Compact convergence and the order bidual for<i>C</i>(<i>X</i>)

Feldman, William; Porter, J. F.

doi:10.2140/pjm.1975.57.113

Cited by 8 publications

(4 citation statements)

References 11 publications

(16 reference statements)

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Order By: Relevance

“…For the sake of emphasis, Proposition 3 is incorporated into the next theorem. (2) we remark that it follows from part (1) that the order-theoretic and Banach duals of (©", 2) n +, | ' || n ) coincide and that (2>' n , 3>' n+ , ||' || n ) is an L-space. That 2i' n is indeed the space of nth order distributions on T follows from Proposition 3, which also implies that | ' |j n and |||" \\\' n are equivalent.…”

Section: Thusmentioning

confidence: 95%

“…We recall that the Banach space S) o of real continuous functions on F with the norm ||/||o = s i i p { | / ( x ) | : * 6 r } for / i n 2> 0 and positive cone @ 0+ of pointwise non-negative members of 3) 0 is an order unit space (and an Af-space). Where S>' 0+ denotes the set of linear functionals 4> on 3> 0 satisfying <f>(f) k 0 for all/in 3> 0+ and (0 O ', || -\' o ) is the Banach dual of (0 O , | " ||o)> the system (2)'Q, $O + , I ' ||o) is a base norm space (and an £-space). The order unit 1 in 3i Q+ can be taken as the generator of ||' || 0 (i.e., ||' || 0 is the Minkowski functional on the order interval The Riesz Representation Theorem states that (S>' 0 , |j ' ||o) can be identified with the space of regular, finite Borel measures on F with total variation norm and that @' 0+ can be identified with the non-negative measures in %.…”

Section: Partial Order Structures On 2)mentioning

confidence: 99%

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Order and Schwartz distributions

Feldman

Porter

1977

Glasgow Math. J.

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The space of Schwartz distributions on the unit circle Г in the plane is topologically a considerable generalization of the space of regular, finite Borel measures on Г. However, the order structure of is usually taken to be the same as that of : there are no “positive” distributions which are not measures. This perhaps warrants consideration, since the order structure of generates its topology. In this paper we construct a system of order structures for which is a more natural complement in the intermediate stages to the topology of and which provides an interpretation of with its Schwartz topology as a quotient of a generalized base norm space V′. Where denotes the space of continuous functions on Г with its supremum norm topology, V′ is the dual of . The space contains the infinitely differentiable functions on Г with their usual topology, and (via the pointwise ordering on ) in its product ordering is realized as a generalized order unit space. Some consequences for harmonic functions are discussed.

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Section: Thusmentioning

confidence: 95%

Section: Partial Order Structures On 2)mentioning

confidence: 99%

Order and Schwartz distributions

Feldman

Porter

1977

Glasgow Math. J.

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“…Such elements ex (called semiorder-units) were studied in [4] and [5]. In the presence of the weak order unit \/ex this stronger condition implies that the set Ex = [x E X: ex(x) > 0} is both open and compact (see the proof of Theorem 1).…”

mentioning

confidence: 99%

A Vector Lattice Topology and Function Space Representation

Feldman¹,

Porter²

1978

Transactions of the American Mathematical Society

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Abstract. A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called "order partition spaces," an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact A', and conversely each such CC{X) is an order partition space. {Ce{X) denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of CC{X) for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.A classical theorem of Kakutani (see [10]) states that each order unit space with its order unit topology is lattice isometric to a dense subspace of C(X) containing the constant functions with supremum norm for some compact X and, conversely, each such C(X) is an order unit space. (Here C(X) denotes the space of real-valued continuous functions on X.) In § 1 we generalize the concept of an order unit and the topology which it generates and provide an extension of Kakutani's theorem for this setting. Specifically, we consider a vector lattice having a weak order unit e and a partition of e which, in analogy to the order unit case, defines a locally convex topology. Such a space, called an "order partition space," is shown (Theorem 1) to be lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact X. (Here CC(X) denotes C(X) with the topology of compact convergence.) Conversely, CC(X) for X locally compact is an order partition space. The authors obtained a similar result in Theorem 1 of [5] for the restricted case that X is a countable union of open compact sets. In the remainder of §1, sufficient conditions are discussed for the order partition topology to be lattice-theoretic-i.e., independent of the choice of e

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“…We remark that condition (*) in the definition of an order partition is similar to the stronger condition v A nex < ßex(n = 1,2,...). Such elements ex (called semiorder-units) were studied in [4] x > l/k one obtains a system {ek} satisfying the weaker condition but not (*), with e -\/ek = 1. We note that by defining e'k to be 0 for x < (k + l)~l and 1 for x > l/k, together with the connecting line segment, one obtains an order partition for C[0,1] having carrier space (0, 1].…”

mentioning

confidence: 99%

A vector lattice topology and function space representation

Feldman¹,

Porter²

1978

Trans. Amer. Math. Soc.

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A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called "order partition spaces," an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact A', and conversely each such CC{X) is an order partition space. {Ce{X) denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of CC{X) for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.

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Compact convergence and the order bidual forC(X)

Cited by 8 publications

References 11 publications

Order and Schwartz distributions

Order and Schwartz distributions

A Vector Lattice Topology and Function Space Representation

A vector lattice topology and function space representation

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