FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz

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1 Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as (V a) c 0, by introducing a new kind basis notion for general Topological vector spaces. Where A is an infinite set, each V a S is a Banach space and (V a) c 0 is the linear space of all functions x: A V a such that, for each ε > 0, the set {a A : x a > ε} is finite or empty. This is especially important for the vector-valued sequence spaces (V i ) i N because of its fundamental place in the theory of the operator spaces (see, for example, [12]). 1. Introduction The absence of a basis notion in vector-valued function spaces causes many deficiencies in the structural and geometric characterizations of the spaces. In those spaces having infinite dimension some important topics such as the approximation property, separability, representation of operators and every topics in which the bases are used cannot be investigated sufficiently. Especially, in the representations of the spaces of continuous linear operators on some Banach spaces bases play a crucial role. For example, if Y is an arbitrary Banach space, then the operator space L(l 1, Y ) is equivalent, by the mapping u {u(e n )}, 2000 Mathematics Subject Classification. Primary 46A35, 46B15, 47B38; Secondary 46B28, 46B45, 46E28, 42C05. Key words and phrases. Biorthogonal systems, Schauder bases, generalization of bases, operators on function spaces, vector-valued function spaces, representation of operators. c 2009 Juliusz Schauder Center for Nonlinear Studies 335

2 336 Y. Yılmaz to the space l (Y ) = { } {y n } Y : y n <, 1 n< endowed with the norm {y n } = 1 n< y n where {e n } is the unit vector basis of l 1. Further, for example, a continuous linear operator T from l into l 1 is equivalent, by the mapping T {T (f n )}, to a sequence {y n } ba(n, ) = l such that n y n is weakly unconditionally Cauchy, [13, pp. 167], where {f n } is the sequence of associate coordinate functionals to the unit vector basis {e n } of l 1 and T is the adjoint operator of T. A reasonable questions to ask at this point is what is the extension of this results to the vector-valued function spaces l 1 (A, X) and (A, X)? More generally, what is the characterizations of the operators between the Banach function spaces λ(a, X)? Where λ is the any of the spaces l, or l p, p 1, A is an infinite set, X is a Banach space and the definitions of the spaces λ(a, X) are given in the next section. We shall seek an answer to this questions for more general cases. The answer contains many difficulties since the vector-valued function spaces such as l p (A, X) and (A, X) doesn t have a basis in general. Such questions on the structural and geometric properties of these spaces lead us to seek an extended definition of the classical basis notion. Geometric and other structural properties of some vector-valued function spaces has been studied intensively in the last twenty years. Especially, the space of all bounded X-valued functions has many literature. Some important references on l (A, X) are [2] [5], [7]. In general, these spaces have derived properties from X. For example, there are many examples, [3], [5], of normed spaces of l (A, X) which are barrelled whenever X is barrelled. Also, unordered Baire-like properties of these spaces and of its subspaces also depend on the space X (see [4] and [7]). Further, Ferrando and Lüdkovsky, [6], have considered the function space (A, X) which is an important extension of the classical Banach space. They proved again that (A, X) is either barreled, ultrabornological or unordered baire-like if and only if X is, respectively. An analogous result for the X- valued sequence space (X) = (N, X) was obtained earlier by J. Mendoza [11]. In [16], for a locally convex space X, we deal with the function spaces l 1 (A, X) and (A, X) and study some geometric and structural properties such as the separability and linear functionals on them. Furthermore, in [17], we characterized the continuous operators from an arbitrary Banach spaces to the any of these spaces by introducing a new notation as relative adjoint operators. Our main interest in this work to give a systematic procedure on the structural investigation of some improved forms of mentioned function spaces and to characterize certain operators between them by means of the new basis notion.

3 Function Bases Prerequisites We use the notations N, C and R for the sets of all positive integers, complex numbers and real numbers, respectively. For normed spaces X and Y over C, L(X, Y ) denotes the space of all continuous linear operators from X into Y while L(X) = L(X, X) and X denotes the continuous dual of X. We shall write L SOT (X, Y ) whenever we consider L(X, Y ) with the strong operator topology instead of uniform (norm) topology. Further, we denote by B X and S X the closed unit ball and sphere of X, respectively. Let us now introduce the notion of (unordered) infinite sums in a topological vector space (TVS, for short). Let A be an infinite set, {x a : a A} be a family of vectors in a topological vector space (TVS) X and let F denote the family of all finite subsets of A. F is directed by the inclusion relation and, for each F F, we can form the finite sum S F = x a. If the net (S F : F) converges to some x in normed space X, then we say that the family {x a : a A} is summable, or that the sum x a exists, and we write x a = x in X. We mean by the convergence of the net (S F : F) to x in X that, for each ε > 0, there exists an F 0 = F 0 (ε) F such that S F x < ε whenever F 0 F. This also implies that the net (S F : F) is bounded. Note that, some convergent nets may not be bounded in general. The definition of summable family does not involve any ordering of the index set A, and we may therefore say that the notion of a sum thus defined is commutative (unconditional). In case A = N, that the family {x n : n N} is summable to x is equivalent to the series n N x n is unconditionally convergent to x. Note that, a series n N x n in a TVS is said to be unconditionally convergent if and only if for each permutation σ of N, n N x σ(n) is convergent. The definition of a convergent series, essentially, involves the order structure of N. If the series n N x n is convergent, and if σ is a permutation of N, then the series n N x σ(n) may not be convergent, that is, {x n : n N} may not be summable [1, p. 270]. Hence summability of the family {x n : n N} in X is stronger than the convergence of the series n N x n in this sense. Let a family {V a : a A} of topological spaces be given. The product V a is the set of all functions x: A V a such that x(a) V a, for each a A. Usually, we use the notation x a for x(a) since it is more convenient. By V A we mean V a with V a = V for each a A. Let us define P a : V a V a by P a (x) = x a. This is called the projection on the a-th factor, or briefly, is called a-th-projection. Let each V a be a normed space over C. Some important linear subspaces of V a is given as follows; The space (V a ) l is the linear space of all functions x: A V a such that { x a : a A} <.

4 338 Y. Yılmaz Moreover, (V a ) l is a normed space with the norm x = { x a : a A}. We denote by (V a ) l p, 1 p < the set of all functions x: A V a such that { x a p : a A} is summable, i.e. x a p <, and it is a normed space with the norm ( ) 1/p x p = x a p. Also, the function space (V a ) is the linear space of all functions x: A V a such that, for each ε > 0, the set {a A : x a > ε} is finite or empty. It is a normed space with the norm. A special case of (V a ) λ (λ = l, or l p ) is λ(a, X) corresponding V a = X for each a A. These function spaces are Banach spaces if and only if each V a is a Banach space. Moreover, if A is a directed set then (V a ) λ is a linear space of all nets with corresponding property. They are usually denoted by λ(x) in the case A = N and V a = X for each a A and are called X-valued sequence spaces. 3. Function bases One of the our main results is the following definition. It brings a new point of view to the basis notion. Definition 3.1. Let (X, T ) be a TVS, A be a set and {V a : a A} be a family of TVSs. A family {η a : a A} of continuous linear functions η a : V a X is called an unordered (or unconditional) function basis for X with respect to the family {V a }, or briefly, is called a uf-basis for X (whenever {V a } and A are clear from context) if the following condition is satisfied. There exists a unique family {R a : a A} of linear functions R a from X onto V a such that for each x X the family {(η a R a )(x) : a A} is summable, that is, the associate finite sums net (π F (x) : F) converges to x in X. Where, for each F F, π F (x) = (η a R a )(x), and F is the directed family (by the inclusion relation ) of all finite subsets of the index set A. Furthermore, {η a } is called a uf-schauder basis for X whenever each R a is continuous.

5 Function Bases 339 Remark 3.2. In the special case A = N, convergence of the net (π F (x) : F) to x corresponds to the unconditional convergence of the series n=1 (η n R n )(x) to x. In this case, summability of {(η n R n )(x) : n N} is independent of the usual ordered structure of N. If there exists a unique sequence {R n } such that the series n=1 (η n R n )(x) converges to x (in the usual manner) then, by throwing the letter u, we say that {η n } is an f-basis. In this case the convergence of the series may not be unconditional. Remark 3.3. In Definition 3.1, the word uf-basis for X is replaced by the word up-basis whenever X is a vector subspace of the product space V a. Also, we use the word up-schauder basis whenever each R a is continuous. Definition 3.4. The family {R a : a A} is called associate family of functions (A.F.F.) to the uf-basis {η a : a A}. Let {η a : a A} be a uf-basis for X with respect to the family {V a }. Clearly, the finite summation π F (x) defines an operator π F on X. This operator is called F -projection on X and it is continuous whenever {η a } is a uf-schauder basis. Theorem 3.5. Let A be a set and each V a be a normed space. Then (V a ) and (V a ) l p has up-schauder bases for every choice of the family {V a : a A}. Proof. Define I a : V a (V a ) ρ, ρ = or l p, by I a (t) = y such that y(δ) = t if δ = a otherwise y(δ) = 0. Then the family {I a : a A} is an up-schauder basis for (V a ) ρ. We prove the assertion only for (V a ) since the proof for (V a ) l p almost is the same. Choose the family {R a : a A} as R a : (V a ) V a ; R a (x) = P a (x) = x a. Then, we must show that the finite sums net (π F (x) : F) converges to x in (V a ) where π F (x) = (I a P a )(x) = I a (x a ). But, for some x (V a ), π F (x) is the function x F such that x F (a) = x a if a F and x F (a) = 0 if a / F. Now, consider an arbitrary ε > 0. We must find a finite subset F 0 = F 0 (ε) F such that, for each finite F F 0, x x F ε. Since the set {a A : x a > ε} is finite or empty, taking F 0 as F 0 = {a A : x a > ε},

6 340 Y. Yılmaz we have x x F = {x a : a A F } ε, for each finite F F 0. For the uniqueness of the family {P a } pose a P a )(x) = (I (I a R a)(x) and π F (x) = (I a (P a R a))(x), F F. Remember that π F (x) = (P a R a)(x) and π F (x) π G (x) for F G. Since lim F F π F (x) = 0, we have that (P a R a)(x) = 0 for each a and for every x (V a ). This implies, P a = R a for each a. Further, each P a is continuous because x a x. Remark 3.6. Let X be a TVS over the field C possessing a basis {x n } (in the classical manner). Then the sequence {η n } of the functions η n : C X : η n (t) = tx n is an f-basis for X with respect to (the family) C. Indeed; taking {R n } as the sequence of associate coordinate functionals (f n ) to the basis {x n }, we obtain that (η n R n )(x) = f n (x)x n, n=1 and so, it converges to x for each x X. Moreover, {η n } is a uf-basis for X if and only if {x n } is an unconditional basis of X. It is classical that a TVS having a basis is separable. An evolution of this result is: Theorem 3.7. Let X be a TVS for which a family {η a : a A} be an ufbasis with respect to the family {V a } of separable TVSs. Then, X is separable if A is countable. Proof. Let A be countable and W a be countable dense subspace of V a for each a. Then η a(w a ) is countable dense subspace of X since the image of countable sets under a function is also countable, and a union of a countable family of countable sets is countable [8]. Furthermore, for some x X, there exists a net {x a δ } δ in W a such that x a δ R a(x) in V a. So n=1 η a (x a δ) (η a R a )(x)

7 Function Bases 341 in X for each a. This implies a (x η a δ) (η a R a )(x) = π F (x) for each F F. Hence each neighbourhood U πf (x) of π F (x) includes an element η a(x a δ ) of η a(w a ). On the other hand, each neighbourhood U x of x includes a neighbourhood U πf0 (x) of some π F0 (x) since the net (π F (x), F) converges to x in X. Consequently, U x includes an element of η a(w a ). This shows that η a(w a ) is dense in X. Another important attribute of classical bases is the fact that a TVS having a basis is considered as a sequence spaces. The analogue of this conjecture in the new definition is that if {η a : a A} is a uf-basis for X with respect to the family {V a } then we consider X to be a subspace of V a. This is the content of the following theorem. Theorem 3.8. Let X be a TVS and {η a : a A} be a uf-basis for X with respect to the family {V a } of TVSs. Then, X is linearly homeomorphic with a subspace Y of V a such that {I a } is a up-basis for Y (with the family {P a } of a-th-projections as the A.F.F. to {I a }). Where I a is defined by I a (t) = ω such that { Ra (x) for some x X, if a = b, ω(b) = 0 otherwise, and {R a } is the A.F.F. to the uf-basis {η a }. Further {η a } is Schauder basis if and only if {I a } is. Proof. Let Y = { y } V a : y a = R a (x) for some x X, and define T : X Y by T x = y. Clearly T is a linear isomorphism. Further, by imposing on Y the quotient topology QT by the operator T, we identified X with Y as topologically since T is continuous in QT. Moreover, y = T x = (T η a R a )(x) = (T η a )(R a (x)) = (T η a )(y a ) = (T η a P a )(y) = (I a P a )(y) where P a is the a-th-projection on the product space V a and I a = T η a. Hence {I a } is a up-basis. Further, P a is continuous from (Y, QT ) to V a if and only if R a is continuous from X to V a. An outstanding contribution of the classical bases to the structural investigation of Banach spaces is the approximation property (AP, for short). Note

8 342 Y. Yılmaz that, a Banach space X is said to have AP if, for every compact set K in X and every ε > 0, there exists an operator T : X X of finite rank (i.e. T (x) = n i=1 x i (x)x i, for some {x i } n i=1 X and {x i }n i=1 X ) so that T (x) x ε, for every x K. It is classical that a Banach space having a basis also has the AP. However, this is directly depend on the family {V a } here. Theorem 3.9. Let X be a Banach space for which {η a : a A} be a uf- Schauder basis with respect to the family {V a } of Banach spaces. Then X has the approximation property if and only if each V a has. Proof. It follows from the above theorem that we may consider X as to be a subspace of V a with the up-basis {I a } and with the A.F.F. {P a }. Let X has the AP and fix some a A. Suppose some K V a is compact and let ε > 0. Since I a (K) is also compact in X, there exists a finite rank operator T on X such that for every y K. This implies T (I a (y)) I a (y) ε (P a T I a )(y) y = (P a T I a )(y) (P a I a )(y) P a T (I a (y)) I a (y) ε. Hence P a T I a is the desired finite rank operator (corresponding K and ε). Conversely, let each V a has the AP. It isn t hard to see that any finite direct sum of Banach spaces has the AP if and only if each component space has this property. Let K X be compact and ε > 0. There exists an F 0 (ε) F, such that π F (x) x ε/2, for each finite F F 0, since the net (π F (x), F) converges to x in X. Further, for each F F, π F (K) is a compact subset of V a and so there exists a finite rank operator T on V a such that T (π F (x)) π F (x) ε/2, for every x K. Thus, for each finite F F 0, (T π F )(x) x = T (π F (x)) π F (x) + π F (x) x T (π F (x)) π F (x) + π F (x) x ε 2 + ε 2 = ε. This implies, each T π F such that F F 0 is the desired finite rank operator. Now, let us introduce the notion f-biorthogonal systems in Banach spaces. Definition Let {V a : a A} be a family of Banach spaces, X be an arbitrary Banach space, {η a : a A} L(V a, X) and {R a : a A}

9 Function Bases 343 L(X, V a). If R a η b = I Va (identity of V a ) whenever a = b, and R a η b = 0 (zero operator from V b to V a ) whenever a b, then the family {(η a, R a ) : a A} is called an f-biorthogonal system for X with respect to the family {V a }, or briefly, is called an f-biorthogonal system for X. Further, we use the term p-biorthogonal system for X whenever X is a subspace of V a. Proposition Let {η a : a A} be a uf-basis for X. Then, the family {(η a, R a ) : a A} is an f-biorthogonal system for X where {R a } is the A.F.F. to the uf-basis. Proof. For an arbitrary v V b, η b (v) X, and so η b (v) = (η a R a )(η b (v)) = a [(R a η b )(v)]. η Hence, by the uniqueness of the family {R a }, this equality holds if and only if R a η b = I Va for a = b, and R a η b = 0 for a b. Theorem Let X be a Banach space for which {η a } be a uf-basis for X with respect to the family {V a } of Banach spaces. Then x = { π F (x) } F F defines a norm on X, equivalent to the initial norm of X, and for some K > 0. x x K x, x X, The proof is quite similar to the its classical analogue (see [9] and [13]). Definition The number F F π F is called the basis constant of the uf-basis {η a : a A} and a uf-basis whose basis constant is 1 is called monotone. Example Up-bases of (V a ) and (V a ) l p given in Theorem 3.5 are monotone. Let us prove the assertion only for (V a ). Fix some v a S Va and define x a : A V a, for some a A, by { va if b = a, x a (b) = 0 if b a. Then, for some F F containing a, we have π F (x a ) = b F(I b P b )(x a ) = x a.

10 344 Y. Yılmaz Thus, for some x a B (Va), π c F (x a ) 0 = x a. Also, for every x B (Va), π F (x) = (I a P a )(x) (I a P a )(x) = x, whence, we have π F = 1 for each F F. 4. Function bases and operators spaces In this section we deal with Banach spaces and continuous operators between them. The object is to give a general conjecture characterizing certain operator spaces acting on some important vector-valued function spaces such as (V a ) and (V a ) l p. This is especially important for the vector-valued sequence spaces because of its fundamental place in the theory of the operator spaces (see, for example, [12]). Following theorem which is a generalization of the result given in [13, p. 162] is one of the main yields of this work. (V i ) i N Theorem 4.1. Let X be a Banach space for which {η a : a A} be a uf- Schauder basis with respect to the family {V a } of Banach spaces and Y be an arbitrary Banach space. Then L(X, Y ) is topologically isomorphic by the mapping to the subspace Λ = { χ = {χ a } : T {T η a : a A} F F x B X } (χ a R a )(x) of the product space L(V a, Y ), endowed with the norm (4.1) χ = (χ a R a )(x), F F x B X and there exists a constant K such that T χ K T. Remark 4.2. By a topological isomorphism between Banach spaces X and Y we mean a linear mapping Ψ from X onto Y such that there exists some K > 0 with the property x Ψ(x) K x, for every x X. Note that a topological isomorphism is automatically one to one and, sometimes, it is called a topological isomorphism into if it isn t onto. Further, a topological isomorphism is called an equivalence or isometric isomorphism whenever K = 1.

11 Function Bases 345 Proof of Theorem 4.1. Define the linear operator Ψ: L(X, Y ) Λ by Ψ(T ) = χ such that χ a = T η a. Let us first show that the function in (4.1) is well-defined. Pick some χ Λ and say π F (x) = (χ a R a )(x), for F F. It is continuous linear operator on X, whence, (χ a R a )(x) = π F < F F x B X by the uniform boundedness principle. Let χ = 0, then x BX π F (x) = 0 for each F F, in particular, for F = {a} (a A). This implies (χ a R a )(x) = 0 for each a and x B X. This implies χ a = 0 for R a 0, and so χ = 0. Other properties of norm can easily be verified. Let T L(X, Y ) and say χ a = T η a. Then ( ) T x = T (η a R a )(x) = (T η a R a )(x) F F = (χ a R a )(x) = lim F F π F (x) for x X. Thus, F F π F (x) < so that F F π F < by the uniform boundedness principle. This means χ Λ, whence, Ψ is well-defined. Further T π F = (χ a R a )(x) = χ. F F F F x B X On the other hand, for each F F, ( (χ a R a )(x) = T (η a R a )(x)) T (η a R a )(x) = T π F (x) T x K T x by the Theorem 3.12, whence, χ K T. This shows that Ψ is a topological isomorphism into. Now let us prove that Ψ is also surjective. Let χ Λ be arbitrary and write T x = (χ a R a )(x), for x X, We shall show that this formula defines a bounded linear operator from X into Y. Let us first show that it is well-defined. This, obviously, equivalent to the summability of the family {(χ a R a )(x) : a A} to T x, and so is equivalent

12 346 Y. Yılmaz to the convergence of the net (π F (x) : F) to T x for each x X. From the fact that the family {(η a, R a ) : a A} is an f-biorthogonal system, we have π F F 1 (π F (x)) = π F (π F (x)), F, F 1 F. This implies (π F (π F (x)) : F) is a Cauchy net in X, and so it has a limit. For the set of all π F (x) is dense in X, and Y is complete we have (π F (x) : F) converges to T x, whence, T is well-defined. Linearity is obvious and T is continuous since x BX T x χ. Remark 4.3. It is clear from the proofs of Theorem 4.1 and the following Theorem 4.11 that the topological isomorphisms are equivalences whenever the uf-bases are monotone. Corollary 4.4. Let Y be an arbitrary Banach space, 1 < p < and 1/p + 1/q = 1. Then the operator space L(l p (A, X), Y ) is equivalent by the mapping T {T I a : a A} to the Banach space E q ; the space of all χ = {χ a } L(X, Y ) A such that χ = ( ) 1/q g χ a q <. g B Y Proof. Take X = l p (A, X) with the monotone up-basis {I a : a A} in Theorem 4.1. Remember that R a = P a, whence, χ = (χ a P a )(x) F F x B lp(a,x) Y ( = g a P a )(x)) (χ F F x B lp(a,x) g B Y = F F g B Y x B lp(a,x) = F F g B Y (g χ a )(x a ) ( ) 1/q g χ a q = ( ) 1/q g χ a q. g B Y We also use the fact that l p (A, X) = l q (A, X ) in the last two rows. It must be noted that l q (A, L(X, Y )) E q. This is clear since ( ) 1/q ( ) 1/q ( ) 1/q g χ a q g χ a q = χ a q. g B Y g B Y Let us show now that the above inclusion relation may be strict.

13 Function Bases 347 Example 4.5. Let X =, Y = l p and A = N. Take χ = (χ n ) as χ n = (a n ij ): a n nn = 1, a n ij = 0 otherwise, that is, χ = , , Previously, let us prove that χ E q. Recall that each continuous operator from into l p must have an infinite matrix form [10]. Obviously, each χ n L(, l p ). Further, for x = (x n ) l p (N, ) = l p ( ) χ n (x n ) = (0,..., 0, x n n, 0,... ), n N, where each x n = (x k n) k=1. On the other hand, for each g = (g n ) l p = l q and for y B c0, Also, for y = e k, Hence, This implies (g χ n )(y) = g(0,..., 0, y n, 0,... ) = g n y n g n. (g χ n )(x n ) = g n. g χ n = g n. ( ) 1/q ( ) 1/q g χ n q = g n q = 1, g B l p g B n N lq n N that is, χ = (χ n ) E q. However, n N χ n q = since each χ n = 1. This means χ / l q (L(, l p )). Corollary 4.6. Let each V a and Y be Banach spaces. Then L((V a ) l 1, Y ) is equivalent, by the mapping T {T I a : a A}, to the space (L(V a, Y )) l, where (I a ) is the up-basis of (V a ) l 1. Proof. Take X = (V a ) l 1 in Theorem 4.1 with the monotone up-basis (I a ). Hence, we have a R a )(x) = (χ χ a (x a ) = T (I a (x a )), F F for x (V a ) l 1. This implies χ = F F P xa 1 T (I a (x a )) = n=1 T I a = χ a.

14 348 Y. Yılmaz However, L((V a ), Y ) is not equivalent to the space (L(V a, Y )) l 1. The former may contain strictly the latter. Corollary 4.7. Let each V a and Y be Banach spaces. Then the operator space L((V a ), Y ) is equivalent by the mapping T {T I a : a A} to the Banach space E 1 ; the space of all χ = {χ a } L(V a, Y ) such that g χ a < for each g B Y, endowed with the norm χ = g B Y g χ a <. Proof. Taking X = (V a ) in Theorem 4.1 with the monotone up-basis (I a ), one can complete the proof by using the same technique in Corollary 4.4 with the fact [(V a ) ] = (Va ) l 1. Example 4.8. Clearly, (L(V a, Y )) l 1 E 1 in Corollary 4.7 and the inclusion relation may be strict. let A = N, V n = l 1+1/n and Y =. Take χ = (χ n ) as is in Example 4.5, that is, χ = , , Clearly, each χ n L(l 1+1/n, ) and χ n =1. This means χ / (L(l 1+1/n, )) l 1. However, the same discussion as in Example 4.5 shows that g χ n = g n = 1, g B c g B l1 0 whence, χ E 1. n N Following notion was introduced in [17] and was used in the characterization of certain operator spaces. Here, we use it in Theorem 4.11 to give a general conjecture to characterization of operators from an arbitrary Banach space to the another one which has an uf-basis. Definition 4.9. Let X, Y, Z be an arbitrary Banach space and let T L(X, Y ). Then we define Z-adjoint operator of T by n N T Z : L(Y, Z) L(X, Z) : R T Z(R) n=1 such that (T Z(R))(x) = R(T x) for each x X.

15 Function Bases 349 This is reduced to classical definition of adjoint operator whenever Z = K(R or C), the scalar field of both X and Y. Z-adjoint operators satisfies the usual attributes of the adjoint operators. For example TZ is linear continuous and T = TZ [17]. Further, it can be proved also as in the classical case that (λt + βa) Z = λt Z + βa Z and (T A) Z = A ZT Z for all scalars λ, β and for all T, A L(X, Y ). However, there exists important structural differences between adjoint and Z-adjoint operators. Let us now present an example illustrating dimensional differences between these two adjoint types. This example is taken from [17]. Example Take X = Y = Z = in Definition 4.9 and consider the difference operator from into such that x = (x k x k+1 ) k=1 for x = (x k ) k=1. Clearly, is given by the infinite matrix = It is clear that the adjoint is given by = However it is illustrated in [17] that the -adjoint of is the following four-dimensional infinite matrix (a ijkl ) i,j,k,l N, such that 1 if i = k and j = l. a ijkl = 1 if i = k, j = l 1 and l 1, 0 otherwise. Unfortunately, we cannot associate the self-adjointness to new adjoint type in the case Z K, even if X and Y are the same Hilbert space. Now, we are ready to state another fundamental result of this work. Theorem Let X be a Banach space for which {η a : a A} be a ufbasis with respect to the family {V a } of Banach spaces and Y be another arbitrary Banach space. Then L(Y, X) is isomorphic by the mapping T {T V a (R a ) : a A}

16 350 Y. Yılmaz to the subspace { Φ = {ϕ = ϕ a } L(Y, V a ) : {(η a ϕ a )(y) : a A} is summable in X, } for each y Y, endowed with the norm (4.2) ϕ = F F y B Y and there exists a constant K such that (η a ϕ a )(y), T ϕ K T. Proof. As is similar to the proof of Theorem 4.1, we have (4.2) is finite by the uniform boundedness principles, and satisfies the norm conditions. Let T L(Y, X) and say ϕ a = T V a (R a ). Then, for y Y, T y = (η a R a )(T y) = a [R a (T y)] η = [η a (TV a (R a ))](y) = a ϕ a )(y), (η whence, ϕ = {ϕ a : a A} Φ and T = (η a ϕ a )(y) (η a ϕ a )(y) = ϕ. y B Y F F y B Y Further, since π F (x) K x, for some K > 0(x X, F F), ϕ = (η a ϕ a )(y) F F y B Y = [η a (TV a (R a ))](y) = (η a R a )(T y) F F y B Y F F y B Y = π F (T y) K T y K T. F F y B Y y B Y This shows that the mapping T {T V a (R a ) : a A} is a topological isomorphism into. Furthermore, it is not hard to see that for some ϕ Φ, the formula T y = (η a ϕ a )(y) defines an operator T L(Y, X). Thus, the mapping T {TV a (R a )} is a topological isomorphism from L(Y, X) to Φ.

17 Function Bases 351 Corollary Let each V a and Y be Banach spaces. Then L(Y, (V a ) ) is equivalent by the mapping T {T V a (P a ) : a A} to SOT (L(Y, V a )) ; the Banach space of all functions ϕ = {ϕ a } L(Y, V a) such that {ϕ a (y) : a A} (V a ), for each y Y, where (P a ) is the associate projections to the up-basis {I a : a A} of (V a ) ; endowed with the norm ϕ = ϕ a. Proof. Take Y = (V a ) in Theorem 4.11 with monotone up-basis (I a ). The summability of the family {(I a ϕ a )(y) : a A} in (V a ) is equivalent to (I a ϕ a )(y) = ϕ a (y) = ϕ a (y) < F F F F for each y Y. Further, ϕ = (I a ϕ a )(y) = to F F y B Y F F y B Y ϕ a (y) = ϕ a. Corollary L(Y, l p (A, X)) is equivalent, by the mapping T {T V a (P a ) : a A} l p (A, L SOT (Y, X)); the Banach space of all functions ϕ = (ϕ a ) L(Y, X) A which are pointwise absolutely p-summable, that is, such that (ϕ a (y)) l p (A, X) for each y Y, endowed with the norm ( ) 1/p ϕ = ϕ a (y) p where p 1. y B Y Proof. Take X = l 1 (A, X) in Theorem 4.11 with monotone up-basis (I a ). The family {(I a ϕ a )(y) : a A} is summable in l p (A, X) if and only if ϕ a(y) p < for each y Y. Also the norm ϕ = (I a ϕ a )(y) F F y B Y in Theorem 4.11 is reduced to the form ( ) 1/p ϕ = ϕ a (y) p. y B Y

18 352 Y. Yılmaz Example It is clear that l p (A, L(Y, X)) l p (A, L SOT (Y, X)) and the inclusion relation may be strict. Let us show this for p > 1. Take A = N, X = C, Y = l p and ϕ n = f n, n N; the associate coordinate functionals to the unit vectors basis {e n } of l 1. Remember that f n (x) = x n and each f n l p = l q since f n = 1. In this case (I n f n )(y) = y n e n for each y l p. Further {f n } n=1 = f n (y) = P y n = 1, F F y B l1 n F F F y n 1 n F n N whence, {f n } n=1 l p (A, L SOT (Y, X)) = l p (l p, weak ). However, {f n } n=1 / l p (A, L(Y, X)) = l p (l q ) since f n = 1 for each n. A similar example may be given for the case p = 1. Remark The characterizations of L((V a ) l p, Y ) and L(Y, (V a ) l p ), p > 1, also can be derived from Theorems 4.1 and We don t give here all characterizations but only some instances for mentioned function spaces. These theorems provide a fundamental tool in the representations of the operators between some vector-valued function spaces. References [1] N. Bourbaki, General Topology, Reading Mass., Addison Wesley, [2] L. Drewnowski, et al., On the Barelledness of spaces of bounded vector functions, Arch. Math. 63 (1994), [3] J. C. Ferrando, On the barelledness of the vector-valued bounded function space, J. Math. Anal. Appl. 184 (1994), [4], Unordered Baire-like vector-valued function spaces, Bull. Belg. Math. Soc. 2 (1995), [5], On some spaces of the vector-valued bounded functions, Math. Scand. 84 (1999), [6] J. C. Ferrando and S. V. Ludkowsky, Some Barelledness properties of (Ω, X), J. Math. Anal. Appl. 274 (2002), [7] J. C. Ferrando and L. M. Sánchez Ruiz, Barrelledness in l (Ω, X) subspaces, J. Math. Anal. Appl. 303 (2005), [8] J. L. Kelley, General Topology, Van Nostrand, Princeton, N. J., [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces 1, Springer Verlag, Berlin, Heidelberg, New York, [10] I. J. Maddox, Elements of Functional Analysis, second ed., Cambridge Univ. Press, Cambridge, New York, [11] J. Mendoza, A barelledness criterion for C 0 (E), Arch. Math. 40 (1983), [12] H. P. Rosenthal, The complete separable extension property, J. Oper. Theory 43 (2000), no. 2, [13] I. Singer, Bases in Banach Spaces 1, Springer Verlag, Berlin, Heidelberg, New York, [14], Bases in Banach Spaces 2, Springer Verlag, Berlin, Heidelberg, New York, 1981.

19 Function Bases 353 [15] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw Hill, New York, [16] Y. Yilmaz, Structural properties of some function spaces, Nonlinear Anal. 59 (2004), [17], Characterizations of some operator spaces by relative adjoint operators, Nonlinear Anal. 65 (2006), Manuscript received April 4, 2006 Yılmaz Yılmaz Inonu University Department of Mathematics 44280, Malatya, TURKEY address: TMNA : Volume N o 2