• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.119-126
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• 2016

In this paper, we study both the numerical index and the polynomial numerical index. First, we give a sufficient condition for a Banach space X to have lushness. Second, we study the relation between the renormings of a Banach space and the k-order polynomial numerical index. This shows that every real Banach spaces of dimension greater that 1 can be renormed to have 2-order polynomial numerical index ${\alpha}$ for any ${\alpha}{\in}[0,1/18)$.