Cx4 AFTER I
PRELIMINARIES
In this chapter, we collect together, for convenience, all the known concepts and results which are required, in the subsequent chapters. These are divided into sections bearing titles of the topics to v^ich they relate.
1.1 TOPOLOGICAL VECTOR SPACES
Definitions and results in Topological vector spaces Tiriiich are used in this dissertation can be found in [4] or [14].
We begin with some examples that are useful later in the dissertation. Throughout IR and (J will denote the f ielda of real and conqplex numbers respectively.
Example 1.1.1, ; Let X be any set. We consider the set C^U) s {f ; X - ()l / for every e > 0, there exist only finitely many x e X suah that jf(x)| > e }, It is easily verified that C (A) is a Banach space with respect to the usual operations and the norm defined on it as tU ii - swp If(x)| (i.ia) X e X
Example 1.1.2 : Let B be any Banach space and X any set. We define L(X, B) as follows :
L(X, B) = {f ; X - B j there exists M > 0 such that for any sequence (x ) in X,
* l|f(Xn)ll < M } . nail For any f e L(X, B) , we then define
llfll = sup E j|f(x„)|| . (xJc=X n n'
L(X, B) becomes a Banach space with respect to usual operations and )|f || defined above.
Definition 1.1*1 : If A is a subset of the dual E of a locally convex space Cl.c.s,, in the sequel) E, then the set
A° - { X e £ / ||f(.x)jli 1 for all f e A }
is called the polar of A,
Definition 1.1.2 i If E is an l.c.s, then the polars of all the singleton sets in E form a base for the neighbourhood, system at zero for a locally convex topology on 2, called the weak topology on E, We denote this topology by cT (E, E ), In a similar way we can define the weak topology cT (E , E ) on E , Also, the family {A°/A c A} where is the collection of all balanced convex cT (E , E) - compact subsets of E gives a topology on E called the Mackey topology on E, denoted by '^ (E, E*^ ), Thirdly, the family {A°/A e»A}, where ^ is the collection of all balanced convex (fCE , E) - bouaded subsets of E gives a locally convex topology on E, called the strong topology of E, which we denote by p(E , E ) ,
Definition 1.1.3 i A family *1* of line ar functionals on a t.v.s, E is said to be equicontinuous at a point x e E if for every r > 0 , there exists a neighbourhood U of x^ such that |f(x) - f(x )| <. r for all £ e y- and x e U , T is said to be equi continuous on E if T- is equicontinuous at every point on E*
The concept of inductive limits is useful, in general, in the study of bornological spaces. Definition 1.1.4 t Let I be a directed set and
(E.). -J. , a family of l,c.s.*s. Suppose that for every p4±r (i, d) of indices such that i ^ j we have a continuous linear aaap f ^^ t E. - E. and that these a^ps satisfy the following two conditions :
f is the identity map for each i e I (1,1,2)
f - f . f i4 for i i d i X (1.1.3) i^ l«c.s, F is said to be an inductive limit of the inductive family (E., f . .^ if the following conditions are satisfied. For every i e I there exists a conti nuous linear map f. j E. -• F such that
f^ « f . f .^ for i 1 J (1.1.4)
Given an l.c.s, G and a family of continuous linear mapsJ g ; E - G SUCH that g « g o f ^ for i i d, there exists a unique continuous linear map g ; i?" -» G such that
g^ * g o f^ for all i e I (1.1.5)
Theorem 1.1.1 : ([4], Proposition 2, p. 38 ). Let X and Y be normed linear spaces and T a linear operator on A into Y, Then T is continuous on X if and only if there is a constant M such that l|T(x)|| i ^^||xj| for all x in X.
1,2 tHS SPACE OF ENTIRE FUNCTIONS Cy£R (J .
Let r denote tne set of all complex entire functions. It is well known that any entire function a can be repre- sented. by a power series I a iP' v^ere the coefficients a_*n s are such that la^ln ' -•0 as n-»«. For any entire function a we define the real number Hall by
Hall - sup { la^l , la^l^/'^ } (1.2.1) nil
It is easily verified that il ail satisfies the following conditions \
Hall 1 0 and ||a|| - 0 if and only if a » 0. (1.2,2)
lla + pjl < ||a||+ lUll (1.2.3)
llXall i A(>) . II all where A e (f
and A(A) = max (1, I )<| ) , (l.2.'4)
It follows from (1,2.2) and (1.2.3) that d(a, (i) = lla - pll defines a metric on \ • Theorem 1.2.1 ; ([5], p. 16) T is a complete separable linear metric space*
Theorem 1.2.2 ; ([5], p. 17) ^ contains no bounded open set and so V is not normable.
Theorem 1,2«3 : ([5], p. 19) Every continuous linear functional f (a) defined for a e T is of the 00 OO form f (a) » 2 a^^ c^ , a • S ^ ^^ where n<>0 n^ 1 /n { |cjjl > o i 1 } is bounded.
Theorem 1.2.^ ; (C5]f P« 21). T , the dual of the space C is a complete metric space. More precisely, if f, g e r , then
f + g is continuous (1,2.5) if ^n " ^ *^®^ ^n ** ®^ • - ^ ^ (1.2.6)
If (c ) is a sequence of complex numbers tending to c th^ in general (c„f) does not coverge to cf. (1.2.7)
Definition 1.2.1 t ([8], p. 646). A linear transformation T from T to T is said to be metrically bounded if there exists ^ > 0 such that l|T(a)|| i M . jl all for all a e f .
The»rem 1.2,5 s ([8] p, 646). If T is a linear transformation such that
l|T(a)|l 1 M . Hall for all a e T , then
T(U - a^o^o * ^01 ^1 »
^'n n^ n * *• where a. . , i, ;) = 0, 1 are complex numbers with moduli! not exceeding M, |k | i M° , n 1 2 , and E^ » Z^ for n«Of If 2, ••• •
Rerprk X^P,1 t The converse of the above theorem also holds. In fact the metrically bounded linear transformations are of the above form. i.e. BCD . { T ; f - V/Ti^^) - a^J^ * a.^^ S^ ,
T(^l) - &^^i^ + a^^^cTi »
where a^^^, i, j » 0, 1, are complex numbers with 8
moduUb. not exceeding k and
1/n Ik^l i k for n ;^ 2 } .
naf^nition 1>2.2 '* ([20]) • A linear functional
F I r "* ^ will be called a metrically bounded functional if there exists a positive real number M such that
I F(a) I < M . Hall for all a cT .
Theorem 1«2.6 t ([20]) A linear functional F J r - (J; is metrically bounded if and only if there are complex numbers a and b such that if
n?^ case j|F|| » jaj + |bl • In fact F((f^) » a,
FiS^) - b.
theorem 1.2>7 : ([5], p. 23). The notions of strong and weak convergence in V are equivalent.
Theorem 1,2.8 ([5], Theorem 5), The statement that a "*a asp-«»in the space • is XT equivalent to the statement that a(z) -* a(z) uniformly in any finite circle. i
However, a corresponding result does not hold good for the space \ ([4]),
oo Definition 1.2>^ ([6], p. 87). Given a • I a^ za' naO " in r » ^or each real muiber r > 0 , we define a map J . ; r I : T •• H such that
I a s r j - I |a„| . r"^ (1.2.8) n-0 "
It is easily seen that for each r > 0, ja s r | defines a norm on T . The noraeci linear space so obtained is denoted by T (r) and the dual of T (r) is denoted by r (r).
Theorem 1.2.Q ( [6] , Theorem 2 ) r - u rcr) r>0
Theorem 1.2.10 ; ([6], Lemtna 2). Every functional in \ (r) is of the form
f (a) • Z c^ a„ , a « I a^ z" where n»0 "^ ** n-0 '^
{ |c l/r"^ } is boundeo and conversely. 10
Following ^. G. Iyer [7], we denote by T(R^ - R^) a continuous linear transformation from { (R-) into V (Rp) and the family of all such transformations by F(R.^ - R ) , Consistent with this notation we denote by T(«»-«>) a continuous linear tr«nsfca*mation of Y" into \ and the family of such transformations by
Theorem 1.2.11 t ( [7] , Theorem 5) .
The following relation is valid,
F(oo -co) , n { U F( R^ - R ) } . R2>0 R^>0 ^ •*
We recall that if (f(n)") an* (0(n)) are two sequences, 0(n) > 0 then the statement f(n) « O(0(n)) means that |f(n)| < A 0(n) for all n where A denotes a constant.
Theorem 1.2.12 » ([s], Theorem 5) . With every homomorphism T of T ($) into itself we can assoliate a unique sequence H^^, n « 0, 1, 2, ,.. of disjoint finite subsets (i)ossibly empty! of the set I of non- negative integers having the following properties : 11
T(x**) • Z x^ , the right hand side being peH„ interpreted as zero element when H is empty, (1.2.9)
If p is the greatest integer in H , then p « 0 (n) as n -* «» , writing p » 0 whenever H^^ is empty. Under these conditions,
T(a) - I ajtix^) for all a - 2 a^^^ x?^ e V (J (1.2.10)
Conversely, if the disjoint finite (possibly empty) subsets of I be preassigned satisfying (1.2.10), T(x'^) and T(a) defined above, then T is a homomorphism of \ (O into itself*
Theorem 1.2.15 ; ([s] , Theorem 6) • Let T be a homomorphism of T (^) into itself and H^^ be as in Theorem 1»2.12. Then the following statements are true*
T maps f (C) onto T(r(t)) in a one-to-one aanner if and only if no H is empty. (1.2.11)
If in addition, n » O(pj^) , then T is a closed 12
transformation, that is, transforms closed sets of Via) into closed sets. In particular T[ TCc)] is closed in ^ (c) and so is complete and consequently T is an isomorphism (one-one, bicontinaous) of f(c) onto T[r (c) ] . (1.2.12)
1.3 THE SPACES A(X. (t. s) and A(X. (t . sf .
Definition 1,5.1 ; ([lO], [l3]) . For any set X,
s ; X - IN be a map from X to the set IN^Cf all non- negative integers. We define
t ; X -• ( 0, 1 ] as
t(x) » l/s(x) if s(x) /i 0
« 1 if s(x) . 0 .
Then
A(X, (f, s) • {a s X - (f / for every e > 0, t(x) |a(x)j < e except for
finitely many x e X } (l#3.l)
For each a e A(X, (J , s) we define a real number H a|| by 13
t(x) Hall = SFP |a(x) | (1.3.2) xeX
Then it is easily seen that || all satisfies the conditions (1.2,2) , (1.2.3) and (1.2.4) .
Example 1.3.1 8 ([lO]) . A(IN^, (f , Id) - f ,
IN^ being the set of non-negative integers, Id denoting the identity map.
Example 1.3.2 : ([lO]) A(]N, ()! , \y^ ) is the
space of entire functions of k variables, wher« IN^
denotes the cartesian product of k copies of IN^and S^ : DJ^ - Bf^ is defined as
« k Isl
Theorem 1.3.1 : ([lO], [13]) J A(X, $, s) is
a complete linear space.
Theorem 1.5.2 ; dlO], [13]). A(X, (j:, s) is
normable if and only if s is bounded.
Remark 1.3.1 : Throu^out the thesis we assume
that s is unbounded so that A(X, ^ , s) is non- normable. 14
Theorem 1.3.3 : ([lO], [l3]). The space
A(X, ^,8) is a locally convex topological vector space.
The lemma 1.3»1 below gives a representation of the elements of A(X., ((; , S). The proof of the lemma is useful in the proof of Theorem 4,2«4,
Lemma 1.3.1 : ([l3]» Lemma 2.1) • Let a e A(X, ^ , a) and let for any x e X, a/^ ^\ denote the following element of A(X, (JJ , s) •
«(x,l) ' ^ - ^ X •• 1 y - 0 if y / X .
Then there exists a sequence m <.ffl^i»«»<.ini.<.«.« of natural numbers and a sequence (x^^) in X su^ that lim oiv^ a » Z a(x.) • a/„ -,\ • i»l ^ ^^i» ^^
Proof : Without loss of generality we can assume that a / 0, the case a » 0 being trivial. Let n lie such that Hail > 1/n • By (l,3.l) we choose 15
t(x) X, , x_, ..,, X in X sucn that |a(x) ) > 1/n , X ei VBLQ O if and only if x e {x|,, x-,^,,, ^m ^ • Otjvio^^ly o m ^ !• Similarly we choose x-, 3t:2,«..,x , t(x9 x^ ^^ ,,.,, x^ in X such that |a(x)| >l/(nQ+l) if and only if x e {x^, Xp, ...| x }• Obviously, m ^ m, • By applying induction, we arrive at a sequence of integers ni- ^ m^ <. m^ <. • .. <. m,^ <. . ,. and a sequence (x, ) in X with the property that ^ k=l
|a(x)| 2L -5—iric— ^ ^^ °^y ^ o
X e {x.,•.., X } for k « 1, 2, ...
Let a^ denote the restriction a (x-j^ Xjjj } for 'k k = 1, 2, 3, ... . Clearly a^^ e A(A, i|; , s) and
ttj^ = 2 aCXj^) a(x^, 1) (k 1 1 ) . i=l
Therefore by (I.3.I), (l.3,2) and (l,3«3) we have 16
that II ajj - aJI •* 0 as k -• <», Hence the lemma.
Remark 1.5.1 : ([ll]. Remark 2.2, [l3]. Remark 3.1.1) We note that tne map x "* a/ -, \ from X into
A(X, 4t s) is an infective map. Identifying a/^j. 2) with
X the lemma 1,3.1 now reads as : for any a e A(X, |, s), there exists a sequence ol' natural numbers m <. m, <. •..
<. m. <. »,, and a sequence (x. ) in X such that
Hm m,k a » Z a(x.) , X. i=l ^ ^
Remark 1.3.2 ; ([11], Remark 2.?, [l33. Remark 3.1.1.3)
Without loss of generality we can represent an element a of A(X, (): , s) by
00 a » 2 a(x ) , X. (1.3,4) i=l ^ ^ if we assume the possibility of a(x ) being zero for some i*s.
Theorem 1.3.3 : ([H], Theorem 2.2).
If the function s satisfies the conditions
s"-^ (p) is finite for all p ^0 (1.3.5) 17
n 1/P The set { [ord s (p) ] /p ii 1} is a bounded set 5 ord s (p) denoting the number of elements in the set s" (p) . (l»3«6)
A(X, $,S) = {p:X-(l;/ there exists a real number t(x) F such that |p(x)j <, P for all x e X}
Remark 1.3>3 : ([13], Remark J.I.1,5).
We observe that even if s does not satisfy conditions
(1.3.5) and (1.3.6) ,
A(X, (f, s) c{p:X-»(l;/ there exists P > 0 such that t(x) |(3(x)i <. P for any x e X } .
Consequently as in the case of A(X, (j;, s), the function
I] .0 : A(X, ij; , s ) - m"^ defined by
t'vX) II ^ ji » sup |p(x)| satisfies the conditions (1.2.2), X If; (1.2.5), (1.2.4). A(X, (J:, s) is only a topological
group ; not a t.v.s. unlike A(X, (f, s).
Remark 1.3.4 : ([l3], Remark 3.1.1.4) .
If s satisfies the conditions (1,3.5) and (1.3.6) la
then oljviously X becomes a countable set and every element in A(x, $, s) as well as A(X, $, s) can be written as Z a(x.) • x. , To each continuous linear functional on the space A(X, $, s), there corresponds a unique function p : X -• (j! in a natural way as follows :
For any f t A(X, (jf, s) •* (j: we define p j X - (f as
fj(x) » f (x). However if p : X - (j: is given, then there need not exist, in general, a continuous linear
functional f on A(X, (f , s) such that f (x) = p(x).
This follows from theorem 1.2.3 • However, if a function
p : X -• (f gives a continuous linear functional f on
A(X, (J , s) such that f (x) = p(x), then the series
Z a(x ) . p(x ) i=l ^
converges for every
a = Z a(x^) • Xj^ e A(X, I , s) ,
In other words f can be written as
f « Z f (x^) . x^ formally. 19
Theorem 1.3.4 : ([ll]. Theorem 3.1) .
The notions of weak and strong convergences are equivalent in the case of A(X, (f , s).
Theorem 1.3.5 ([ll], Theorem 3.2').
The notions of weak and strong convergence are not equivalent in the case of A(x, (f , s) ,
1.4 METRICALLY BOUNDED FUNGTICNABS AND TRANSFORMATIONS
ON THE SPACES V AND A(X. (t. s) .
Let B(r) denote the set of all metrically bounded ti'ansformations ([l2], p. 209) of T into
\ and \ the set of all metrically bounded functional on f ,
Theorem 1.4.1 : ([l2], Theorem 3.2) .
B( r ) is isometrically isomorphic to f + T + B where
B»{k«(k),_| k e ^ and tnere exists p > 0
such that |k^| < P for all n 2. 2 } . 20
Remark 1.4,1 s ([12], Remark 3»l) •
B( r) is a group under addition. Infact B(f ) is a topological group. But BCD is not metric linear like f as the scalar multiplication is not continuous in the scalar variable,
Theorem 1.A.2 : ([l2], Theorem 2.2j, The space
A(X, ();, s)"^ of metrically bounded linear functionals on
A(X, (J, s) is isometric isomorphic to L(Y) >*iere
Y = s"^(o) U s~^(l) and by L(Y) we mean the following Banach space.
L(Y) • { p : Y -» (); / there exists a real number
M having the property that
^ lp(y ) I <. M for every sequence nal
(y^) in Y } .
Norm on L(Y) being defined as
lip II - , sup Z |i3(y„) 1 . (y^)cY n.l n
Theorem 1,4.5 : ([l2]. Theorem 3.6)•
If T is a metrically bounded linear transformation 21
from A(X, ();, s) to A(X, (^, s), then for any x e X, there is an associated sequence (,x/) in X and a sequence (a ) in (f depending on both x and (x.) sudi that
oe T(x) « I a„_ . X. (1.4.1) i-1 ^i ^ t(x),t(x^) l/t(x) |a_, I < M for all i 2 1 (l.A.2) i l^xxi I •* 0 as i -*«o (1.4.3)
Let, for any n 2. 0» Bj, denote a Banach space over (J and let
00 D < B^, nlO>- { Z ^n^n'^n^^^n ^^^1^
1/n and II bl) -• 0 as n - «» } ,
Then it is easily seen that D < B^, n 2 0 > is a complete metric linear space with respect to the metric arising from I Z b 6 jj defined, on D < B^, n > 0> jj^ n n n as follows : 22
II "^ b^f^ll - sup {jlbjl, llb^ll^^''} (1.A.4)
If we define D < B^, n > 0> as follows t
D < B^ , n ^ 0 > m { 2 b^ e B f or n > 0 and ° n-0 ^ " there exists M > 0 such that 1/n II bj^ll i M for all n 1 1 }
Then clearly ^ < B , n 2. 0 > isa unique maximal metric linear subspace of D < B , n ^ 0 > where
II Z b 5 ^ 11 is defined on D
Theorem 1>4.4 : ([l2]. Theorems 3»8 and 3.9).
For any metrically bounded transformation T on A(X, (1;, sh
E T 5 e D
(See example 1.1,1 and 1,1.2 for the definitions of
CQ(X) and L(X, B)),
Theorem 1,4.5 : ([12], Theorem 3.10) .
The space V of all metrically bounded transformations from A(X, ^, s) to A(X, ^, s) is isometric iso^,mopphic to S < L(fl(l/p)), C^(f^(l/p)), P i 1 > .
1,5 BORWOLOGf
In this section we collect all th« known concepts and results in bornology niiich are required in the subsequAat chapters. The definitions and most of the results listed here can be found in [2J and [3] •
Definition 1,5.1 : Let X be a set, A family IB of subsets of X is called a homology on X if it covers X, is hereditary for inclusion i,e, A e ^ , B c A implies B e ^ , and is closed for finite unions. The pair (X,® ) is then called a bornolcgical set and members: of IB are sailed the bounded sets of X,
Example 1.5,1 : For two topological vector spaces E and F we denote by L(E, F) the vector space of all continuous linear maps of E into F, A subset H of 24
L(E, t) is called an equicontinuous subset if for every neighbourhood V of zero in F the set
J —1 H(V) m 0 u(v) isa neighbourhood of zero in E. ueH The family ^ of equicontinuous subsets of L(E, F) is a vector homology on L(E, F). This homology is called the equicontinuous homology of L(£, F) and is a conv^ex homology if F is locally convex.
Definition 1.5.2 : Let E and F be locally convex spaces and let L(E, F) be the space of continuous linear maps of E into F. A subset H of L(E, F) is said to be simply bounded if for every x e E , the set H(x) m u u(x) is bounded in K the scalar field, ueH
Definition 1.5.3 : A map from one bornological set
to another is said to be bounded if it transforms bounded sets into bouided sets.
Definition 1.5.4 t A homology ^^ ^ on X is
said to be finer (weaker) than a bornology Bp ©n X
if ^^^ c IBp i^2 ^^1^ o^ ^^ other words the identity
map (X, JB^) - (X, IB^) i{X,lB^)'*ix,JB^)) Is bounded. 25
Definition 1.5.5 : A subfamily IB ^^ of a bornology
3B on a set X is said to be a base for B if every member of IB is contained in some member of IB^^ •
A family IB of subsets of X is a base for a bornology on X if and only if E covers X and every finite union of elements of IB is contained in a o member of IBo_ . Then the collection of those subsets of X which are contained in an element of IB defines o a bornology IB on X having IB^ as a base.
Remark 1.5.1 : In a t.v.s. £ , the collection of all sets which are absorbed by every neighbourhood of zero forms a bornolo^. This homology is called the Ven Neumann or the canonical bornology of E,
Definition 1.5.6 : A bornology IB on a vector space E over IK is said to be a vector bornology if the two maps (x, y) -• x + y of E x E into E and ( }v, x) •* Xx of UC x E into E are bounded. The pair (£, IB) is then called a borno- logical vector space or a b.v.s,. m
Remark 1>5«2 t Subspaces, quotient spaces, direct sums, inductive limit and ppojective limit of a b,v«s» are defined in the usual manner.
Definition 1.5«7 ' A vector bornology is said to be convex if it has a base consisting of convex sets, A b.v.s, >^ose bornology is convex is callea a covex borno- logical space or a b,c,s.
Definition 1.5.8 t A subset A of a b,«.s, E is callec a bornivorous set if it absorbs all bounded sets of E.
With every I.Q.S, S we can associate its Von - Neumann bornology which is convex, E with this convex bornology is denoted by b-,* Also to every b.c.s, E we can associate a l,c,s, for which the balanced convex bornivoroxis sets of E form a base for the neighbourhood of zero. This l.c.s, is denoted by tg •
Definition 1,5.9 s If for a b,c,s. E, we have E m btg, then E is said to be a topological b.c.s.
Definition 1.5>10 : A b.c.s, E is said to be separated ir" the only bounded subspace of E is (0). 27
Definition 1.5>11 « A sequence (x^) in a b,v,s» E is said to Macicey converge or M-converge to x e E if there exists a decreasing sequence ( An^ ^^ positive reals tending to zero such that { (x_ - x)/^ } is M " " bounded. We then write x_ - x and x is called the n M-limit of (x ), A sequence (x ) will be called an M-Cauchy sequence if the sequence (x - x ) M-convegges to zero as n, m -* «> • It is clear that the image of a bornologically convergent sequence uider a bounded linear map is again a bornologically convergent sequence. Proposition 1«5«1 : In a metrizable t,v,s,, every topologically convergent sequence is bornologically convergent,
Proposition 1.3.2 : A b.v.s. E is separated if and only it' every bornologically convergent sequence in ii has a unique limit.
Definition 1.5.12 ; Let 3* be a filter on a b.v.s, E, We say that J* Mackey converges to zero or M-converges to zero if there exists a bounded set 28
B such that '^ => 1KB »{ABJ AeK}.
Proposition 1.5.5 : (L2]. Proposition 1, p, 10)•
IC E is a b.v.s. and i? is a topological b.v.s, then the rollowing are equivalent for a linear map f ; S. •* a',
f is boundea. (1,5.l)
The image of every sequence M-convergent to zero in E under r is bounded in F, (l,5»2)
Definition 1.5.15 : Let E be a b.v.s. The M- convergence in £ is called topologisable if there exists k topology *^ on E such that the set of M-convergent filters is exactly the same as tne set of filters con verging for J •
Proposition 1.5.4 ; ( [2], Proposition 1, p. 12).
Let £ be a b.v.s. The M-convergence of E is topolo gisable if and only if E possesses a bounded bornivorous set.
Definition 1.5.14 : A balanced, convex subset B of a separatee b.c.s,, E is said to be completant if 29
E„ the vector space generated by B is a Banach space B when it is equippea with the norm given by
Jlxjig - inf |X| • A b.c.s. is said to be complete XEXB ir its homology has a base of completant sets.
Definition 1.5.15 : A b.v.s, E is said to be Mackey - complete or M - complete if E is separated and every M-Gauchy sequence in E has an M-limit in E,
Definition 1.5*16 J A Subset A of a b,v,s« E is said to be I - iisked if the sum of the series
2 X 4 3c. (whenever exists) belongs to A for (x.} i-1 ^ ^ ^ in A and { A^} a sequence of scalars for which
oo ^ I ^ii 1 !• "^^e bornology IB of E which has a base i«l "" of i -disked sets JLa called an Jt -disVted bornology.
Theorem 1,5.1 ; ([2], Theoreme 1, p, 35) • Let
E be a b.c.s. Then the following are equivalent.
E is complete. (1.5.3)
E is M complete and its bornology is C -disked,
(1.5.A) 30
Definition l.'>«17 : The collection 'j of all subsets 0 of a b,v,s. E such that
0-x « {a-xJaeO} is bornivorous for all
X e 0 , forms a topology, called the Mackey - closure topology of E.
We say that a subset F of E is M-closed (open) if it is closed (open) in this Mackey-closure topology,
By EX we denote the bornological dual, i.e. the space of all bounded linear functionals on the b»v,s. E,
We define B° = { f e EV|f(B)| i 1} • The collection
{BVB is bounded in E } is a base for a topology T on E , The topological dual (E ) becomes a b«v«s, when equipped with the bornology of all equicontinuous sets for the pair (E^ , (E^) ).
Definition 1.5-18 : A b.c.s, E is said to be regular if the l.c.s. tg is separated or equivalently EX separates E,
Definition 1«5.19 i A regular b.c.s. E is said to be reflexive if E =(E ) bornologically. 31
Definition 1.5.20 J If E is a b.c.s., then the Von - Neumann homology of (T" (E, E'^) (respectively
(3 (E, E )) is called the weak (respectively strong) homology of E where (T and p denote the weak and strong topologies respectively.
Remark 1.5.? : ([2], p. 48). The weak homo logy of a regular b.c.s. is precisely the Von-Neumann homology of the l.c.s. tg •
Definition 1.5,21 : Let E be a b.c.s, and
E its dual. A sequence (x ) in E is said to converge weakly to a point x in E if f (x ) - f (x) for all f e B^ .
1.6 BORMOLOGy Oi^" THE SPACES P AND T .
M. D, Fatwardhan ([l8]) definei a homology on ^
and \ as follows.
The collection of all subsets of \ (respectively
r ) of the form (f j |jf|| ir } , r real, forms 32
a base for a bornology B (respectively IB ) on i (respectively on l )• She proved the following results.
Theorem 1.6.1 ([18])» IB contains no bornivorous set.
Corollary 1.6.2 ([18]) « The M-convergence of \ is not topologisable.
Theorem 1.6.2 i ([l8])« A linear functional f B I c 3C on \ is bounded if and only if n=o » lim |c^| » 0 i.e. f c \ , n - ~
Remark 1.6.1 : The statement of the above theorem 1.6.2 is presented here as given by M, D,
Patwardhan [l8]. However it does not follow from this statement that ^ » n''^ (see definition of P )•
The theorem: sb-oull rsad as : A linear functional on
r is bounded if and only if it is of the form » 1/n Y Z c^ X where lim |c„j =0 so that p"^ «p. n=0 " n -*« ' 33
We give a proof of this, in a more general form later
(See Theorem 2,1.2).
Theorem 1.6.3 : IB = U ^r. » where IB reIR is the Vdn Neumann bornology of the normed space i (t ) Theorem 1.6.4 t ( \ » ^ ) is the completion of (f, IB ).
if a-,(x) -• 0 uniformly in some finite circle.
We remark here that there is no corresponding result for the topology of \ (see Theorem 1,2.8).
Theorem 1.6.6 : In ( I , TB ) Mackey conver gence and weak convergence of sequences coincide*
Theorem 1.6.7 : ( \ , IB ) is reflexive.
By T( oo -• <» ) we denote a bounded linear transformation of \ into \ and by T(r^ -* ^2) we denote a bounded linear transformation of J (r,) into r (^2)* Similarly we denote by F(rj^ - r2) the collection of all T(r^ - r2) and by F(oo - «) the collection of all T(<» - «>) , 3A
F (oo ^ «) =, n U F (r -* r,) •
Theorem 1.6«9 : A necessary and sufficient condition that there exists a T = T(«» - «>) with
T(x") » Ojj , n - 0, 1, 2, •.. is that for every r > 0, there exists R > 0 such that the sequence
{ Ujj : 1/2R i 2*^ r'^ } is bounded.
Theorem 1.6.10 : A bounded linear functional F/0 on (lp, IB ) is a bounded scalar homomo- rphism with respect to Hadamard multiplication if and only if F • x'^ for some n»
Theorem 1.6.11 : With every boyinded homomorphism T of \ (C) into itself it we can associate a unique sequence {H } ,na:0, 1, 2, •.. of disjoint finite (l^ossibly empty) subsets of the set I of non-negative integers having the following properties t
(1) Tix^) « Z x^ , the right hand side peH^ being interpreted as zero ndien H is empty* 55
(2) If q^ is the smallest integer in H^, then writing a • 0 vrtienever H is empty, we have n - O(q^) , qn /^ 0.
Under these conditions T(a) « I a^ T(x°) for all a - I a^ x^ in \ (C).
Conversely, if the disjoint finite (possibly empty) subset of I be preassigned satisfying (2), TCX'*) and T(a) be defined as above, then T is bounded homomorphism of r'(C) into itself.
Theorem 1.6.12 t With the same notation as in Theorem 1,6.11 , we have
(1) T maps r (C) onto T( f (C)) in a one- one manner if and only if no H is empty.
(2) If in addition q^ « 0(n) , then T"^ is a bounded linear homomorphism of T( \ (C)) onto
f (C) , i,e, T is an automorphism of y (C) onto T(r (O),