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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset A of D, $(K, d_{A})$ is...

The mapping $ν_{x, y}$ in normed linear spaces with applications to inequalities in analysis.

Dragomir, S.S., Koliha, J.J. (1998)

Journal of Inequalities and Applications [electronic only]

The Menelaus Property characterizes real rotund Banach spaces

J. Valentine (1978)

Fundamenta Mathematicae

The moduli of smoothness and convexity and the Rademacher averages of the trace classes $S_{p}$ (1≤p<∞)

Nicole Tomczak-Jaegermann (1974)

Studia Mathematica

The $n$ -dual space of the space of $p$ -summable sequences

Yosafat E. P. Pangalela, Hendra Gunawan (2013)

Mathematica Bohemica

In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an $n$ -normed space, we are interested in bounded multilinear $n$ -functionals and $n$ -dual spaces. The concept of bounded multilinear $n$ -functionals on an $n$ -normed space was initially intoduced by White (1969), and studied further by Batkunde et al., and Gozali et al. (2010). In this paper, we revisit the definition of bounded multilinear $n$ -functionals, introduce the concept of $n$ -dual spaces, and...

The semi- $M$ property for normed Riesz spaces

Ep De Jonge (1977)

Compositio Mathematica

The topology of the Banach–Mazur compactum

Sergey Antonyan (2000)

Fundamenta Mathematicae

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A \subseteq ℝ^{n}$ , n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let $J_{0} (n)$ be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) $J_{0} (2) / S O (2)$ is an Eilenberg-MacLane space $𝐊 (ℚ, 2)$ ; (4) $B M_{0} (2) = J_{0} (2) / O (2)$ is noncontractible;...

The uniform bounded approximation property with respect to the Haar basis

Tomaszewski, B. (1980)

Abstracta. 8th Winter School on Abstract Analysis

Théorèmes de factorisation dans les espaces $L^{p}$

B. Maurey (1972/1973)

Séminaire Équations aux dérivées partielles (Polytechnique)

Théorèmes de factorisation dans les espaces $L^{p}$ (suite)

B. Maurey (1972/1973)

Séminaire Équations aux dérivées partielles (Polytechnique)

Three-space problems for the approximation property

A. Szankowski (2009)

Journal of the European Mathematical Society

It is shown that there is a subspace $Z_{q}$ of $ℓ_{q}$ for $1 < q < 2$ which is isomorphic to $ℓ_{q}$ such that $ℓ_{q} / Z_{q}$ does not have the approximation property. On the other hand, for $2 < p < \infty$ there is a subspace $Y_{p}$ of $ℓ_{p}$ such that $Y_{p}$ does not have the approximation property (AP) but the quotient space $ℓ_{p} / Y_{p}$ is isomorphic to $ℓ_{p}$ . The result is obtained by defining random “Enflo-Davie spaces” $Y_{p}$ which with full probability fail AP for all $2 < p \leq \infty$ and have AP for all $1 \leq p \leq 2$ . For $1 < p \leq 2$ , $Y_{p}$ are isomorphic to $ℓ_{p}$ .

Topological balls

Michael Barr, Heinrich Kleisli (1999)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

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