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The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space. (English) Zbl 1107.46302

Summary: Let \(E\) and \(F\) be Hilbert spaces with unit spheres \(S_1(E)\) and \(S_1(F)\). Suppose that \(V_0S_1(E)\to S_1(F)\) is a Lipschitz mapping with Lipschitz constant \(k=1\) such that \(-V_0[S_1(E)]\subset V_0 [S_1 (E)]\). Then \(V_0\) can be extended to a real linear isometric mapping \(V\) from \(E\) into \(F\). In particular, every isometric mapping from \(S_1(E)\) onto \(S_1(F)\) can be extended to a real linear isometric mapping from \(E\) onto \(F\).

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B04 Isometric theory of Banach spaces
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