Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL , . In this paper we present our formalization of this theory in Mizar .
First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space , a real normed linear space  and a linear topological space  with the concept of the limit of an image filter .
Then, following Bourbaki ,  (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in ), using the notion of filters.
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