A remark on the ball-covering property of product spaces

Abstract Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

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Fixed point iterations for Prešić-Kannan nonexpansive mappings in product convex metric spaces

Acta Universitatis Sapientiae Mathematica ◽

10.2478/ausm-2018-0005 ◽

2018 ◽

Vol 10 (1) ◽

pp. 56-69

Author(s):

Hafiz Fukhar-ud-din ◽

Vasile Berinde

Keyword(s):

Fixed Point ◽

Banach Spaces ◽

Hilbert Spaces ◽

Metric Spaces ◽

Unique Fixed Point ◽

Nonexpansive Mappings ◽

Uniformly Convex ◽

Product Spaces ◽

Convex Metric Spaces ◽

Fixed Point Iterations

Abstract We introduce Prešić-Kannan nonexpansive mappings on the product spaces and show that they have a unique fixed point in uniformly convex metric spaces. Moreover, we approximate this fixed point by Mann iterations. Our results are new in the literature and are valid in Hilbert spaces, CAT(0) spaces and Banach spaces simultaneously.

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DAUGAVET PROPERTY IN TENSOR PRODUCT SPACES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474801900063x ◽

2019 ◽

pp. 1-20 ◽

Cited By ~ 2

Author(s):

Abraham Rueda Zoca ◽

Pedro Tradacete ◽

Ignacio Villanueva

Keyword(s):

Tensor Product ◽

Banach Spaces ◽

Tensor Products ◽

Symmetric Tensor ◽

Continuous Functions ◽

Product Spaces ◽

Projective Tensor ◽

Spaces Of Continuous Functions ◽

Daugavet Property

We study the Daugavet property in tensor products of Banach spaces. We show that $L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$ has the Daugavet property when $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$ has the Daugavet property provided $X$ and $Y$ are $L_{1}$ -preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

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