The generalized Day norm. Part I. Properties

Budzyńska, Monika; Grzesik, Aleksandra; Kot, Mariola

doi:10.17951/a.2017.71.2.33

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Introduction Recently E Maluta Presented a Reflexive Lur2

The Generalized Day Norm and Renorming Of Banach Spaces1

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2017

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“…is an element of c 0 (here the k-th coordinate of x is repeated exactly k times and • p is the standard norm in lp). So we can apply the Day norm |||•||| β,p ( [3]) to the element u(x) and set…”

Section: The Generalized Day Norm and Renorming Of Banach Spacesmentioning

confidence: 99%

“…• L is such a Banach space. Next in [3] the authors introduced the generalized Day norm |||•||| β,p in c 0 and showed its properties. In this paper applying the generalized Day norm, we prove that for each 1 < p,p < ∞ in the Banach space (lp, • p ) there exists an equivalent norm • L,α,β,p,p such that (lp, • L,α,β,p,p ) contains diametrically complete set with empty interior.…”

Section: Introduction Recently E Maluta Presented a Reflexive Lurmentioning

confidence: 99%

“…Throughout this paper all Banach spaces are infinite dimensional and real. We will also use the notations, assumptions and facts from [3]. Additionally, let us recall the following definition.…”

Section: Introduction Recently E Maluta Presented a Reflexive Lurmentioning

confidence: 99%

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The generalized Day norm. Part II. Applications

Budzyńska

Grzesik

Kot

2017

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In this paper we prove that for each \(1< p, \tilde{p} < \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

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Section: The Generalized Day Norm and Renorming Of Banach Spacesmentioning

confidence: 99%

Section: Introduction Recently E Maluta Presented a Reflexive Lurmentioning

confidence: 99%