The space of vector-valued integrable functions under certain locally convex topologies

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

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Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

Journal of Function Spaces ◽

10.1155/2016/3763649 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8

Author(s):

P. Rueda ◽

E. A. Sánchez Pérez

Keyword(s):

Type Theorem ◽

Classical Case ◽

Function Spaces ◽

Dual Systems ◽

Banach Function Spaces ◽

Infinite Dimensional ◽

Local Compactness ◽

Finite Dimensional ◽

Integrable Functions ◽

Vector Valued

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

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Duals of vector-valued Köthe function spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070845 ◽

1992 ◽

Vol 112 (1) ◽

pp. 165-174 ◽

Cited By ~ 2

Author(s):

Miguel Florencio ◽

Pedro J. Paúl ◽

Carmen Sáez

Keyword(s):

Function Space ◽

Normed Space ◽

Function Spaces ◽

Integrable Functions ◽

Nikodym Property ◽

Köthe Dual ◽

Vector Valued

AbstractLet Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

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Operators on locally convex spaces of vector-valued continuous functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700026538 ◽

1987 ◽

Vol 36 (2) ◽

pp. 267-278

Author(s):

A. García López

Keyword(s):

Hausdorff Space ◽

Continuous Operator ◽

Continuous Functions ◽

Special Treatment ◽

Locally Convex Spaces ◽

Uniform Topology ◽

Zero Sequence ◽

Vector Valued ◽

Locally Convex ◽

Convex Spaces

Let E and F be locally convex spaces and let K be a compact Hausdorff space. C(K,E) is the space of all E-valued continuous functions defined on K, endowed with the uniform topology.Starting from the well-known fact that every linear continuous operator T from C(K,E) to F can be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.As a consequence we characterise the spaces E for which an operator T defined on C(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) in C(K,E).

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The strict topology on a space of vector-valued functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500027784 ◽

1979 ◽

Vol 22 (1) ◽

pp. 35-41 ◽

Cited By ~ 13

Author(s):

Liaqat Ali Khan

Keyword(s):

Scalar Field ◽

Topological Space ◽

Vector Space ◽

Convex Space ◽

Locally Convex Space ◽

Strict Topology ◽

Locally Compact ◽

Vector Valued Functions ◽

Vector Valued ◽

Locally Convex

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

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Multipliers of Modules of Continuous Vector-Valued Functions

Abstract and Applied Analysis ◽

10.1155/2014/397376 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Liaqat Ali Khan ◽

Saud M. Alsulami

Keyword(s):

Compact Hausdorff Space ◽

Hausdorff Space ◽

Normed Spaces ◽

Locally Compact ◽

Continuous Vector ◽

Vector Valued Functions ◽

Vector Valued ◽

Locally Compact Hausdorff Space ◽

Locally Convex

In 1961, Wang showed that ifAis the commutativeC*-algebraC0(X)withXa locally compact Hausdorff space, thenM(C0(X))≅Cb(X). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing thatHomC0(X,A)(C0(X,E),C0(X,F))≃Cs,b(X,HomA(E,F)),whereEandFarep-normed spaces which are also essential isometric leftA-modules withAbeing a certain commutativeF-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

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