The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces

Volume 140 / 1992

M. Randall Holmes Fundamenta Mathematicae 140 (1992), 199-223 DOI: 10.4064/fm-140-3-199-223


This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry. The question of what Banach spaces can be embedded in a linear isometric fashion in this uniquely determined closed linear span of U is investigated.


  • M. Randall HolmesDepartment of Mathematics
    Boise State University
    1910 University Drive
    Boise, Idaho 83725

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