Continuous Norms on Locally Convex Spaces

Vítor Neves

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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