Spectrum of bounded continuous functions specified on a unit sphere in Banach space

1969 ◽  
Vol 3 (2) ◽  
pp. 137-146 ◽  
Author(s):  
V. D. Mil'man
Keyword(s):  
Banach Space ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Bounded Continuous Functions

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 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.

New Almost Periodic Type Functions and Solutions of Differential Equations

1996 ◽  
Vol 48 (6) ◽  
pp. 1138-1153 ◽  
Author(s):  
Bolis Basit ◽  
Chuanyi Zhang
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Continuous Functions ◽  
Parabolic Partial Differential Equations ◽  
Almost Periodic ◽  
Almost Automorphic Functions ◽  
Indefinite Integral ◽  
Pseudo Almost Automorphic ◽  
Bounded Continuous Functions ◽  
Automorphic Functions

AbstractLet X be a Banach space and . Let Π and Π0 be two subspaces of , the Banach space of bounded continuous functions from 𝕁 to X. We seek conditions under which Π + Π0 is closed in . This led to introduce a general space, which contains many classes of almost periodic type functions as subspaces. We prove some recent results on indefinite integral for the elements of these classes. We apply certain results on harmonic analysis to investigate solutions of differential equations. As an application we study specific spaces: the spaces of asymptotic and pseudo almost automorphic functions and their solutions of some ordinary quasi-linear and a non-linear parabolic partial differential equations.

scholarly journals Separable measures and the Dunford-Pettis property

1991 ◽  
Vol 43 (3) ◽  
pp. 423-428 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Regular Space ◽  
Weakly Compact ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions ◽  
Locally Convex Topology ◽  
Locally Convex

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.

scholarly journals Preimages of Residual Sets of Continuous Functions under Operation of Superposition

2005 ◽  
Vol 12 (4) ◽  
pp. 763-768
Author(s):  
Artur Wachowicz
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Real Functions ◽  
Second Category ◽  
Residual Sets

Abstract Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

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