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Some observations on local uniform boundedness principles. (English) Zbl 0791.46012

The uniform boundedness principle for continuous linear maps from Banach spaces into normed spaces is one of the first major consequences of the Baire category theorem. One way to do this is to first prove Osgood’s theorem, sometimes known as the local uniform boundedness principle, which states that a pointwise-bounded family of continuous maps of a complete metric space into a metric space must be uniformly bounded on some open subset.
Uniform boundedness principles play an important role in automatic continuity. A basic variation introduced by Pták and extended by others enables one to derive interesting results for systems theory.
This paper investigates several different aspects of local uniform boundedness principles. In the first section, we prove versions of the gliding hump theorem from automatic continuity for complete metric spaces, locally compact Hausdorff, and sequentially compact spaces; including versions based on variations of the Mittag-Leffler inverse limit theorem. The conclusions of the theorems are weaker when the spaces are sequentially compact. In the second section, we show that the weaker conclusions for sequentially compact spaces reflect the fact that Baire spaces can be characterized by the equivalence between the Baire category theorem and a specific version of the local uniform boundedness principle, and thus optimally strong uniform boundedness principles for sequentially compact spaces are unattainable.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
54E52 Baire category, Baire spaces
54E50 Complete metric spaces
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References:

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