Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces
S. Heinrich, P. Mankiewicz (1982)
Studia Mathematica
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S. Heinrich, P. Mankiewicz (1982)
Studia Mathematica
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J. A. Johnson (1979)
Colloquium Mathematicae
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Nigel J. Kalton (2008)
Revista Matemática Complutense
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A. Jiménez-Vargas, J. Mena-Jurado, R. Nahum, J. Navarro-Pascual (1999)
Studia Mathematica
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Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.
P. Enflo (1975-1976)
Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")
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P. Mankiewicz (1983)
Fundamenta Mathematicae
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Heiko Berninger, Dirk Werner (2003)
Extracta Mathematicae
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C. Finet, W. Schachermayer (1989)
Studia Mathematica
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Catherine Finet (1988)
Studia Mathematica
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G. Godefroy, N. J. Kalton (2003)
Studia Mathematica
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We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y, then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded...
G. Androulakis (1998)
Studia Mathematica
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Let (x_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑_n |x*(x_{n+1} - x_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x_n) which spans Y and...