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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Several theorems on boundedness and equicontinuity


Author: J. D. Stein
Journal: Proc. Amer. Math. Soc. 26 (1970), 415-419
MSC: Primary 46.10
MathSciNet review: 0270124
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Abstract: This paper presents several results concerning equicontinuity of a pointwise-bounded family of linear transformations on a Banach space. The first is the following generalization of the Banach-Steinhaus Theorem: Let $ \{ {T_\alpha }\vert\alpha \in A\} $ be a pointwise-bounded family of linear transformations from a Banach space $ X$ to a normed linear space $ Y$, and assume that, for each $ \alpha \in A,{T_\alpha }$ is continuous on a closed subspace $ {S_\alpha }$ of $ X$. Then $ \exists {\alpha _1}, \cdots ,{\alpha _n} \in A$ such that the family is equicontinuous on $ \bigcap\nolimits_{k = 1}^n {{S_{\alpha k}}} $. The second theorem deals with a pointwise-bounded family of linear transformations from a Banach space $ X$ to a normed linear space with a continuous bilinear mapping into another normed linear space. The others deal with homomorphisms of Banach algebras.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0270124-1
PII: S 0002-9939(1970)0270124-1
Keywords: Functional analysis, Banach spaces
Article copyright: © Copyright 1970 American Mathematical Society