Uniform boundedness in metric spaces
James D. Stein
Proc. Amer. Math. Soc. 32 (1972), 299-303
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Abstract: Let be metric spaces, a family of maps from X into Y. Let is said to be pointwise-bounded if for each is finite, and is uniformly bounded if for each is finite. The major result of this paper is to place a sufficient condition on the maps in 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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