Representation Theorems for Operators on Free Banach Spaces of Countable Type

2019 ◽  
Vol 11 (1) ◽  
pp. 21-36 ◽  
Author(s):  
J. Aguayo ◽  
M. Nova ◽  
J. Ojeda
Keyword(s):  
Banach Spaces ◽  
Representation Theorems ◽  
Countable Type

scholarly journals Order preserving functions on ordered topological vector spaces

1999 ◽  
Vol 60 (1) ◽  
pp. 55-65 ◽  
Author(s):  
J.C. Candeal ◽  
E. Induráin ◽  
G.B. Mehta
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Reflexive Banach Spaces ◽  
Representation Theorems ◽  
Mackey Topology ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Total Preorder

In this paper we prove the existence of continuous order preserving functions on ordered topological vector spaces in an infinite-dimensional setting. In a certain class of topological vector spaces we prove the existence of topologies for which every continuous total preorder has a continuous order preserving representation and show that the Mackey topology is the finest topology with this property. We also prove similar representation theorems for reflexive Banach spaces and for Banach spaces that may not have a pre-dual.

scholarly journals AN ELEMENTARY PROOF OF JAMES’ CHARACTERIZATION OF WEAK COMPACTNESS

2011 ◽  
Vol 84 (1) ◽  
pp. 98-102 ◽  
Author(s):  
WARREN B. MOORS
Keyword(s):  
Integral Representation ◽  
Banach Spaces ◽  
Elementary Proof ◽  
Weak Compactness ◽  
Representation Theorems

AbstractIn this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.

scholarly journals The Invariant Subspace Problem for Non-Archimedean Banach Spaces

2008 ◽  
Vol 51 (4) ◽  
pp. 604-617 ◽  
Author(s):  
Wiesław Śliwa
Keyword(s):  
Banach Space ◽  
Invariant Subspace ◽  
Banach Spaces ◽  
Continuous Operator ◽  
Linear Continuous Operator ◽  
Infinite Dimensional ◽  
Countable Type ◽  
Closed Invariant Subspace ◽  
Subspace Problem ◽  
Invariant Subspace Problem

AbstractIt is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

scholarly journals Non-Archimedean t-Frames and FM-Spaces

1992 ◽  
Vol 35 (4) ◽  
pp. 475-483 ◽  
Author(s):  
N. De Grande-De Kimpe ◽  
C. Perez-Garcia ◽  
W. H. Schikhof
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Type Theorem ◽  
Countable Type ◽  
Nuclear Spaces

AbstractWe generalize the notion of t-orthogonality in p-adic Banach spaces by introducing t-frames (§2). This we use to prove that a Fréchet-Montel (FM-)space is of countable type (Theorem 3.1), the non-archimedeancounterpart of a well known theorem in functional analysis over ℝ or ℂ ([6], p. 231). We obtain several characterizations of FM-spaces (Theorem 3.3) and characterize the nuclear spaces among them (§4).

New Examples of Non-Archimedean Banach Spaces and Applications

2012 ◽  
Vol 55 (4) ◽  
pp. 821-829 ◽  
Author(s):  
C. Perez-Garcia ◽  
W. H. Schikhof
Keyword(s):  
Banach Spaces ◽  
Supremum Norm ◽  
Valued Fields ◽  
Orthogonal Bases ◽  
Valued Field ◽  
Countable Type

AbstractThe study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have t-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm ║ · ║ on c0, equivalent to the canonical supremum norm, without non-zero vectors that are ║ · ║-orthogonal and such that there is a multiplication on c0 making (c0, ║ · ║) into a valued field.

GENERALIZED COMONOTONICALLY ADDITIVE OPERATORS: REPRESENTATIONS BY CHOQUET INTEGRALS

2002 ◽  
Vol 10 (04) ◽  
pp. 329-346
Author(s):  
HEINZ J. SKALA
Keyword(s):  
Artificial Intelligence ◽  
Decision Theory ◽  
Banach Spaces ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Integral Representations ◽  
Representation Theorems ◽  
Projection Property ◽  
Choquet Integrals

Operators which behave (sub-, super-) additive on comonotonic functions occur quite naturally in many contexts, e.g. in decision theory, artificial intelligence, and fuzzy set theory. In the present paper we define comonotonicity for Riesz spaces with the principal projection property and obtain integral representations (in terms of Bochner integrals) for comonotonically additive operators acting on Riesz with the principal projection property and taking values in certain Riesz- or Banach spaces. As easy corollaries we obtain essential generalizations of representation theorems à la schmeidler, Proc. Am. Math. Soc. 97 (1986), 255 – 261. The existence of the necessary convergence theorems makes it possible to extend our results to set-valued operators. This is the topic of a further paper.

Sign in / Sign up

Export Citation Format

Share Document