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.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

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 Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

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.

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.

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 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.

scholarly journals On some optimization problems in not necessarily locally convex space

2008 ◽  
Vol 18 (2) ◽  
pp. 167-172
Author(s):  
Ljiljana Gajic
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Convex Space ◽  
Optimization Problems ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Equilibrium Existence ◽  
Cooperative Equilibrium ◽  
Locally Convex

In this note, by using O. Hadzic's generalization of a fixed point theorem of Himmelberg, we prove a non - cooperative equilibrium existence theorem in non - compact settings and a generalization of an existence theorem for non - compact infinite optimization problems, all in not necessarily locally convex spaces.

scholarly journals The H∞-optimization in locally convex spaces

2002 ◽  
Vol 15 (2) ◽  
pp. 91-103
Author(s):  
Chuan-Gan Hu ◽  
Li-Xin Ma
Keyword(s):  
Control Theory ◽  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Model Matching ◽  
Matching Error ◽  
Locally Convex ◽  
Convex Spaces

In this paper, the ordinary H∞-control theory is extended to locally convex spaces through the form of a parameter. The algorithms of computing the infimal model-matching error and the infimal controller are presented in a locally convex space. Two examples with the form of a parameter are enumerated for computing the infimal model-matching error and the infimal controller.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

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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