CHAPTER II

THE BORNDLOGfY Off THE SPACES ACX.I.S) AND A(X.(t.s) •

In this chapter we introduce a homology on

A(X, (JJ, S) in a natural way and present some of its basic and useful properties. It is observed that this homology is not only different from the Yon-Neumann homology of the space A(X, (f, s) but it is not topologisable even. We also study the properties of a homology introduced on the dual space A(X, (f , s) in a manner analogoios to the one on A(X, C^, s). Finally, as an application, we make some interesting remarks on the topological as well as the bornol'ogical versions of the closed graph Theorem,

2.1 THE BORNOLOGg OF THE SPACE A(X. t. s) ,

We begin by defining a homology on A(X, (f, s) with the help of jj Jj introduced iji (1.3.2), For each r = l, 2, 3,,.. we denote by B^ the set 37

{a e A(X, (j:, s) / Ij alii r } . Then the family

IB* »{BJral, 2, 3»..»} forms a base (see

Definition 1,5«5) for a bornology B on A(X, C{ , s).

IB thus consists of those subsets of A(X, (jl , s) vrtiioh are contained in some B , It is straightforward that

(A(X, (JI, S), IB) is a separated convex bornological vector space (b.c.s, in short) with a countable base. In the sequel we shall mean by a bouinded set a set boxinded in this homology, unless stated to the contrary.

Theorem 2.1.1 : IB contains no bornivorous set.

Proof : Suppose IB contains a bornivorous set A, Then there exists a basic bounded set B^ e IB * such that A c: B. and consequently B. is also borni­ vorous. \ie now assert that if i^ > i then

XB. ^ B, for any A £ $ which leads to a contra­

diction. Since i-, > i , it is easy to see that

XB. V B. for all Xe ^ such tnat j A j 11 .

•fie nave only to prove that "X Bj^ a B. for any

y e I such that | "X | < 1 also. 38

Let thus I X1 < 1 . Since s is unbounded

(cf • Remark 1,3.1) given n we can choose x^^ such that s(x ) > n. Further as j Xi < 1 we can choose an n such that

1 < 1/1 >1 < (i^/i)^ < (i^/i)^^''"^ ,

Now let a e (t be such that n s Cx i s(x I i "" / 1 > I < la^l < (i^) " , and let t(x„) a = a„ x^ . Then ||a||= |a j " <. i. n n n -i- and hence a e B , Now ^1

II >a 11 = II >a^ x^ II

- I X a^ I ^

and hence >a 4 B , Thus ^B ^ B for any 1 1^ 1 Xe (): . Corollary 2«1.1 : The M-convergence in

A(X, (j: , S) is not topologisable. 59

Proof : Suppose the M-convergence in A(A, (|, s) is topologisable. Then by proposition 1,5.4, A(X.()!,s) would possess a bounded bomivorous set and this will contradict Theorem 2,1.1.

Remark 2«1.1 : The canonical bornology IB^

on A(X, (JJ, s) arising out of the metric of A(X, ^, s) is strictly finer than the bornology IB, Indeed, the set {a / jl ajji r } is bounded in IB but not in

IB , r being positive real number.

Theorem 2>1.2 : (cf, Remark 1,6.1) • If s satisfies conditions (l.3,5) and (l.3,6) , then a linear functional f on A(X, (): , s) is bounded if and only if f » I f (x^) • x^ where t(x„) X lim If(x^) 1^.0 i.e. A(X, (J: , s) = A(X,(|;,s).

Proof ; We observe that if s satisfies conditions (1,3•5) and (1,3.6) then X is countable

say X = {xjj} « Without loss, by rearranging the x^ s,

if necessary, we can assume that t(x ) ••0,

First we prove that if f is a bounded linear AO

functional then f is of the form f = Z f (x^) • x^, isl Let a e A(X, (|; , s). Then by Remark 1,3»2 we can write a « 2 a(x.) . X. . i=l ^ ^ n i»e. a = lim a v*iere a = I a(x.) , x n - oo n " i=l ^ ^

Since A(x, (|; , s) is metrizable, by Proposition 1,5.1, M a ••a implies a "* a in the Von - Neumann n n M bornology and consequently, a^^ •• a in IB, Again, in view of the remark preceeding Proposition 1«3*1» as M f is bounded f(a ) "* f(a) • But n n f(ttj^) = I a(x^) • f(x^). Therefore, ial M ~ f(a ) •• ^ a(x.) • f(x.) . But IB is a separated ** i»l ^ ^ boriiology. Hence

oo f(a) = Z a(x^) . f(x^) i=l '08 i.e. f = Z f(x.) , X,, . i=l ^ ^ We have further to prove that if f is bounded, then 41

t(x„) |f(x)| -0 asn-~ n' t(x ) Let ii- possible lim |f(x )| . ^ > 0 , n - »

Then , given '^ > 0 such that **1 < ^ » there exists a divergent sequence of integers (n ) such that

jf(xj^)| > "i] for i # nq • C3ioose n e H such that u > 1 and % "fyy 1 . Consider (a^^) in AUA, S) where a_ « w • X • n n • l/t(x^) Define X ^ - ^A • Then X ^ "* ^ ^^

2/t(x„) 5 II a/\J - II It ^ . X II - n2

and l/t(x„^)

> ( 1 --A) 42

which is tinbounded, by Proposit4«n 1«5«3, f is not bounded, t(x ) Conversely let |f (x^^) | ^ - 0 and (a^) M be a sequence in A(X, | , s) such that a •• 0,

Then by definition of M-convergence there exists a constant k and a sequence X n 1 *^^ sclalars such that ^° l/t(xj ilaq/>q B < k. i.e. |aq(x^)| < >q • k ^ , i ^l.

Further by condition (1,3.6), there exists an M 1 1/P such that [ ord (s p) ] <. M for all p 1 1 •

Since lim I f (x ) I ^ • 0 , there exists an n^ n -co n o t(x ) such that I f(Xjj) | ^ <. l/2kM for all i 1 n^.

Hence l/t(x„) |f (Xjj) < (l/2kM) " n 2. n^

Now, 43

|r(a )| - 1 2 a (x^) . f(x^) |

i 2 I a (x.) I If(x ) j

%-^ l/t(x ) 1-1 ** ^

oo l/t(x.) l/t(x.) + Z X . k ^ • (l/2kM) ^

n o -1 l/t(x,) , >q k i . |f(x^)i

oo P + S [2 X„ (1/2M) ] p=nQ s(Xjj)=p V^ l/t(xj - Z X„ .k ^ .IfCxJj + 2 >^ . I/2P i.l ^ p»n^ ^ o < <» (independent of q) •

Thus {f(a )} is bounded. Hence f is bounded on every sequence which M-converges to zero and again by Proposition 1.5.3 t is bounded.

Given a e A(X, $ , s), for each real r > 0, 44

we define a map | • j r | t A(X, I, s) - H by

l/t(x) . I a : r i - sup i a(x) | • r (2.1«l) X la t ri Is well defined. For if a e A(X, (|, s) then t(x) for every ^ > 0 , |a(x) j ± e for all but a finitely many x»» • Given r > 0 we can take an e > 0 such ^at r e < 1 • For this e > 0 there exists x^t Xpf •••» Xj^ such that tor every x / x. ,

.t(x) I a(x) I i e

t(x) i.e. 1 a(x) I • r 4 r . e , , , , lA(x) l/t(x) i.e. j a(x) I . r i (re)

£ 1 .

Therefore,

»up (ja(x)l.r "" ) 1 max (l,max |a(xj l.r^'^*^^^^

It is easily seen that for each r > O , ja J r| 45

defines a norm on A(X, (J! , s). The normed linear space so obtained will be denoted by A(x, ():, s, r) and the dual of AU, (JJ, S, r) by A(X, (f, s, r) «

Now we denote by 3B the Von Neumann homology on A(X, $, s, r) . The Theorem 2.1.3 shows that

(A(X, (|;, S), IB ) is the bornological inductive limit of the normed spaces A(X, (f, s, r)*s equipped with their Von-Neumann homologies.

Th^9rem g,l,? : IB - U IBj. reIR*

Proof : Let B e IB » Then there exists a constant k such that ji ail <. k for all a e B , t(x) i.e. sup |a(x) | i Ic • X l/t(x) i.e. |a(x)j / k i 1 for all x e X , l/t(x) i.e. sup |a(x) | . (l/k) < 1. 46

i.e. j a : 1/k I < 1 .

Hence B c U , 3B re if ^

For tile reverse inclusion , if B e B -ttien there exists a constant k such that |a s rj <. k for all a e B . l/t(x) i.e. sup ia(x) j . r 1 k .

Without loss take k > 1 , Then t(x) t(x) |a(x)| , r i k i k for all x, t(x) i.e. |a(x)| 1 k/r for all x,

i.e. II all < k/r .

i.e. B e IB ,

Hence U B c B reXR"*" 47

X Theorem 2.1.4 : The bornological dual A(X,(l;,s,r) of A(X, ();, s, r) is the same as its topological dual

A(X, (t, s, r)^.

Proof j The proof follows immediately from the fact that a linear functional on a normed linear space is continuous if and only if it is bounded,

On A(X, (t, s, r) we now define a map

I , : l/r I : A(X, (| , s, r) - IR * as l/t(x) la : l/r I = sup |a(x) j . (l/r) (2.1.2) X

Lemma 2.1.1 ;

A(X, (|:, s, r) c {a » X - (): / la : l/r| < « } .

Consequently ja J l/r j is well-defined on A(X,():, s,r) ,

Proof : Let, if possible, there exist a l/t(x) p : X - (J sucn that { |p(x) | , (l/r) } is not boundeoL, Then there exists a sequence (x , p ^ 1) p 48

l/t(x ) in X such that jp(x )| • (l/r) ^ > P

1 l/t(x ) ^•®* T:7~TI • "* p < i/p •

We define a : X - (j: by

y -* 0 if y /^ Xp

Xp - l/p(Xp) for p i 1 .

Then, a e A(x, (|;, s, r) and a - Z ITx"! " ^p*

Let now f,, be a continuous linear functional on

A(X, (J, s\ defined by toM = {i(x) for every

X in X . (see Remark 1.3.4). Then

i.e. fg(a) diverges.

Hence § i A(x, (j:, s, r) • A(X, (|;, s, r) now becomes a normea linear space relative to ]« t l/r| •

We denote oy IB- / the Von Neumann homology of

A(X, I, s, r) witn this norm which we shall call the l/r norm. 49

Remark 2^1.1 ; If s satisfies conditions (1.3.5) and (l,3.6), one can define another norm on A(X,ij:,s) by

« l/t(x ) la : rl » Z |a(x^) I . (r/M) » (2.1.3) 1 nsl It can be easily verified that these two norms on A(X, (^, S) are equivalent. In other words for every r > 0 , there exists t > 0 such that

j a J rj <_ s implies ja J t| <. s and vice-versa.

Similarly on A(X, ^, s, r)'^ also, one can define a norm as follows,

j a : l/2r| = Z |a^| . (l/2r)^'^*^'^^ 1 n=l ^ (1.2.4)

It can be easily verified again that the norms (2.1.2) and (2.1.4) on A(A, 4» s, r) are equivalent,

2.2 THE BORNOLOCy OF THE SPACE A(X. (t. s) .

A convex vector bornology B can be defined on

A(X, (j;, s) with the help of the function

j] .]| I A(x, $, s) -» 3R (see Remark 1,3.3) in a 50

manner similar to that on A(X, |, s) • We note that

B when restricted to A(A, (J, S) gives IB. Moreover,

A(X, (j:, s) » U . A(X, (j;, s, r) algebrat eally and rem *" as in the proof of theorem 2.1,3, we have

B « U IB ,

rem* ^/'

Theorem 2>2.1 x (A(X, (f, S) , IB ) is

M-complete,

Proof : Let {a_} be an M-cauchy sequence in

A(X, ():, sf.

Then there exists a sequence (A^nm^ ^^ positive

reals tending to zero such that II a - o^m/Mnmll- ^

where k is some fixed real number. We now choose a

sequence {>,„«} of positive reals such that

X n-a k. /A ^n m-. for all n, m and further such that

X n^a^ - ^^2=^2 "^^""^^'^ ^ 1 ^i^ and m^ 1 m2 •

For this, since P^^^ "• 0 » without loss we can 51

assume that A^ < 1 for every n and m« Now set nm lu 3 1 , m, « 1 and choose (n^, m.) inductively such that n.i > n.1— -1 , m.1 > m.i- .1 and M-_' n„m < 1/i for n 2. n^ t ^ 1 n»i* ^^® no^^ define {X^^ } as

^ nm " l/n»ln{i,o} if n^^ <. n < n1+ 1

and m. <. m < m • J j+x It is easily seen that {X } i3 the required sequence « Moreover, ^ „„ "• 0 and nm

Ji ^^n - ««>/>nm'l ^ ^ «n " VA^nm" ^ ^*

t(x) i.e. |(ajj(x) - agj(x))/A^ | < k for all x.

l/t(x) l/t(x) i.e. lajj(x) - ttjjjCx) I i Xnm • ^ ^ ^

Thus for every fixed x, (a„(x)) is a Cauchy sequence

in (f and hence there exists a(x) in $ such that

a (x) - a(x) as n - «> for every x. Now, 52

I {a„(x) - ap(x)>/X„,„il*^"^ W^M - <^^M/\/^^^

for all p 2. n+1 • t(x) i.e. I{a^(x) - ap(x)}/X^^^^^| i k f or all p ^ n+1.

Hence , as p - «> we get that t(x) Koj^Cx) - a(x) }/)^ ^^^\ 1 k for all x M and \^ ^^^ - 0 , Hence Oj^ - a • Now,

t(x) t(x) I a(x) I » I a(x) - Oj^Cx) + a^^Cx) 1 t(x) t(x) i |a(x) - a^ix)\ + lojjCx)! t(x) t(x) i ^ • 5>n,n+li * l«n^^>i

i k + ||an„ •j•l <' ' because an M-Gauchy sequence is bounded. Hence a e A(X, <$, s) and therefore A(X, (j:, s) is M complete,

Corollary 2.2.1 i A(X, (f, s) is complete.

In view of Theorem 1,5.1 it is enougji to show 53

that IB is I -disked. For tbis we show that each

B^ e IB is C -disked. Let thus 1^^} be a sequence of scalars such that I I X^ I < 1 and i=l ^ {a^} a sequence in B . t(x) II I > . a li » sup j Z A. a.(x) I i=l ^ i X -^ •»• t(x) 1 sup [ S 1 X^l • laj^(x) j]

t(x) <. r , sup C 2 I X^j ] X

J Hence Br^ is JL disked and the assertion follows,

Theorem 2.2.2 » (A(X, $,3), IB ) is not complete.

groof I In view of Theorem 1.5.1 again it is enough to show that (A(X, ij;, s), IB ) is not M- complete,

f\c s is unbounded there exists a sequence (x^) in X such that sCx^^) ^ n so that tjj(x^) <, 1/n 54

for all n 2. 1» Consider the se^[uence n l/t(x ) o^ =1 (1/2) ^ . x^. Then

l/t(x ) { (ttjj - ttjj,) / (1/2) ^^ , n i m } is bounded in

A(X, ({,3), In other words ia } is an M-Cauchy

sequence in A(X, (j!, s) and hence in A(x, (|: , s). But

in view of Remark 1,3»4

A(X, ():, s) c A(X, (1;, s) , As (A(X, 4, s), IB )

is M-complete the M-limit of (a } exists in A(X,(l;,s) <

In fact the M-limit of {a^ in A(X, ()l, s) is *» t(xj , l/t(x ) a - Z (1/2) ^ . X. as {(a„ - a)/(l/2) ^ ' n^} i=l ^ ^ is bounded in A(X, (|; , s ) ,

Further we note that a i A(x, (j!,s). Now

we claim that the M-limit of {a } does not exist in n M A(X, (J!, s). For otherwise, suppose that o^ "* P M f. in A(X, (p, s). Then a - p in A(X, $ , s) ,

Proposition 1,5.4 now gives that a = P • This

contradicts the fact that a 4 A(X, $, s). 55

Theorea 2.2.5 J Let s satisfy conditions

(1.3.5) and (1.3.6), Then (A(X, (J!, S)^ , B*)

is the M-corapletion of (A(X, $, s), IB ) • In other words, every a e A(X, ^, s) can be written as the

M-limit of a sequence in A(x, (^, s).

Proof t Let f = E f(x.) , x, e A(X, $, s) i»l ^ ^

(See Remark 1.3.^)• Then tnere exists a number d t(x„) such that |f(x^)| < d for all n 1 1. Further

since t(x_) -• 0 as n - «• , for every q there

exists n such that t(x ) < 1/q whenever n > n • q ^ " n ^ "q Now, consider the sequence (f_ « Z f(x.), q=l,2,,,,) ^ 1=1 ^ in A(X, (J, s). Then II l^ II - II I f(x ) . X /(i/a)"^ II (1/2) "^ n^+l ^^ ^ t(x.) q-t(x.) - sup |f(x.)| ^ . 2 ^

< 2d < « . 56

i.e. (f - f^) / (1/2) e B^^

M i.e. fq - f.

Hence the proof.

The Corollary 2,2.2 below is immediate from

Corollary 2.2,1 and Theorem 2,2,3.

Corollary 2.2.2 t (A(X, ():, s) , B ) is the completion of (A(X, (j:, s), IB ) .

The proof of the following Theorem runs on similar lines as that of Theorem 2.1,2.

Theorem 2.2,4 t The bomological dual of (A(x, (J , s)^ , B^) is ( A(x, (^, s), ne ) .

Corollary 2.2.5 : A(X, (JJ , s) is regular,

(cf. Definition 1,5.18).

Theorem 2.2.5 : (A(X, (J, s) , B ) is reflexive, (cf. Definition 1,5.19).

Proof ; By Theorem 2.2.4 above, we have that

[A(X, (J, s)^ ] = A(X, (j;, s). Now we claim that 57

t the topology ']f of uaiform convergence on bounded sets of A(X, (t, s) ([4], p. 195) is the same as the metric topology !J induced by || «!) on A(X,(j!, s).

Let us consider the set

B^^ » { p = Z b^ x^ e AU, 4, s)* /ll pU < k} .

It is straight forward that ^i/hkM ^ ^k ^ ^1/k where B^ = {a « Z a^ x^ e A(X, (J:, s) /|| a|| < r} and B^ denotes the polar of B. , On A(X, (f, s) , we now define the bornology of equicontinuous sets

(cf. Example 1,5.1), and as above we can show that it coincides witn B , Hence

A(X, (f, s) = A(X, (|, s) and consequently

A(X, ()!, s) is reflexive.

Theorem 2.2.6 t (A(X, ^ , S) , IB ) is a regular topological b.c.s.

Proof ; We observe that if F c A(X, (|, s) is such that a(f.) is bouided in I for every 58

a e A(X, 4» s), then F is bounded in A(x, (j:, s) .

In other words, every weakly bounded subset of A(x,(Jl,s)

is bounded in IB , Indeed, every weakly bounded set

in A(X, $, s) is e qui continuous * and hence bjF

Theorem 2,2,5 above it is bounded in JB , Thus the

bornology IB on A^X, ((, s; is the weak homology

(TgCACx, ();, s) , A(x, (j:, s)) on A(X, it, s) , Further,

it can be easily seen that (A(X, ^, s) , IB ) is a

regular b,c.s, as its dual A(A, (J, s) separates its

points. Hence in view of the Remark 1,5,3, we observe

that the bornology IB on A(x, ij!, s) which is the

He

same as (S^g (A(x, I, s) , A(x, $, s)) is precisely

the Von Neumann homology of "tfACx (t s)^ 1 Hence

btr^^ A gx*-j - IB and thus (A(x, ();, s) , IB )

is a topological b,c,s. By a similar argument it can

be seen that (A(X, (t, s), IB) is also a regular topo­

logical b.c.s.

KOTHE, G, ; Topological VectorSpaces I, Springer Verlag 1969, Theorem (2), p, 169. 59

Remark 2.2.1 : In the proofs of the Theorem

2,2.5 and 2•2,6 aiwve we saw that the bcrnology IB of

A(X, ((:, s) is the same as the homology

(f*g{A(it, ()J, s) , A(A, (f, s)) and is also the equicon- tinuoiois homology on A(x, (|l, s) . Thus A(X, ^, s) is a separated l.c.t.v.s, for which every subset of

A(X, ();, s) bounded for 6lA(x, ^, s) , A(X, (f, s)) is equicontinuous. In other words, A(x, ij;, s) is barrellea, (see [3], Proposition (4) and Definition

(l) p. 73). Consequently every simply bounded subset

(Definition 1.5»2) of A(X, I, s) is equicontinuous.

This is nothing but the well-known Banach Steihfeaus

Theorem , ([3], Theorem 1, p, 73).

2.3 CON^/EBGENCfi IN A(x. (fc. s) .

The following theorem gives an alternative

The author is thankful to Prof. Dr Taqdir Husain for suggesting this problem. m

characterisation of the Mackey convergence in A(x,(|!,s), when s satisfies the conditions (l.3,5) and (l,3«6),

M % Theorem 2.5.1 : f^ - 0 in A(A, (f, s) if

and only if ^n^^) "* ^ uniformly in some finite

circle i| a| <. r, (Compare witti-Theorem 1,2,8), M Proof ; Only if part s Suppose f ^ "* 0

oo and f„ B L c^ x^. Then there exists a constant q n-1 ^" ^ k and a sequence \ in $ tending to zero such that t(x ) llV^q'^ ^ ^ for all q. i.e. ICq^Xql "" <^ ^ t(x^) for all q, i.e. |tiq^n/>q| <. k for all n. lA(x„) i.e. |G I < |> I . k "^ for all n.

Now, if a e AU, 4» S) , a = 2 ax is such that

II all < l/2kM , then

i^(«) ' " ' n'l ""^^ • ^^ '

n«^l l^q^n^ 1 • 1 ^ J 61

l/t(x ) ,

"'^ (2kM) "

« l/t(x ) 1 l\l . 2 (1/2M) ^

ord(s (o)) ~ p = |\ |[ — + E ( Z (1/2M) ) ] 1 2M p=l s(Xjj)=p

ordCs"* (oj) °° P «iKi [ 2M ^ ^^/^r^ ^^y (1.3.S)). p=»i

Thus jf (a) j ^0 uniformly for all a such that

Hall < l/2kM.

If part : Suppose there exists an r such that

|fq(a)| - 0 unifornay for all a such that || a|| < r

i.e. sup |f_(a)| -* 0 as q - <»• Now, if we

define a by

aCx) =x r for x = x^ n = 0 otherwise, we get 62

lfq(a)| - jC^J . r'^'^"^^

sup I f (a) j , lUIIlr

t(x) i.e. [ i ^QIL,33 I ] ^ l/j, foj, all n, sup If (c) I lUllir

Let X = sup |fn(a) 1 • Then V - 0 as Nlir q - «> and

M Henoe f - 0. q

The Theorem 2,3,2 below establishes the equi­ valence of weak and Mackey convergenee in (A(X,$,S; ,IB ),

Theorem 2.3.2 : Let s satisfy conditions

(l»3«5) and (1.3.6). Then tiie Mackey convergnece and weak convergence of sequences in A(X<

Proof : Let f be a sequence in A(x, (J, s) it >

0 < II all i 1/2LM. Say a « 2 ^n *^n ' "^^^ nasi t(x„) 0 < sup \a^\ i 1/2LM

l/t(x„) i*e. |a^i < (1/2LM) ° for all n.

Consider Kfq-f) (a) I . I J^

oo < ^^ I Cqn - ^n I • i^n' ' 64

oo Choose n such that Z 1/2^ < e/4 • no + 1 Then n )(f^.f)(a)j < ^Z^ |C^^-CJ .!aj .^^ l^dn-^ni-^ni

(2.3.1)

Now, as Ijf IliL and | f i| £ L,

'^qn' i L and \C^\ '^ 1 L for all n.

1 l/t(Xj.) i.e. |G^^ - 6^1 < 2 , L ^ for all n, . , l/t(x^) ^ l/t(x„) ^•^- ^V " ^n' • '^n' i 2 . L ' ^\ (1/2LM) «' l/t(x ) - (1/2B ) ^ l/t(x ) n +1 ^" n n *- n +1 o o ~ s(x ) « 2 Z [ Z (1/2M) " ] p=n^+l s(x^)=p

12. Z 1/2^ . P=n^+1 i.e. Z JG_ - C„ I . |a„j < e/2 (2.3.2) n +T1 ^q n n ' • n' 0 65

When 1 < n < n , since C -» C as q - <• , we •~""Q qn n ' can choose q^ such that for all q ^ q, o

^ I'^nn - ^J (1/2LM) " < e/2 (2.3.3) n»l

From (2.3.1) , (2.3.2) and (2.3.3) we see that

|(f - f) (a) I <. e for all q 1 q and for all a such that Hall < 1/21M . i.e. fQ - f uniformly for all a with |j a|| 1 1/2LM and hence M ^q -^ ^'

\ 2.4 REMARKS ON CLOSED GRAPH THBDREM .

Let X and Y be two vector spaces and u

a linear map from X to Y, The graph of u is the

set of all pairs (x, y) in X x Y such that y = u(x).

The graph of u is a subspace of X x Y . If X and

Y are separated bornological vector spaces (respectively

separated topological vector spaces) and if u is

\ The author is thankful to Prof. Dr Taqdir Husain

for suggesting the problem of closed graph theorem for

these spaces. 66

linear and bounded (respectively linear and continuous) then the graph of u is b- closed (respectively closed).

The closed graph theorem is in a sense, the converse of the above assertion. It states as follows :

Theorem 2.4.1 : (Bornological Version) [3]»P»58).

Let E and F be convex bornological spaces such that E is complete and F has a net. Then every linear map u : E -• F with a bornologically closed graph in

E X F is bounded.

Corollary 2.4.1 i Let E and P be conqplete convex bornological spaces and suppose that the homology of F has a countable base. Every linear map u : E •• F , whose graph is b-closed in E x F is bounded.

Theorem 2.4.2 : (Topological Version) ( [4] , p, 301, Theorem 3). Let E and F be two metrizable

complete topological vector spaces, f : E -• F a linear map and let us assume that the graph G of f is

closed in the product space E x F, Then f is

continuous• 67

In Sections 2,1 and 2.2 we have seen that

A(X, ^9 s) is a complete c,b«s with countable base.

Consequently, by Corollary 2.4.1, the closed graph theorem

(Bornological vesion) holds for (A(X, (JJ, s) , IB ) ,

Kankurikar [lO] has proved that A(x, <^, sy is a metrizable, complete tvs« Hence closed graph theorem

(topological version) holds for A(X, ^, s) • For the

same reason the theorem holds good for A(A, ^, s) also.

However, since (A(x, ()!, s), IB ) is not complete

(Theorem 2,2.2), it is natural to expect that the borno-

logical closed graph theorem does not hold good in

A(X, (f, S). And this is, in fact, vtoat happenj . We

give a counter example.

Example 2.4.1 ^ Recall that V denotes the

Iyer»s space of entire functions and is a special case

of A(X, $, s). (see example 1.3.1). Let

u : r ** P ^ defined by

« - ^ ^n^n - ( 2na^ ) J^ 68

and consider B "{^„» n^l}, where ^ » z*^,

B is bounded in the bornology of v . However

{u(5jj)/fjj eB} s (no^ n^.!} is not bounded.

Thus u is a linear map which is not bounded. But

yet the graph of u is b-closed, i?or let

00 M "P "nfo ^"^^n •* « - ^ ^n^^n ^"^ M uCttp) - P . M Now, as a "* a , there exists a sequence IT \ -* 0 and a bounded set B such that ' p r

ffp - a e Xp B^ for all B 1 1 .

i.e. (oCp-aVXp e \ •

i.e. ii ap - a/>p II < r •

i.e. sup I (a^P^ - a^)A | 1 r. n

i.e. |a^P,(P^) " *n ' ^ Xp • ^" ^°^ ®^®^y t±-x.e^ n.

Thus a P' - a as p - «» for all n. Also as a^ n n *^ p, 69

and a are entire functions, 2 n a^^' and I n a n n are convergent. Hence 2 n aA^ "* 2na asp-~,

i.e. I Z n a^P^ - 2 n a^ | - 0 i.e. II uCttp) - u(a) || - 0

i.e. Ilu(ap) - u(a) 1] < l/p (say)

i.e. u(a ) - u(a) e l/p . B^

M i.e. u(0 - ^(a) •

But u(ap) - p and (A(X, (|;, s), IB ) is separated.

Therefore, u(a) • P . Thus tne graph of u is b-closed.

Interestingly A(X, (J, s) satisfies all the other

conditions of Corollary 2.4.1 and only the condition 6f

completeness is violated.