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<oo . n '^n n M ^/^^^n^ Consequently o^j - 0. But f (0^) = 71 -^C^) 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)^.