Linear Topological Spaces and Linear Operators

1982 ◽  
pp. 170-203
Author(s):  
A. A. Kirillov ◽  
A. A. Gvishiani
Keyword(s):  
Linear Operators ◽  
Topological Spaces

scholarly journals Positive operators and approximation in function spaces on completely regular spaces

2003 ◽  
Vol 2003 (61) ◽  
pp. 3841-3871 ◽  
Author(s):  
Francesco Altomare ◽  
Sabrina Diomede
Keyword(s):  
Natural Extension ◽  
Function Spaces ◽  
Positive Operators ◽  
Linear Operators ◽  
Topological Spaces ◽  
Approximation Properties ◽  
Completely Regular ◽  
Borel Measures ◽  
Korovkin Type Theorems ◽  
Locally Convex

We discuss the approximation properties of nets of positive linear operators acting on function spaces defined on Hausdorff completely regular spaces. A particular attention is devoted to positive operators which are defined in terms of integrals with respect to a given family of Borel measures. We present several applications which, in particular, show the advantages of such a general approach. Among other things, some new Korovkin-type theorems on function spaces on arbitrary topological spaces are obtained. Finally, a natural extension of the so-called Bernstein-Schnabl operators for convex (not necessarily compact) subsets of a locally convex space is presented as well.

Compact Linear Operators in Linear Topological Spaces

1954 ◽  
Vol s1-29 (2) ◽  
pp. 149-156 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Linear Operators ◽  
Topological Spaces

scholarly journals Vector Measures on Topological Spaces

2007 ◽  
Vol 14 (4) ◽  
pp. 687-698
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Topological Spaces ◽  
Vector Measures ◽  
Completely Regular ◽  
Measure Representation ◽  
Integrable Functions ◽  
Linear Maps ◽  
Vector Case ◽  
Vector Valued

Abstract Let 𝑋 be a completely regular Hausdorff space, 𝐸 a quasi-complete locally convex space, 𝐶(𝑋) (resp. 𝐶𝑏(𝑋)) the space of all (resp. all, bounded), scalar-valued continuous functions on 𝑋, and 𝐵(𝑋) and 𝐵0(𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀𝑡(𝑋,𝐸), 𝑀 τ (𝑋,𝐸), 𝑀 σ (𝑋,𝐸) of tight, τ-smooth, σ-smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶𝑏(𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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