The Urysohn space embeds in Banach spaces in just one way (Q935251)

\textit{C. H. Dowker} [Duke Math. J. 14, 639--645 (1947; Zbl 0029.42203)] proved in 1947 that the Hilbert space \(\ell_2\) is universal for separable metrizable spaces. This means that every separable metric space can be topologically embedded into \(\ell_2\). Prior to this, in 1925, \textit{P. S. Urysohn} [Bulletin. Sc. Math. (2)51, 43--64, 74-9-0 (1927; JFM 53.0556.01)] constructed a universal separable metric space \(U\) up to isometry, i.e. \(U\) contains an isometric (consequently topological) copy of every separable metric space. Let us point out a more recent result which says that \(U\) is homeomorphic to the Hilbert space \(\ell_2\), cf. \textit{V. Uspenskii} [Topology Appl. 139, 145--149 (2004; Zbl 1062.54036)]. On the other side the space \(C[0,1]\) of continuous real functions with the sup metric is also a universal metric space up to isometry, in fact it is the universal separable Banach space up to linear isometry. So \(U\) contains \(C[0,1]\) up to isometry and \(C[0,1]\) contains \(U\) up to linear isometry. The paper under review treats embeddings of \(U\) into Banach spaces proving the following theorem: The linear closure of an isometric copy of \(U\) in \(C[0,1]\) which contains 0 is a uniquely determined separable Banach space \(\overline{U}\) up to linear isometry (and so, because of the known universality of \(C[0,1]\), the linear closure of an isometric copy of \(U\) containing 0 in any Banach space is uniquely determined up to linear isometry).