Isomorphisms of spaces of bounded continuous functions

1983 ◽  
Vol 22 (6) ◽  
pp. 1860-1862
Author(s):  
T. E. Khmyleva
Keyword(s):  
Continuous Functions ◽  
Bounded Continuous Functions

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.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals Generalized Numerical Index of Function Algebras

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Han Ju Lee
Keyword(s):  
Banach Space ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Function Algebras ◽  
Bounded Complex ◽  
Closed Subalgebra ◽  
Bounded Continuous Functions ◽  
Complex Valued ◽  
Numerical Index

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A,  x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

scholarly journals Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

scholarly journals Local invertibility in subrings of C*(X)

1992 ◽  
Vol 46 (3) ◽  
pp. 449-458 ◽  
Author(s):  
H. Linda Byun ◽  
Lothar Redlin ◽  
Saleem Watson
Keyword(s):  
Continuous Functions ◽  
Closed Sets ◽  
Complete Ring ◽  
Maximal Ideals ◽  
One To One ◽  
Local Invertibility ◽  
Bounded Continuous Functions ◽  
Ring Of Functions ◽  
Uniformly Closed

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

scholarly journals Approximation of Bounded Continuous Functions by Linear Combinations of Phillips Operators

2014 ◽  
Vol 47 (3) ◽  
Author(s):  
Gancho Tachev
Keyword(s):  
Continuous Functions ◽  
Approximation Properties ◽  
Linear Combinations ◽  
Direct Estimate ◽  
Durrmeyer Operators ◽  
Rate Of Approximation ◽  
Phillips Operators ◽  
Bounded Continuous Functions

AbstractWe study the approximation properties of linear combinations of the so-called Phillips operators, which can be considered as genuine Szász-Mirakjan-Durrmeyer operators. As main result, we prove a direct estimate for the rate of approximation of bounded continuous functions f E C[0,x), measured in C|\[0,x)-norm and thus generalizing the results, proved earlier by Gupta, Agrawal, and Gairola in [3]. Our estimates rely on the recent results, obtained in the joint works of M. Heilmann and the author-[10, 11]

scholarly journals Operators on Spaces of Bounded Vector-Valued Continuous Functions with Strict Topologies

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Marian Nowak
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Representation Theorems ◽  
Completely Regular ◽  
General Integral ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Continuous Linear Operators

LetXbe a completely regular Hausdorff space, and let(E,‖·‖E)and(F,‖·‖F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions defined onX, equipped with the strict topologiesβz, where  z=σ,∞,p,τ,t. General integral representation theorems of(βz,‖·‖F)-continuous linear operators  T:Cb(X,E)→F  with respect to the corresponding operator-valued measures are established. Strongly bounded and(βz,‖·‖F)-continuous operatorsT:Cb(X,E)→Fare studied. We extend to “the completely regular setting” some classical results concerning operators on the spacesC(X,E)andCo(X,E), whereX  is a compact or a locally compact space.

scholarly journals Weakly compact operators and the strict topologies

1989 ◽  
Vol 39 (3) ◽  
pp. 353-359 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

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