such that {fn} c [x~]*, [gn] c [Yn]* and {gn} is total on [Yn], then the above conditions are equivalent to the following.

2 2 2 GUPTA, KAMTHAN: DOMINATING SEQUENCES

(v) For every x ~ [Xn], the system of equations

(3.4) fn(x)--gn(y) ( n = 1 ,2 . . . . )

has a unique solution y in [yn].

P R O O F .

(1) The implicat ion (i) ~ (ii) of (1) is obvious. For proving (ii) ~ (i), let P ~ sp {Xn} and define a l inear map T 0 on P b y

To a~xt : ~ a ~ Y i for ~ i x t E P-

Obvious ly T O is well def ined and cont inuous b y condit ion (3.2). Therefore, i t can be ex tended to a map T on P ~ [Xn] such tha t T(xn) ~ Yn (cf. [7], Proposi t ion 5, p. 129). Thus (i) ~ (ii).

(2) To p rove the equivalence be tween ( i )and (iii), we assume first t ha t (i) is true. Therefore, there is a cont inuous l inear map T f rom [xn] to [g,] such tha t T(xn) ~-- y, , n ~ 1. I f T* denotes the aAjoint of T, mapping [y,]* into [Xn]*, the e lement T*(g) in [Xn]* for every g E [Yn]*, is the desired unique solution. Thus (i) ~ (iii).

For the implicat ion (iii) ~ (i), define a map T O on P as before, namely ,

n

for every ~ a ~ x i E P . We now show tha t T o takes bounded sets in P to i-~l

bounded sets in [Yn]" So, let B be a bounded set in P and wri te J~ : (T0(b): b E B}. The boundedness of B in P implies t ha t for any f E [Xn]*, there is a cons tant K -~ K(f) such t ha t If(b)[ ~ K for all b E B. For an a rb i t r a ry g E [Yn]*, there is an f E [Xn]* such tha t f(Xn) ~ g(Yn), n ~ 1. Therefore, for a ny

n

b = ~ , ~ixt E B,

one has

Ig(To(b))l : lg (l~=l atyt }l : f [2alXl} ~ K

and so B is a([yn], [yn]*)-bounded and tl~us bounde d in [Yn] (cf. [7], Theorem 3, p. 209). Since P is bornological, T o is cont inuous (see [7], Proposi t ion 1, p. 220). Therefore To can be ex tended to a cont inuous linear mapping T on [Xn] such tha t T(Xn) -~ Yn" Thus (i) ~ (iii).

(3) In .proving (i) ~=* (iv), we need prove (iv) ~ (i), as the other implica- t ion follows obvious ly for an y sequences {Xn) C X and {y,) C Y. Therefore,

GUPTA, KAMTHAI~: DOMINATING SEQUENCES 223

n

assuming (iv) to be val id , let us wri te P = sp {x.} a n d for p = ~ a i x i in P i=1

def ine mapp ings T 1 and T 2 f r o m P to [y~] as follows:

min (n,no-- 1)

TI(P) = ~ ~i Yi i=l

and

[,~ a iy i i f n ~ n o T2(p) = i=n.

[ 0 i f n ~ n o - - 1,

where n o is the in teger as in t he hypothes is . Since {xn} is min ima l in X, we

can f ind a sequence {fn) C X * such t h a t fn(Xm) ~ ~nm" H e n c e there ex is t in a na tu r a l fashion two pro jec t ions $1 and S 2 f rom P on to P ( n , - i ) =

= [x 1 . . . . . Xn,_l] a n d on to

p(.O = [x,~., xn.+l . . . . ],

n respect ive ly , such t h a t for x = ~ f i ( x ) x i in P ,

i=1

rain (n,n,- 1)

S l ( z ) = fl(z) i

a n d

&(x) =

n

i f no

0, r/, <~ ~o"

L e t R1 a n d R 2 be the m app i ngs f r o m P(n.-!) a n d p(n.) to [ y J def ined as:

and

I~ l ) n,--I

R 1 O~iZ i = ~ o~iYi [i=l i = I

R 2 o~ix i = ~ o~iYi, i=n, i~n,

respec t ive ly . Then R 1, being def ined on a H a u s d o r f f f in i te d imens ional TVS, is con t inuous (cf. [7], P ropos i t ion 2, p. 142) a n d R 2 is con t inuous b y our hypo-

thesis. Since T 1 = R 1 o S 1 a n d T 2 ~ R 2 o $2, the c o n t i n u o t y o f T 1 and T 2 follows. Thus the m a p T o = T 1 ~- T 2 is a con t inuous l inear m a p f r o m P in to [Yn] a n d therefore i t can be ex t ended to a cont inuous l inear m a p T on [Xn] such t h a t T(Xn) = Yn" This comple tes the p r o o f of the equiva lence be tween (i) (iv).

224 GUPTA, KAMTHAN: DOMINATING SEQUENCES

(4) For proving (i) ~ (v), let us first assume the truth of (i). Then there exists a continuous linear map T: [x,] --~ [y,] with T(xn) = Yn. The biorthogo i nality of the pairs {x~, fi} and {Yt, g~} implies

fi(xj) ~- ~}ij : gi(Yj) : gi(T(xj))

n

for i , j = 1, 2 . . . . . Therefore, for x ~ - ~ i x j E P where ai's are scalars, j=i

we have by the linearity of the map T that

/ s l [~(x) = fi % x i = gi T ~ jx j = g~(Tx). t j = l / j =

As {xt} is complete in [xi] and T is continuous, it follows that

f i(x) = gi(T(x)) , x ~ [xi], i = 1 , 2 , . . . .

Thus, for x C [xt], the equations (3.4) has the solution y = T ( x ) in [Yi]. I t is unique because the sequence {gi} is total on [Yn]. Hence (i) ~ (iv) (observe tha t for this implication we have not made use of the metrizability property of the spaces X and Y).

For converse, let (v) be true. I f y is the unique solution of the equatons (3.4) corresponding to a point x E [xn], define a map T from [xn] to [yn] by T ( x ) = y. Since {g~} is total on [Yi], it can be easily proved that T is linear. For proving the continuity of T, let us consider a sequence {Zn) in [x~] such that lim T(zn) = z'. Then, by the continuity of fi and gf, i = 1, 2 . . . . . we have

[~(z) = lim ft(zn) = lim g~[T(z~)] = gi(z'), 12 ---~ ~ r t ~ ~

for every i ~ 1, 2 , . . . . Hence z' = T(z) . Therefore T is continuous by the closed graph theorem [7]. Now the relations T ( x n ) = Yn, n ~ 1, follow from the equality

gi(T(x~)) : - f~(x,~) = ~tn : gi(Yn),

for i, n = 1, 2 . . . . , and the total i ty of the sequence {gt} on [Yn]. Hence (i) is true and the proof of the theorem is completed.

NOTE. The above theorem remains valid if we replace completeness by sequential completeness of Y and restrict X to be a metrizable 1.c. TVS.

REM_~R]~S. We observe tha t the closed graph theorem plays a k e y role in the proof of the implication (v) =~ (i). Thus the above result will be true even if we take the subspace [xn] of X to be barrelled and :Y to be a Pt~k space, by an application of a result proved in ([7], Theorem 4, p. 301). We may mention here that Kamthan and Ray have introduced a generalization

GUPTA, KAMTHAN: DOMINATING SEQUENCES 225

of equivalent sequences to equivalent sequences of subspaces in [14] where they have treated a part of the above theorem (especially (1), (i) ~=~ (ii)) more rigorously. However, this theorem is still subject to its generalization in terms of sequences of subspaces. Further, we observe that the condition (3.2) gives a characterization of a Schauder base when we deal with the same sequence {Xn} in the same TVS X having some additional properties (see [11], for details).

TPIEOREM 3.5. Let (X, ~) be a complete l.c. T V S and {xn) be a sequence in X such that x n ~ 0 for all n ~ 1. Then there exists a complete AK-space A (with its Sehauder base (en}) such that

(3.6) {x.} {e'}; {3.7) {e n) >~ (xn).

t~OOF. Let us introduce

A = {{0~n}: {an} E (D such that 2 s i x , converges in X}. i ~ l

Clearly, �9 c A and it is a sequence space with usual pointwise addition and scalar multiplication. For each p E @x and ~ ----- {~l} E A, define

p*(~) = sup p ~lxl . n ~ l

One can easily verify that p* is a semi-norm on A and the family ~)* = {p*: P E ~)x) separates the space A. Thus we have a unique locally convex Haus- dorff topology z on A, generated b y | Also, simple computations yield that (A, v) is a complete K-space (for instance, given i ~ 1 we can find p E @x such that p(xi) :/= 0 and so [~llp(xl) ~ 2p*(~), that is,

I which in turn implies the continuity of Pi; for completeness of A, one may follow [21]).

Now the AK-ness of (A, v) would follow if we show that (e") is a Schauder base for A. Therefore, consider a = {~-) E A and p* ~ @*. Since

p* ~ - - . ~ e i e i ~ sup p ~ixl ~ 0 , n ~ , i=1 I l ~ m ~ n + l

we find that

= z - lim ~ ~t el" n~r i = l

This representation is clearly unique and hence we conclude that {e n} is a

226 GUPTA, KAMTHAN: DOMINATING SEQUENCES

baae for (A, v). Further, using the duali ty of A with ~, we may write the above expansion in the form

?l

: l im ~ ei(~)e t. n ~ i ~ l

where ei(~) : ~i ~--- Pt(~), i ~ 1, define continuous linear functionals on A. Consequently, {e n, e n} is a Schauder base for (A, r).

In proving {xn} ~ {e"}, we observe that if ~ i x f converges, ~ = {~i} E A

and so ~ = . ~ i e i. Thus {x,} ~ {e"}. For showing {e"} ~ {x,}, a~sume that i~ l

. ~ a t e i converges to a in A. Then for each p* ~ | and c ~ 0, there exists an i~ l N with

p ~ c , for n ~ N ~ sup p ~ixi ~ e .

Therefore {e n} > {x~} and (3.6) follows. To prove (3.7), consider the mapping To: sp {e n} ~ sp {x,}, defined by

Clearly, T O is linear and onto. Further, for arbi t rary/9 E @x,

11 / s l } p To aie i : p ~ix~ ~ sup p :r :qe l . l i = 1 J i ~ j ~ n

Thus T O is continuous. Hence there exists a unique extension T of T O from [c n] to [xn], such that T is a continuous linear operator with T(e n) : z n. This completes the proof the the theorem.

NOT~. Observe that except for proving (3.7), we just make use of the sequential completeness of X in the rest of the proof of the foregoing result.

The above result immediately leads to

COROI~AI~Y 3.8. Let {x,} be not a base for [x,] c X , X being a complete Hausdorff l.c. T V S . Then therc exists a sequcnce {en} in a completc Hausdorff 1.c. T V S A such that > {e"} holds but {e'}.

P~OOF. I f {x,} >~ {ca}, then combining this fact with (3.7) and from the fact that {e"} is a Schauder base for A, we conclude {x,} is base for [x,], which is a contradiction.

GUPTA, KAMTHAN: DOMINATING SEQUENC]~S 227

I t is clear from what has preceded tha t domination and equivalence of sequences do not imply their strict domination and strict equivalence respec- tively. However, if the domination implies strict domination in certain situations, we have a characterization of bases in Frdchet spaces contained in

THEO~]~M 3.9. Let X be a Fr(chet space and {xn} be a complete sequence in X such that x n ~: O, n = 1, 2 . . . . . The sequence {xn} is a base for X if and only if

(3.10) (fin} c K, {xn} > {ft,} ~ {xn) >~ {fin}.

PROOF. Necessity follows from Lemmas 1 and 2 proved by Arsove in [3]. For sufficiency, let the condition (3.10) be true. Consider the space A

as constructed above. Since {e n} is a base for A, we have

g(~) -~ g ( . ~ ~j e i) : ~ ~ig(ei), i~ l i21

where cr : {~.} ( A and g E A*. Hence {xn} > {g(en)} as the convergence of the series ~ a i x ~ means that {~i} ( A. This, in turn, implies that {x~} >>

~>~ >> {g(e")} by condition (3.10). Therefore, there exists a continuous linear operator f: [x.] -+ [g(e')] such that

f(xn) ~- g(en), n -~ 1 ,2 . . . . .

Since [g(en] c K and [x,] = X, f ~ X*. Thus for g ~ A* ~: fen] *, there exists f ( X* --i [xn]* such that g(e n) ---- f(xn) is true for every n ~ 1. Therefore, by the implication (iii) ~ (i) of Theorem 3.1, we have {xn) >~ (e '} . Since {e n} >~ >~ {x,} by the previous theorem, it follows that {e') ~ (xn} and therefore {x ,} is a base for X. This completes the proof.

In the case, when {xn) is a base for the whole space X, we have:

TH]~OREM 3.10. Let X be a complete 1.c. T V S having a base {x,, fn}" Then there exist

(1) a complete l.e. T V S of sequences having a Schauder base (e n, e n} ; and (2) an algebraic isomorphism T from A onto X such that (i) T(en) ----- x n,

and (ii) T is continuous. (3) Further, i /

~ ( x ) = sup p i ( x ) x i , n ~ l

then ~ ~- {p: p E @x} defines a locally convex topology ~ on X such that (X, -~)

is complete and the identity map I: (X, ~) --+ (X, ~) is continuous. Finally,

each fn is continuous on (X, ~).

228 GUt'TA, KAMTHAI~: DOMINATING SEQUENCES

PROOF. Construction of (A, v) as in Theorem 3.5 leads to the part (1) of the result. For the part (2), observe that

and

A = {{f,(x)}: x ~ X}

(2, } p*({fi(x)}) ----- sup p t(x)x i . n~l

The natural mapping T: A - + X, defined as

T({f~(x)}) = T ( . ~ /i(x)e') = . ~ /~(x)x~ = x, i~ l i>l

is clearly an algebraic isomorphism, and satisfies (i) and (ii) of (2).

The proof that the system ~9 generates a complete locally convex topo- logy follows as usual and the identi ty map I is continuous in view of inequality p(x) < p(x), which holds for each x E X and p E @x. Since p*({f,(x)}) : ~o(x),

the continuity of each fn on (X, ~) is a consequence of the K-property of (A, ,).

COROLLa~ 3.11. I f X satisfies the additional hypothesis that it is metriz- able, then each f,, is continuous on X and therefore {an, f , } is a Schauder base for X .

PRoos. Indeed, ~ = ~: by the closed graph theorem as the graph of the map I: (X, ~ ) - ~ (X, ~), is closed.

RE~--~KS. The above corollary is due to 1Newns [20]. The previous results from Theorem 3.5 to Corollary 3.11 are easily seen to be valid even if we take X to be a TVS rather than an 1.c. TVS and in this case Corollary

3.11 is due to Arsove [1]. I f the spaces (X, ~) and (X, ~) satisfy the closed graph theorem hypothesis (property), for instance (X, ~) is barrelled and

(X, ~) a B-complete 1.c. TVS (l~ts space), then from a result of Robertson

and l%obertson [22] (see also Husain [8], p. 58), it follows tha t ~ = ~ and so each fn is continuous on (X, ~). This observation is also roughly contained in Mityagin's paper ([19a], p. 88). Let us also mention here that McArthur in his unpublished note [16] has proved a more general version of the statement (3) of Theorem 3.10; see also his address [17], p. 885.

GUPTA, KA]KTHAN: DOMIgATING SEQUENCES 229

4. Applications

Let us recall from [12] the following

DJ~FI~ITIO~ 4.1. For a sequence space 2, a series ~ x l in an 1.c. TVS (X, ~) i;>1

is said to be weakly k-unconditionally Cauchy (abbreviated as w. 2-u.C.) provided the series ~ t x l converges in X for each {~t} E 2t.

For an AK-space 2, the above definition can be reformulated in the following form:

PROPOSlTIO~ 4.2. For an AK-space 4, a series ~ x t in an l.c. T V S (X, ~) i~ l

is w.2-~.C, if and only q {e n} > {xn}, where {e n} is the ~nit vector basis for 2

P~OOF. Straightforward.

NOTE. I f 2 is an AK- space with unit vector basis {en}, one can write its fl-dual in the following form:

Z : {~n}: (fin} E m and (e n} > {fin}}~ Making use of this representation of fl-duai and the results of the preced-

ing section, we prove a more general version of Theorem 3.2 of [12], contained in the following

T~EORE~ 4.3. Let 2 be an FK-, AK-space and (X, ~) a sequentially com- plete 1.c. TVS . Then a series ~ x i in X is weakly k-unconditionally Cauchy

if and only if (e n) > (f(Xn) }, for each [ E X*, where {e n} is the unit vector basis for 4.

P~OOF. Necessity follows by Proposition 4.2 and the continuity of each member of X*.

For converse, let us assume (e n} ~ (f(Xn) } for each f E X*. Since (e n} is a base for 2, applying Theorem 3.9, we get

{e"} v f x * .

Hence, for each f E X*, there exists unique g C 2* such that g(e n) ~ f(xn) , Vn ~ 1. Consequently, by the Hahn- -Banach theorem we can find unique g E 2" ---- [en] * corresponding to each f ~ [xn]* such that

/ ( x n ) = g(en), n = l , 2 . . . . .

Therefore by the implication (iii) ~ (i) of Theorem 3.1 (cf. the note after this result), it follows that (e n} >~ (xn}. Hence the proof is completed.

4 Periodica Math. 15 (3)

230 GUPTA, KAMTHAN: DOMII~ATING SEQUENCES

REMARKS. T h e a b o v e t h e o r e m , in p a r t i c u l a r , i n c l u d e s t h e r e s u l t s o f

M c A r t h u r - - R e t h e r f o r d [18] , p . 117 a n d W e i l l [24] , P r o p o s i t i o n 1,12 for

A - - %. I f ~ - - 1 p (1 < p < ~ ) in T h e o r e m 4.3, w e g e t t h e r e s u l t o f B e n n e t t

[6], p . 21 a n d h e n c e t h a t o f Singer [23], p . 131.

Added i n proof . F u r t h e r r e s u l t s in t h e d i r e c t i o n o I T h e o r e m 3.1 a r e

g i v e n in t h e a u t h o r s ' f o r t h c o m i n g m o n o g r a p h " T h e o r y o] bases and cones"

( P i t m a n , L o n d o n ) .

R E F E R E N C E S

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[2] M. G. AItSOVE, Proper bases and linear homeomorphisms in spaces of analyt ic func- tions, Math. Ann. 135 (1958), 235--243. M R 20 ~ 7215

[3] M. G. ARSOVE, Similar bases and isomorphisms in Fr4chet spaces, Math. A n n 185 (1958), 283--293. M R 20 # 7215

[4] S. B~ACH, Sur une propridt4 caract4ristique des fonetions orthogonales, C. R. Acad. Sei. Paris 180 (1925), 1637--1640.

[5] S. BANACH, Thdorie des operations lindaires, Monografje Matematyczne, Warszawa, 1932. Zbl 5,209

[6] G. BENNETT, Some inclusion theorems for sequence spaces, Pacific J . Math. 46 (1973), 17--30. M R 48 # 9342

[7] J. ttORVATH, Topological vector spaces and distributions, I , Addison-Wesley, Reading, 1966. M R 34 ~ 4863

[8] T. Hus~IN, The open mapping and closed graph theorems in topological vector spaces, Clarendon, Oxford, 1965. M R 31 ~r 2589

[9] P. K. KAMTHAN and M. GUPTA, Expansion of entire functions of several complex variables having finite growth, Trans. Amer. Math. Soc. 192 (1974), 371--382. M R 50 ~= 10299

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[12] P. K. K~Tt t~ -~ and IV[. GUPTA, Weak uncondit ional Cauehy series, Rend. Circ. Mat. Palermo 29 (1980), 364--368. M R 88g: 46011

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[14] 1 ~. K. KA~THZ~ and S. K. t%Ay, Equivalence of sequence of subspuees in topological vector spaces, Math. Japon. 29 (1976), 333--340. M R 54 @ 8216

[15] G. K6THE, Topological vector spaces, I , Springer, New York, 1969. M R 49 @ 1750 [16] C. W. McA~TB~r~, The pro]ective equicontinuoustopology. (Preprint) [17] C. W. McA~THU~, Developments in Schauder basis theory, Bull. Amer. Math. Soc.

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[19] B. S. MITJAGIN, Approks imat ivna ja razmernost ' i bazisy v jadernyh pros t rans tvah (Approximative dimension and bases in nuclear spaces), Uspehi Mat. Naulc 16 (1961), no. 4, 63--132. M R 27 ~ 2837

[19a] B. S. MITYlkGII~ (B. S. ~VIITJA_GIlg), Approximat ive dimension and bases in nuclear spaces, Russian Math. Surveys 16 (1961), 59--127.

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[21] J. R. RETHEI%FOI%D and C. W. M c A ~ T ~ R , Some remarks on bases in l inear topologi- cal spaces, Math. Ann . 164 (1966), 38--41. M R 83 $ 3079

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[22] A. P. ROBERTSON and W. ROBERTSON, On the closed graph theorem, Proc. Glasgow Math. Assoc. 3 (1956), 9--12. MR 18--810

[23] I. SINGER, Some remarks on domination of sequences, Math. Ann. 184 (1970), 113--132. MR 40 ~ 6237

[24] L. J . WEILL, Uncondit ional and shrinking bases in locally convex spaces, Paci]ic J. Math. 29 (1969), 467--483. MR 39 ~ 7389

(Received April 14, 1982)

DEPAI~TMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 208016 INDIA

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