Vol. 56, No. 1, pp. 73-79 (1999)
Continuous Norms on Locally Convex Spaces
Vítor NevesDep. de Matemática, Univ. de Aveiro,
3810 Aveiro - PORTUGAL
Abstract: Given a locally convex space $E$ with nonstandard extension $^*E$ in a polysaturated model of Analysis, we distinguish very large infinite and very small infinitesimal elements of $^*\!E$, show that $E$ is normable if and only if the former do not exist (Theorem 3.1) and show that the existence of continuous norms on $E$ is a necessary condition for validity of Inverse Function Theorems (Theorem 2.2). We use a stronger version of the embedding of standard sets in hyperfinite sets (Lemma 4.1).
Keywords: Locally convex space; polysaturated model; finite; infinitesimal; perturbation.
Classification (MSC2000): 46A03, 46B99, O3H05.
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Electronic version published on: 31 Jan 2003. This page was last modified: 27 Nov 2007.