Closed ideals in the uniform topology on the ring of real-valued continuous functions on a frame

Let E and F be locally convex spaces and let K be a compact Hausdorff space. C(K,E) is the space of all E-valued continuous functions defined on K, endowed with the uniform topology.Starting from the well-known fact that every linear continuous operator T from C(K,E) to F can be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.As a consequence we characterise the spaces E for which an operator T defined on C(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) in C(K,E).

Some notes on densely continuous forms

Mathematica Slovaca ◽

10.2478/s12175-008-0090-3 ◽

2008 ◽

Vol 58 (4) ◽

Author(s):

Peter Vadovič

Keyword(s):

Acta Math ◽

Continuous Functions ◽

Hausdorff Spaces ◽

Graph Topology ◽

Uniform Topology ◽

Pointwise Topology ◽

Topology Of Pointwise Convergence ◽

Nonempty Compact ◽

Cardinal Functions ◽

Special Space

AbstractWe consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLÁ, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLÝ, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLÝ, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D p*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D p*(X) and on real-valued continuous functions C p(X) and a generalization of a sufficient condition for the countable cellularity of D p*(X).

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

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2009.1129 ◽

2010 ◽

Vol 47 (3) ◽

pp. 289-298 ◽

Cited By ~ 2

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.

Almost Somewhat Near Continuity and Near Regularity

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2021-0009 ◽

2021 ◽

Vol 7 (1) ◽

pp. 88-99

Author(s):

Zanyar A. Ameen

Keyword(s):

Continuous Function ◽

Continuous Functions ◽

One To One ◽

Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

Approximating Continuous Functions by Iterated Function Systems and Optimization

SSRN Electronic Journal ◽

10.2139/ssrn.616605 ◽

2002 ◽

Author(s):

Davide La Torre ◽

Matteo Rocca

Keyword(s):

Iterated Function Systems ◽

Continuous Functions ◽

Iterated Function

On existence result of a class of nonlinear integral equation

Filomat ◽

10.2298/fil1711593b ◽

2017 ◽

Vol 31 (11) ◽

pp. 3593-3597

Author(s):

Ravindra Bisht

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Existence And Uniqueness ◽

Existence Result ◽

Nonlinear Integral Equation ◽

Continuous Functions ◽

Concave Functions ◽

Real Numbers ◽

Fixed Point Techniques ◽

Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

ON ε-CONTINUOUS FUNCTIONS

Real Analysis Exchange ◽

10.2307/44153909 ◽

1995 ◽

Vol 21 (1) ◽

pp. 203

Author(s):

Banaszewski

Keyword(s):

Continuous Functions

ON THREADING CONTINUOUS FUNCTIONS THROUGH COMPACT SETS

Real Analysis Exchange ◽

10.2307/44153417 ◽

1982 ◽

Vol 8 (2) ◽

pp. 455

Author(s):

Akemann ◽

Bruckner

Keyword(s):

Continuous Functions ◽

Compact Sets

I-density Continuous Functions

Real Analysis Exchange ◽

10.2307/44151978 ◽

1989 ◽

Vol 15 (1) ◽

pp. 13

Author(s):

Ciesielski ◽

Larson ◽

Ostaszewski

Keyword(s):

Continuous Functions

ON THE MAXIMAL FAMILIES FOR THE CLASS OF STRONGLY QUASI–CONTINUOUS FUNCTIONS

Real Analysis Exchange ◽

10.2307/44152545 ◽

1994 ◽

Vol 20 (2) ◽

pp. 631

Author(s):

Grande

Keyword(s):

Continuous Functions