scholarly journals A generalized inductive limit strict topology on the space of bounded continuous functions

2007 ◽  
Vol 18 (4) ◽  
pp. 485-494 ◽  
Author(s):  
Jose Aguayo ◽  
Samuel Navarro ◽  
Jacqueline Ojeda
Keyword(s):  
Inductive Limit ◽  
Continuous Functions ◽  
Strict Topology ◽  
Bounded Continuous Functions

scholarly journals Some Classes of Continuous Operators on Spaces of Bounded Vector-Valued Continuous Functions with the Strict Topology

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Marian Nowak
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Strict Topology ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Regular ◽  
Strictly Singular ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
The Relationship

LetXbe a completely regular Hausdorff space and letE,·Eand(F,·F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions onX, equipped with the strict topologyβσ. We study the relationship between important classes of(βσ,·F)-continuous linear operatorsT:Cb(X,E)→F(strongly bounded, unconditionally converging, weakly completely continuous, completely continuous, weakly compact, nuclear, and strictly singular) and the corresponding operator measures given by Riesz representing theorems. Some applications concerning the coincidence among these classes of operators are derived.

The Strict Topology in a Completely Regular Setting: Relations to Topological Measure Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 873-890 ◽  
Author(s):  
Steven E. Mosiman ◽  
Robert F. Wheeler
Keyword(s):  
Measure Theory ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Strict Topology ◽  
Completely Regular ◽  
Borel Measures ◽  
Topological Measure Theory ◽  
Locally Convex Topologies ◽  
Bounded Continuous Functions ◽  
Locally Convex

Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology β on C*(X) was proved by Buck [2]: the dual space of (C*(X), β) has a natural representation as the space of bounded regular Borel measures on X.Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures.

Nuclear operators on Cb(X, E) and the strict topology

2018 ◽  
Vol 68 (1) ◽  
pp. 135-146
Author(s):  
Marian Nowak ◽  
Juliusz Stochmal
Keyword(s):  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Strict Topology ◽  
Completely Regular ◽  
Borel Measures ◽  
Nuclear Operators ◽  
Bounded Continuous Functions

AbstractLetXbe a completely regular Hausdorff space,EandFbe Banach spaces. LetCb(X,E) be the space of allE-valued bounded, continuous functions onX, equipped with the natural strict topologyβ. We study nuclear operatorsT:Cb(X,E) →Fin terms of their representing operator-valued Borel measures.

Semigroups of Operators in C(S)

1970 ◽  
Vol 22 (1) ◽  
pp. 47-54 ◽  
Author(s):  
F. Dennis Sentilles
Keyword(s):  
Transition Function ◽  
Continuous Functions ◽  
Linear Operators ◽  
Semigroups Of Operators ◽  
Supremum Norm ◽  
Strict Topology ◽  
Borel Measures ◽  
Transition Functions ◽  
Semigroup Of Linear Operators ◽  
Bounded Continuous Functions

Our study in this paper is two-fold: One is that of a semigroup of linear operators on the space C(S) of bounded continuous functions on a locally compact Hausdorff space S, while the other is that of a transition function of measures in the Banach space M(S) of bounded regular Borel measures on S. It will be seen that an informative and essentially non-restrictive theory of the former can be obtained when C(S) is given the strict topology rather than the usual supremum norm topology and that, in this setting, the natural relationship between semigroups and transition functions obtained when S is compact is maintained, essentially because the dual of C(S) with the strict topology is M(S).

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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