Uniform boundedness in metric spaces
HTML articles powered by AMS MathViewer
- by James D. Stein PDF
- Proc. Amer. Math. Soc. 32 (1972), 299-303 Request permission
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.References
- A. Alexiewicz, On sequences of operations. I, Studia Math. 11 (1949), 1–30. MR 38555, DOI 10.4064/sm-11-1-1-30
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Vlastimil Pták, A uniform boundedness theorem and mappings into spaces of operators, Studia Math. 31 (1968), 425–431. MR 236672, DOI 10.4064/sm-31-4-425-431
- J. D. Stein Jr., Several theorems on boundedness and equicontinuity, Proc. Amer. Math. Soc. 26 (1970), 415–419. MR 270124, DOI 10.1090/S0002-9939-1970-0270124-1
- James D. Stein Jr., Two uniform boundedness theorems, Pacific J. Math. 38 (1971), 251–260. MR 305242
Additional Information
- © Copyright 1972 American Mathematical Society
- 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