Abstract
In the first part of this paper we prove the following theorem: Let X be a set, let Y be a separable metrizable uniform space, and let H be any precompact subset of YX; endow X with the coarsest uniformity uH making H uniformly equicontinuous. Then the covering type of (X,uH) equals the weight of H. As a corollary we get: If A is any precompact subset of a locally convex space, then the closed absolutely convex hull\(\overline {\Gamma A} \) has the same weight as A. In the second part we consider some forms of the closed graph and Banach-Steinhaus theorem depending on some cardinal α. Taking α=χ0 will show that the class ℓ(ζ) of locally convex spaces E introduced by Kalt on [10] can be characterized by the following Banach-Steinhaus condition: If fn, n∈ℕ, and f are linear mappings of E into a separable locally convex space such that each fn is continuous and fn(x)→f(x) for every x∈E, then f is continuous.
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Pfister, H. Über das Gewicht und den Überdeckungstyp von uniformen Räumen und einige Formen des Satzes von Banach-Steinhaus. Manuscripta Math 20, 51–72 (1977). https://doi.org/10.1007/BF01181240
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DOI: https://doi.org/10.1007/BF01181240