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Strictly convex renormings. (English) Zbl 1173.46001

The authors present the following characterization in linear topological terms of the normed spaces which are strictly convex renormable. Let \(X\) be a normed space and \(F\) a subspace of \(X^*\) which is 1-norming for \(X\). Then \(X\) admits an equivalent \(\sigma(X,F)\) lower semicontinuously strictly convex norm if, and only if \(X\) is sigma quasi-diagonal with respect to \((X,\sigma(X,F))\). Here a subset \(M\) of \(X^2\), where \(X\) is a linear topological space, is quasi-diagonal if it is symmetric, and if \(x=y\) whenever \((x,y)\in M\) and \(x,y\in \overline{\text{conv}}(DM)\), where \(D(x,y)=(x+y)/2\). A set \(M\) is sigma quasi-diagonal if it is a countable union of quasi-diagonal sets.
The authors also consider the class of solid Banach lattices made up of bounded real functions on some set \(\Gamma\). They characterize the elements of this class which admit a pointwise strictly convex renorming.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B42 Banach lattices
46E05 Lattices of continuous, differentiable or analytic functions
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