Invariants in abstract mapping pairs

2004 ◽  
Vol 76 (3) ◽  
pp. 369-382 ◽  
Author(s):  
Li Ronglu ◽  
Wang Junming
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Function Series ◽  
Important Improvement ◽  
Sequential Evaluation ◽  
Locally Convex

AbstractIn a topological vector space, duality invariant is a very important property, some famous theorems, such as the Mackey-Arens theorem, the Mackey theorem, the Mazur theorem and the Orlicz-Pettis theorem, all show some duality invariants.In this paper we would like to show an important improvement of the invariant results, which are related to sequential evaluation convergence of function series. Especially, a very general invariant result is established for an abstract mapping pair (Φ, B(Φ, X)) consisting of a nonempty set Φ and B(Φ, X) = {f ∈ XΦ: f (Φ) is bounded}, where X is a locally convex space.

The largest family of subsets satisfying sequential-evaluation convergence

2012 ◽  
Vol 49 (3) ◽  
pp. 315-325
Author(s):  
Aihong Chen ◽  
Ronglu Li
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequential Evaluation ◽  
Locally Convex

Suppose X is a locally convex space, Y is a topological vector space and λ(X)βY is the β-dual of some X valued sequence space λ(X). When λ(X) is c0(X) or l∞(X), we have found the largest M ⊂ 2λ(X) for which (Aj) ∈ λ(X)βY if and only if Σ j=1∞Aj(xj) converges uniformly with respect to (xj) in any M ∈ M. Also, a remark is given when λ(X) is lp(X) for 0 < p < + ∞.

scholarly journals Some properties of vector measures taking values in a topological vector space

1987 ◽  
Vol 43 (2) ◽  
pp. 224-230
Author(s):  
Efstathios Giannakoulias
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Necessary Condition ◽  
Vector Measures ◽  
Topological Vector Spaces ◽  
Locally Convex ◽  
Pettis Integrability

AbstractIn this paper we study some properties of vector measures with values in various topological vector spaces. As a matter of fact, we give a necessary condition implying the Pettis integrability of a function f: S → E, where S is a set and E a locally convex space. Furthermore, we prove an iff condition under which (Q, E) has the Pettis property, for an algebra Q and a sequentially complete topological vector space E. An approximating theorem concerning vector measures taking values in a Fréchet space is also given.

scholarly journals On Locally Convex Topological Vector Space Valued Null Function Space c0 Defined by Semi Norm and Orlicz Function

2015 ◽  
Vol 19 (1) ◽  
pp. 62-68
Author(s):  
Narayan Prasad Pahari
Keyword(s):  
Vector Space ◽  
Function Space ◽  
Topological Vector Space ◽  
Convex Space ◽  
Orlicz Function ◽  
Locally Convex Space ◽  
New Class ◽  
And Function ◽  
Locally Convex ◽  
Class Of Functions

The aim of this paper is to introduce and study a new class c0 (S, T, Phi, Xi, u) of locally convex space T- valued functions using Orlicz function Phi as a generalization of some of the well known sequence spaces and function spaces. Besides the investigation pertaining to the structures of the class c0 (S, T, Phi, Xi, u), our primarily interest is to explore some of the containment relations of the class c0 (S, T, Phi, Xi, u) in terms of different Xi and u so that such a class of functions is contained in or equal to another class of similar nature.Journal of Institute of Science and Technology, 2014, 19(1): 62-68

scholarly journals The strict topology on a space of vector-valued functions

1979 ◽  
Vol 22 (1) ◽  
pp. 35-41 ◽  
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.

scholarly journals Functions with bounded variation in locally convex space

2011 ◽  
Vol 49 (1) ◽  
pp. 89-98
Author(s):  
Miloslav Duchoˇn ◽  
Camille Debiève
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Bounded Variation ◽  
Convex Space ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Compact Set ◽  
Locally Convex Spaces ◽  
Vector Valued ◽  
Locally Convex

ABSTRACT The present paper is concerned with some properties of functions with values in locally convex vector space, namely functions having bounded variation and generalizations of some theorems for functions with values in locally convex vector spaces replacing Banach spaces, namely Theorem: If X is a sequentially complete locally convex vector space, then the function x(・) : [a, b] → X having a bounded variation on the interval [a, b] defines a vector-valued measure m on borelian subsets of [a, b] with values in X and with the bounded variation on the borelian subsets of [a, b]; the range of this measure is also a weakly relatively compact set in X. This theorem is an extension of the results from Banach spaces to locally convex spaces.

Generalized Köthe function spaces. I

1969 ◽  
Vol 65 (3) ◽  
pp. 601-611 ◽  
Author(s):  
Nguyen Phuong-Các
Keyword(s):  
Vector Space ◽  
Function Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Special Cases ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Differentiable Vector ◽  
Locally Convex ◽  
Kernel Theorem

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E-valued functions φ(t) such that for all elements e′ of the dual E′ of E. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E. Therefore one usually starts from some function space which has interesting properties.

On majorizing and cone-absolutely summing mappings

1977 ◽  
Vol 24 (2) ◽  
pp. 245-251
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Riesz Space ◽  
Continuous Linear ◽  
Linear Map ◽  
Space Setting ◽  
Locally Convex

AbstractThe notions of majorizing mappings and cone-absolutely summing mappings are studied in the locally convex Riesz space setting. It is shown that a locally convex Riesz space Y is an M-space in the sense of Jameson (1970) if and only if, for any locally convex space E, every continuous linear map from E into Y is majorizing. Another purpose of this note is to study the lattice properties of the vector space ℒl(X, Y) of cone-absolutely summing mappings from one locally convex Riesz space into another Y. It is shown that if Y is both locally and boundedly order complete, then ℒl(X, Y) is an l-ideal in Lb(X, Y). This improves a result of Krengel.

Open decompositions on ordered convex spaces

1973 ◽  
Vol 74 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Positive Cone ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Closed Cone ◽  
Necessary And Sufficient ◽  
Locally Convex

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?

scholarly journals On Locally Convex Topological Vector Space Valued Paranormed Function Space Defined by Orlicz Function

2014 ◽  
Vol 14 (2) ◽  
pp. 109-116
Author(s):  
NP Pahari
Keyword(s):  
Vector Space ◽  
Function Space ◽  
Convex Space ◽  
Orlicz Function ◽  
Locally Convex Space ◽  
Similar Nature ◽  
New Class ◽  
And Function ◽  
Locally Convex ◽  
Class Of Functions

The aim of this paper is to introduce and study a new class (l∞ (X, Y, Φ, ξ, w , L), HU) of locally convex space Y- valued functions using Orlicz function Φ as a generalization of some of the well known sequence spaces and function spaces. Besides the investigation pertaining to the linear topological structures of the class (l∞ (X, Y, Φ, ξ, w , L), HU) when topologized it with suitable natural paranorm , our primarily interest is to explore the conditions pertaining the containment relation of the class l∞ (X, Y, Φ, ξ, w) in terms of different ξ and w so that such a class of functions is contained in or equal to another class of similar nature. DOI: http://dx.doi.org/10.3126/njst.v14i2.10423   Nepal Journal of Science and Technology Vol. 14, No. 2 (2013) 109-116

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

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