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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniform boundedness in metric spaces


Author: James D. Stein
Journal: Proc. Amer. Math. Soc. 32 (1972), 299-303
MSC: Primary 54.60
DOI: https://doi.org/10.1090/S0002-9939-1972-0290344-1
MathSciNet review: 0290344
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Abstract: Let $(X,d),(Y,d’)$ be metric spaces, $\mathcal {F}$ a family of maps from X into Y. Let ${x_0} \in X.\mathcal {F}$ is said to be pointwise-bounded if for each $x \in X,\sup \{ d’(Tx,T{x_0})|T \in \mathcal {F}\}$ is finite, and is uniformly bounded if for each $p > 0,\sup \{ d’(Tx,T{x_0})|T \in \mathcal {F},d(x,{x_0}) \leqq p\}$ is finite. The major result of this paper is to place a sufficient condition on the maps in $\mathcal {F}$ to ensure that, if X is complete, a pointwise-bounded family of continuous maps is uniformly bounded, and to show that this result is best possible.


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Keywords: Uniform boundedness
Article copyright: © Copyright 1972 American Mathematical Society