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A comparison of two notions of porosity

100%
Strobin F.
Commentationes Mathematicae
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2008
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tom 48
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nr 2
EN
In the paper we compare two notions of porosity: the R-ball porosity \((R \rt 0)\) defined by Preiss and Zajı́ček, and the porosity which was introduced by Olevskii (here it will be called the O-porosity). We find this comparison interesting since in the literature there are two similar results concerning these two notions. We restrict our discussion to normed linear spaces since the R-ball porosity was originally defined in such spaces.
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Topological Properties of Real Normed Space

100%
Nakasho K., Futa Y., Shidama Y.
Formalized Mathematics
|
2014
|
tom 22
|
nr 3
209-223
EN
In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. 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 convergence 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).
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Separability of Real Normed Spaces and Its Basic Properties

100%
Nakasho K., Endou N.
Formalized Mathematics
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2015
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tom 23
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nr 1
59-65
EN
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 [34], and also referred to [7], [14] and [16].
4
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Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

63%
Klee V., Veselý L., Zanco C.
Studia Mathematica
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1996
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tom 120
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nr 3
191-204
EN
For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class of all smooth bodies in X is stable with respect to both ∓ and γ̅. In our paper it is shown that when X is separable, these stability properties of rotundity (resp. smoothness) are actually equivalent to the reflexivity of X. The characterizations remain valid for each nonseparable X that contains a rotund (resp. smooth) body.
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