Let X be a Banach space with a separable dual X * . Let Y ⊂ X be a closed subspace, and f : Y → R a C 1 -smooth function. Then we show there is a C 1 extension of f to X .
Let X be a separable Banach space with a separating polynomial. We show that there exists C ≥ 1 (depending only on X) such that for every Lipschitz function f : X → R, and every ε > 0, there exists a Lipschitz, real analytic function g : X → R such that |f (x) − g(x)| ≤ ε and Lip(g) ≤ CLip(f ). This result is new even in the case when X is a Hilbert space. Furthermore, in the Hilbertian case we also show that C can be assumed to be any number greater than 1.1991 Mathematics Subject Classification. Primary 46B20.
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