More about metric spaces on which continuous functions are uniformly continuous
نویسندگان
چکیده
منابع مشابه
Metric Spaces on Which Continuous Functions Are Uniformly Continuous and Hausdorff Distance
Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and (2) for each £ > 0, the set of points in X whose distance from X' exceeds e is uniformly discrete. We obtain these new characterizations: (a) for each metric space V, the Hausdorff metric on C(X, Y),...
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Let X and Y be Banach spaces. A set ᏹ of 1-summing operators from X into Y is said to be uniformly summing if the following holds: given a weakly 1-summing sequence (x n) in X, the series n T x n is uniformly convergent in T ∈ ᏹ. We study some general properties and obtain a characterization of these sets when ᏹ is a set of operators defined on spaces of continuous functions.
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A basic theorem asserts that a continuous function on a compact metric space with values in another metric space is uniformly continuous. The usual proofs based on a contradiction argument involving sequences or on the covering property of compact sets are quite sophisticated for students taking a first course on real analysis. We present a direct proof only using results that are established a...
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We study classes of continuous functions on R that can be approximated in various degree by uniformly continuous ones (uniformly approachable functions). It was proved in [BDP1] that no polynomial function can distinguish between them. We construct examples that distinguish these classes (answering a question from [BDP1]) and we offer appropriate forms of uniform approachability that enable us ...
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For simplicity, we adopt the following rules: n denotes an element of N, X, X1 denote sets, r, p denote real numbers, s, x0, x1, x2 denote real numbers, S, T denote real normed spaces, f , f1, f2 denote partial functions from R to the carrier of S, s1 denotes a sequence of real numbers, and Y denotes a subset of R. The following propositions are true: (1) Let s2 be a sequence of real numbers an...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 1986
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700003981