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Ferrando and Lüdkovsky (J. Math. Anal. Appl. 274 (2002) 577–585), have investigated some structral properties of the function space for a Hausdorff locally convex space X. In this work, we are mainly interested in the space of all unordered absolute summable functions from a set A into a Hausdorff locally convex space X. The main result of the work is the representation of the elements of and These representations are related with the separability of the spaces and provide us to obtain continuous duals of and for a normed space This improves the Ferrando and Lüdkovsky's investigation with geometric aspects.

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... These spaces are important generalizations of the classical Banach spaces c 0 , 1 and ∞ , and they have no Schauder bases in general. The problem is solved by a representation of elements of 1 (A, X) and c 0 (A, X) given in [8]. ...

... An analogous result for the X-valued sequence space c 0 (X) = c 0 (N, X) was obtained earlier by Mendoza [6]. In [8], for a locally convex space X, we dealt with the structural properties of the spaces 1 (A, X) and c 0 (A, X) where 1 (A, X) is defined in the next section. A basic result in [8] was a lemma introducing a representation for elements of these spaces, similar to those of spaces possesing a basis. ...

... In [8], for a locally convex space X, we dealt with the structural properties of the spaces 1 (A, X) and c 0 (A, X) where 1 (A, X) is defined in the next section. A basic result in [8] was a lemma introducing a representation for elements of these spaces, similar to those of spaces possesing a basis. Hence, we can easily study the separability of the spaces and linear functionals on them. ...

Continuous linear operators from '1 (A,X) and c0 (A,X) into (A,X), = '1, '1 or c0, for a normed space X are investigated. It is shown that such an operator has an operator matrix form whenever A is the set of positive integers.

... An analogous result for the X-valued sequence space c 0 (X) = c 0 (N; X) was obtained earlier by J. Mendoza [10]. In [16], for a locally convex space X, we deal with the function spaces`1spaces`spaces`1 (A; X) and c 0 (A; X) and study some geometric and structural properties such as the separability and linear functionals on them. This work may also be considered a contribution to the e¤orts on the structural investigation of these vector-valued function spaces. ...

... Indeed, it is shown in [16] that this net converges to x. Uniqueness is similar to that of previous result. Further, each R is continuous. ...

We give, in this work, a new basis definition for Banach spaces and investigate some structural properties of certain vector-valued function spaces by using it. By novelty of the new definition, we prove that ℓ∞ has a basis in this sense, and so we deduce as a result that it has approximation property. In fact, we obtain a more general result that the linear subspace P(B,X) of ℓ∞(B,X) of all those functions with a precompact range has an X-Schauder basis. Hence P(A,X) has approximation property if and only if the Banach space X has. Note that P(B,X)=ℓ∞(B,X) for some finite-dimensional X. Further, we give a representation theorem to operators on certain vector-valued function spaces.

... let a family = {( , ): ∈ } be given, (see Definition(11). Then is a -biorthogonal system for if and only if { : ∈ } is -basis for with { } being . . . to { : ...

... An analogous result for the Xvalued sequence space c 0 (X) = c 0 (N, X) was obtained earlier by J. Mendoza [11]. In [16], for a locally convex space X, we deal with the function spaces 1 (A, X) and c 0 (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 is to characterize certain operator spaces acting on some important vector-valued function spaces such as (V a)a∈Ac0 , by introducing a new kind basis notion for general Topological vector spaces. Where A is an infinite set, each Va is a Banach space and (Va) a∈Ac0 is the linear space of all functions x:A → ∪Va such that, for each ε > 0, the set {a ∈ A : ∥xa∥ > ε} is finite or empty. This is especially important for the vector-valued sequence spaces (V i)i∈Nc0 because of its fundamental place in the theory of the operator spaces (see, for example, [12]).

In this work, we introduce the notion of relative adjoint operators and characterize some operator spaces by this notion and by the results presented in [Y. Yilmaz, Structural properties of some function spaces, Nonlinear Anal. 59 (2004) 959–971]. Hence, for example, we prove that the operator space L(ℓ∞(A,X),c0(A,Z)) is equivalent to c0(A,LSOT(ℓ∞(A,X),Z)) in the sense of isometric isomorphism, where A is an infinite set, X, Z are Banach spaces and LSOT(X,Z) is the space L(X,Z) endowed with the strong operator topology. Note that the vector-valued function spaces ℓ∞(A,X) and c0(A,Z), defined in the prerequisites, are important generalizations of the classical Banach spaces ℓ∞and c0.

We provide a representation of elements of the space for a locally convex space and and determine
its continuous dual for normed space and . In particular, we study the extension and characterization of isometries on space, when is a normed space with an unconditional basis and with a
symmetric norm. In addition, we give a simple proof of the main result
of G. Ding (2002).

Assuming Sigma is an algebra of subsets of a non-empty set Omega and X is a normed space, I investigate whether or not certain barrelledness conditions, some of them introduced in the seventies by Saxon and Valdivia, are enjoyed by several subspaces of the linear space of all those bounded X-valued functions defined on Omega which are the uniform limit of a sequence of X-valued Sigma-simple functions equipped with the supemum-norm.

If Ω is a set, Σ a σ-algebra of subsets of Ω, and X a normed space, we show that the space ℓ ∞ (Σ,X) of all bounded X-valued Σ-measurable functions defined on Ω, provided with the supremum-norm, is barrelled if and only if X is barrelled. Assuming X separable, this implies that the space ℓ ∞ (Ω,X) of all bounded X-valued functions defined on Ω, endowed with the supremum-norm, is barrelled whenever X is barrelled.

Assuming that Ω is a nonempty set and X is a normed space, we show that the real or complex linear space c0(Ω,X) of all functions f:Ω→X such that for each ϵ>0 the set {ω∈Ω:‖f(ω)‖>ϵ} is finite, equipped with the supremum norm, is either barrelled, ultrabornological or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological or unordered Baire-like.