scholarly journals Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

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.

scholarly journals Duals of vector-valued Köthe function spaces

1992 ◽  
Vol 112 (1) ◽  
pp. 165-174 ◽  
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.

scholarly journals Compactness in Vector-valued Banach Function Spaces

Positivity ◽  
2007 ◽  
Vol 11 (3) ◽  
pp. 461-467 ◽  
Author(s):  
Jan van Neerven
Keyword(s):  
Function Spaces ◽  
Banach Function Spaces ◽  
Vector Valued

scholarly journals On Generalized Morse-Transue Function Spaces

1959 ◽  
Vol 11 ◽  
pp. 416-426
Author(s):  
H. W. Ellis
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Length Function ◽  
Integral Type ◽  
Banach Function Spaces ◽  
Integrable Functions ◽  
Bounded Functions

Marston Morse and William Transue (6, 8) have introduced and studied function spaces, called MT-spaces, for which the elements of the topological dual are of integral type. Their theory does not admit certain classical Banach function spaces including spaces of bounded functions and spaces. The theory of function spaces determined by a length function (λ-spaces) (4, 5), which depends on a fixed measure, admits many of the maximal MT-spaces, the spaces and spaces of locally integrable functions but does not admit certain maximal MT-spaces including the space of complex continuous functions with compact supports.

scholarly journals Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions

2010 ◽  
Vol 53 (3) ◽  
pp. 466-474 ◽  
Author(s):  
Luis Dubarbie
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Dimensional Case ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Disjointness Preserving ◽  
Vector Valued

AbstractIn this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

scholarly journals The Bishop-Phelps-Bollobàs Property for Compact Operators

2018 ◽  
Vol 70 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Sheldon Dantas ◽  
Domingo García ◽  
Manuel Maestre ◽  
Miguel Martín
Keyword(s):  
Hilbert Space ◽  
Topological Space ◽  
Banach Spaces ◽  
Positive Measure ◽  
Compact Operators ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nikodym Property

AbstractWe study the Bishop-Phelps-Bollobàs property (BPBp) for compact operators. We present some abstract techniques that allow us to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X and Y be Banach spaces. If (c0, Y) has the BPBp for compact operators, then so do (C0(L), Y) for every locally compactHausdorò topological space L and (X, Y) whenever X* is isometrically isomorphic to . If X* has the Radon-Nikodým property and (X), Y) has the BPBp for compact operators, then so does (L1(μ, X), Y) for every positive measure μ; as a consequence, (L1(μ, X), Y) has the BPBp for compact operators when X and Y are finite-dimensional or Y is a Hilbert space and X = c0 or X = Lp(v) for any positive measure v and 1 < p < ∞. For , if (X, (Y)) has the BPBp for compact operators, then so does (X, Lp(μ, Y)) for every positive measure μ such that L1(μ) is infinite-dimensional. If (X, Y) has the BPBp for compact operators, then so do (X, L∞(μ, Y)) for every σ-finite positive measure μ and (X, C(K, Y)) for every compact Hausdorff topological space K.

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