Gruenhage compacta and strictly convex dual norms |
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Authors: | Richard J Smith |
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Institution: | aQueens' College, Cambridge, CB3 9ET, UK |
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Abstract: | We prove that if K is a Gruenhage compact space then admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and , where K is a Gruenhage compact in the w*-topology and |||||| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if is a tree, then admits an equivalent, strictly convex dual norm if and only if is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images. |
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Keywords: | Gruenhage space Rotund Strictly convex Norm Tree Renorming theory Perfect image Continuous image |
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