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**1 - 5**of**5**### DOI: 10.2478/forma-2014-0024 degruyter.com/view/j/forma Topological Properties of Real Normed

"... Summary. In this article, we formalize topological properties of real nor-med spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed sub-space. Then we discuss linear functions between real normed spec ..."

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Summary. In this article, we formalize topological properties of real nor-med spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed sub-space. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve conver-gence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).

### Separability of Real Normed Spaces and Its Basic Properties

, 2015

"... Summary. In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the ..."

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Summary. In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on Separability of Real Normed Space Let X be a real linear space and A be a subset of X. The functor Sums Q A yielding a subset of X is defined by the term (Def. 1) { l, where l is a linear combination of A : rng l ⊆ Q}. Let us consider a real normed space V and a real normed subspace V

### Uniform Boundedness Principle

, 2008

"... Summary. In this article at first, we proved the lemma of the inferior limit and the superior limit. Next, we proved the Baire category theorem (Banach space version) [20], [9], [3], quoted it and proved the uniform boundedness principle. Moreover, the proof of the Banach-Steinhaus theorem is added. ..."

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Summary. In this article at first, we proved the lemma of the inferior limit and the superior limit. Next, we proved the Baire category theorem (Banach space version) [20], [9], [3], quoted it and proved the uniform boundedness principle. Moreover, the proof of the Banach-Steinhaus theorem is added.

### DOI: 10.1515/forma-2015-0022 degruyter.com/view/j/forma Summable Family in a Commutative Group

"... Summary. Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters ” in Isabelle/HOL [21, 7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of limit of a family indexed by a directed set, or a sequence, in ..."

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Summary. Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters ” in Isabelle/HOL [21, 7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of limit of a family indexed by a directed set, or a sequence, in a metric space [29], a real normed linear space [28] and a linear topological space [14] with the concept of limit of image filter [?]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commu-tative group (“additive notation ” in [16]), using the notion of filters.