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Cx4 AFTER I PRELIMINARIES in This Chapter, We Collect Together, for Convenience, All the Known Concepts and Results Which Are Re 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.
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