Introduction to Banach Spaces (original) (raw)
On uniform convergence and some related types of convergence
2013
There are known several types of convergence of nets of functions. The best known types of convergence in the space of functions are pointwise and uniform convergence. One can find many kinds of convergence which are called in different manner. Some of them are weaker than uniform convergence but stronger than pointwise one, some of them are even stronger than uniform convergence. In the work we will consider mainly convergence of nets of functions defined on a topological space with values in a metric space. We investigate connections between different kinds of convergence and preserving of continuity and integrability of limit functions. Throughout the article, (X,) will denote a topological space and (Y, ρ) will denote a metric space. We start from definitions of some types of convergence of nets of functions f : X −→ Y. The best known types of convergence are pointwise and uniform convergence.
Uniform boundedness in function spaces
Topology and its Applications
We begin with some basic information about uniform and function spaces. Our topological notation and terminology are standard (see [5]). By N and R we denote the set of natural and real numbers, respectively. 1.1. Uniform spaces Let X be a nonempty set. A family U of subsets of X × X satisfying conditions (U1) each U ∈ U contains the diagonal ∆ X = {(x, x) : x ∈ X} of X;
Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications
Fixed Point Theory and Applications, 2010
The concept of weakly quasi-nonexpansive mappings with respect to a sequence is introduced. This concept generalizes the concept of quasi-nonexpansive mappings with respect to a sequence due to Ahmed and Zeyada 2002 . Mainly, some convergence theorems are established and their applications to certain iterations are given.
On variations on quasi Cauchy sequences in metric spaces
For a fixed positive i nteger p, a sequence (x n) in a metric space X is c alled p-quasi-Cauchy if (Δ p x n) is a null sequence where Δ p x n = d(x n+p , x n) for each positive integer n. A subset E of X is called p-ward compact if any sequence (x n) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f (x n)) is a p-quasi Cauchy sequence in Y whenever (x n) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f (x n) is p-quasi Cauchy in Y whenever (x n) is a quasi cauchy sequence of points in E.
2021
The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in [7]) that has been used to define several new concepts in recent articles [9, 10]. We first introduce a new notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions [8], as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called “quasi-Cauchy Lipschitz functions” is introduced following the line of investigations in [3, 4, 5, 12] and again several coincidence results are proved along with a very interesting observation that every real valued ward continuous function defined...
(Quasi)-uniformities on the set of bounded maps
International Journal of Mathematics and Mathematical Sciences, 1994
From real analysis it is known that if a sequence {f ne} of real-valued functions defined and bounded on Xc converges uniformly to f, then f is also bounded and the sequence {f nc} is uniformly bounded on X.
On certain notions of precompactness, continuity and Lipschitz functions
arXiv (Cornell University), 2021
The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in [7]) that has been used to define several new concepts in recent articles [9, 10]. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions [8], as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in [3, 4, 5, 12] and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real valued ward continuous function defined on a metric space can be uniformly approximated by real valued quasi-Cauchy Lipschitz functions.