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Smooth partitions of unity in Banach spaces. (English) Zbl 0853.46015
Summary: We show that if a Banach space \(X\) has an LUR norm, and if every Lipschitz convex function on \(X\) can be approximated by \(C^k\)-smooth functions, then \(X\) admits \(C^k\)-smooth partitions of unity, and thus every continuous function on \(X\) is a uniform limit of \(C^k\)-smooth functions.

MSC:
46B20 Geometry and structure of normed linear spaces
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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