scholarly journals A Riesz representation theorem for cone-valued functions

1999 ◽  
Vol 4 (4) ◽  
pp. 209-229
Author(s):  
Walter Roth
Keyword(s):  
Hausdorff Space ◽  
Inductive Limit ◽  
Riesz Representation Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Borel Measures ◽  
Riesz Representation ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

We consider Borel measures on a locally compact Hausdorff space whose values are linear functionals on a locally convex cone. We define integrals for cone-valued functions and verify that continuous linear functionals on certain spaces of continuous cone-valued functions endowed with an inductive limit topology may be represented by such integrals.

Measures Defined by Gages

1992 ◽  
Vol 44 (6) ◽  
pp. 1303-1316 ◽  
Author(s):  
Washek F. Pfeffer ◽  
Brian S. Thomson
Keyword(s):  
Set Theory ◽  
Measure Space ◽  
Hausdorff Space ◽  
Outer Measure ◽  
Riesz Representation Theorem ◽  
Locally Compact ◽  
Riemann Integral ◽  
Riesz Representation ◽  
Intuitive Concept ◽  
Locally Compact Hausdorff Space

AbstractUsing ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

scholarly journals Completely superharmonic measures for the infinitesimal generator A of a diffusion semi-group and positive eigen elements of A

1981 ◽  
Vol 83 ◽  
pp. 53-106 ◽  
Author(s):  
Masayuki Itô ◽  
Noriaki Suzuki
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinitesimal Generator ◽  
Hausdorff Space ◽  
Inductive Limit ◽  
Semi Group ◽  
Radon Measures ◽  
Countable Basis ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space

Let X be a locally compact Hausdorff space with countable basis. We denote byM(X) the topological vector space of all real Radon measures in X with the vague topology,MK(X) the topological vector space of all real Radon measures in X whose supports are compact with the usual inductive limit topology.

scholarly journals On the representation of strictly continuous linear functionals

1981 ◽  
Vol 24 (2) ◽  
pp. 123-130 ◽  
Author(s):  
Liaqat Ali Khan ◽  
K. Rowlands
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Hausdorff Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Completely Regular ◽  
Totally Bounded ◽  
Hausdorff Topological Vector Space ◽  
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; when E is the real or complex field this space will be denoted by C(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. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

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 Linear mappings between topological vector spaces

1972 ◽  
Vol 14 (1) ◽  
pp. 105-118
Author(s):  
B. D. Craven
Keyword(s):  
Inductive Limit ◽  
Vector Spaces ◽  
Continuous Linear ◽  
Additional Structure ◽  
Normed Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

If A and B are locally convex topological vector spaces, and B has certain additional structure, then the space L(A, B) of all continuous linear mappings of A into B is characterized, within isomorphism, as the inductive limit of a family of spaces, whose elements are functions, or measures. The isomorphism is topological if L(A, B) is given a particular topology, defined in terms of the seminorms which define the topologies of A and B. The additional structure on B enables L(A, B) to be constructed, using the duals of the normed spaces obtained by giving A the topology of each of its seminorms separately.

A nonstandard proof of the Riesz representation theorem for weakly compact operators on C(Ω)

1989 ◽  
Vol 105 (1) ◽  
pp. 141-145
Author(s):  
Yeneng Sun
Keyword(s):  
Representation Theorem ◽  
Compact Operators ◽  
Riesz Representation Theorem ◽  
Linear Functionals ◽  
Weakly Compact ◽  
Vector Measures ◽  
Riesz Representation ◽  
Weakly Compact Operators ◽  
Measure Technique

AbstractAn easy way to construct the representing vector measures of weakly compact operators on C(Ω) is given by using the Loeb measure technique. This construction is not based on the Riesz representation theorem for linear functionals; thus we have a uniform way to treat the scalar and vector cases. Also the star finite representations of regular vector measures follow from the proof.

scholarly journals Multipliers of Modules of Continuous Vector-Valued Functions

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.

A PROJECTIVE DESCRIPTION OF THE SPACE OF HOLOMORPHIC GERMS

2001 ◽  
Vol 44 (2) ◽  
pp. 407-416 ◽  
Author(s):  
P. Laubin
Keyword(s):  
Inductive Limit ◽  
Extension Theorem ◽  
Inductive Limit Topology ◽  
Projective Description ◽  
Whitney Extension ◽  
The One ◽  
Whitney Extension Theorem ◽  
Locally Convex ◽  
Suitable Sequence

AbstractA natural topology on the set of germs of holomorphic functions on a compact subset $K$ of a Fréchet space is the locally convex inductive limit topology of the spaces $\mathcal{O}(\sOm)$ endowed with the compact open topology; here $\sOm$ is any open subset containing $K$. Mujica gave a description of this space as the inductive limit of a suitable sequence of compact subsets. He used a set of intricate semi-norms for this. We give a projective characterization of this space, using simpler semi-norms, whose form is similar to the one used in the Whitney Extension Theorem for $C_\infty$ functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non-quasi-analytic spaces.AMS 2000 Mathematics subject classification: Primary 46A13; 46F15

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

scholarly journals A generalized inductive limit topology for linear spaces

1973 ◽  
Vol 14 (2) ◽  
pp. 105-110 ◽  
Author(s):  
S. O. Iyahen ◽  
J. O. Popoola
Keyword(s):  
Inductive Limit ◽  
Locally Convex Spaces ◽  
Closed Graph Theorem ◽  
Linear Maps ◽  
Inductive Limit Topology ◽  
Definition Of ◽  
Object Of Study ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Convex Spaces

In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem.

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