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# Structural properties of some function spaces

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## Abstract

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. ...
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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. ...
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... 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. ...
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