The generalized Day norm. Part II. Applications

Monika Budzyńska, Aleksandra Grzesik, Mariola Kot

Abstract


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.

Keywords


Diametrically complete set; Day norm, fixed point; Kadec-Klee property; LUR space; nonexpansive mapping; non-strict Opial property; 1-unconditional Schauder bases

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References


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DOI: http://dx.doi.org/10.17951/a.2017.71.2.51
Date of publication: 2017-12-18 20:31:33
Date of submission: 2017-12-16 23:00:58


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