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Displaying similar documents to “A bilinear version of Holsztyński's theorem on isometries of C(X)-spaces”

Topology of the isometry group of the Urysohn space

Julien Melleray (2010)

Fundamenta Mathematicae

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Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space ℓ²(ℕ). The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise,...

Linearly rigid metric spaces and the embedding problem

J. Melleray, F. V. Petrov, A. M. Vershik (2008)

Fundamenta Mathematicae

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We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows...

A metric on the space of projections admitting nice isometries

Lajos Molnár, Werner Timmermann (2009)

Studia Mathematica

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Motivated by the concept of separation between propositions in quantum logic, we introduce the so-called separation metric or Santos metric on the space of all projections in a Hilbert space. We show that the resulting metric space has only "nice" surjective isometries. On the nontrivial projections they are all unitarily or antiunitarily equivalent to the identity or to taking the orthogonal complement. We relate this result to Wigner's classical theorem on the form of quantum mechanical...

On continuous surjections from Cantor set.

Félix Cabello Sánchez (2004)

Extracta Mathematicae

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It is a famous result of Alexandroff and Urysohn that every compact metric space is a continuous image of a Cantor set ∆. In this short note we complement this result by showing that a certain uniqueness property holds. Namely, if (K,d) is a compact metric space and f and g are two continuous mappings from ∆ onto K, the, for every e > 0 there exists a homeomorphism phi of ∆ such that d(g(x), f(phi(x))) < e for all x∆.

On metric σ-discrete spaces

Szymon Plewik, Marta Walczyńska (2016)

Banach Center Publications

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By studying dimensional types of metric scattered spaces, we consider the wider class of metric σ-discrete spaces. Applying techniques relevant to this wider class, we present new proofs of some embeddable properties of countable metric spaces in such a way that they can be generalized onto uncountable metric scattered spaces. Related topics are also explored, which gives a few new results.