Let \(E\) be a real inner product space of dimension at least 2 and \(V\) a linear topological Hausdorff space. If \(\operatorname{card}E\leq \operatorname{card} V\), then the set of all orthogonally additive injections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology. If \(\operatorname{card}V\leq \operatorname{card}E\), then the set of all orthogonally additive surjections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology.
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Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.
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Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.
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