Distortion problem (original) (raw)
In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called λ-distortable if there exists an equivalent norm |x| on X such that, for all infinite-dimensional subspaces Y in X,
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| dbo:abstract | In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called λ-distortable if there exists an equivalent norm |x | on X such that, for all infinite-dimensional subspaces Y in X, (see distortion (mathematics)). Note that every Banach space is trivially 1-distortable. A Banach space is called distortable if it is λ-distortable for some λ > 1 and it is called arbitrarily distortable if it is λ-distortable for any λ. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by and . James proved that c0 and ℓ1 are not distortable. Milman showed that if X is a Banach space that does not contain an isomorphic copy of c0 or ℓp for some 1 ≤ p < ∞ (see sequence space), then some infinite-dimensional subspace of X is distortable. So the distortion problem is now primarily of interest on the spaces ℓp, all of which are separable and uniform convex, for 1 < p < ∞. In separable and uniform convex spaces, distortability is easily seen to be equivalent to the ostensibly more general question of whether or not every real-valued Lipschitz function ƒ defined on the sphere in X stabilizes on the sphere of an infinite dimensional subspace, i.e., whether there is a real number a ∈ R so that for every δ > 0 there is an infinite dimensional subspace Y of X, so that | a − ƒ(y) |
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| dbo:wikiPageWikiLink | dbr:Annals_of_Mathematics dbr:Lipschitz_function dbr:Functional_analysis dbr:Banach_space dbr:Acta_Mathematica dbr:Distortion_(mathematics) dbr:Tsirelson_space dbr:Israel_Journal_of_Mathematics dbr:Geometric_and_Functional_Analysis_(journal) dbr:Hilbert_space dbc:Functional_analysis dbr:Sequence_space dbr:Russian_Mathematical_Surveys dbr:Separable_metric_space | ||
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| rdfs:comment | In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called λ-distortable if there exists an equivalent norm |x | on X such that, for all infinite-dimensional subspaces Y in X, (en) | |
| rdfs:label | Distortion problem (en) | ||
| owl:sameAs | freebase:Distortion problem wikidata:Distortion problem https://global.dbpedia.org/id/4iu7D | ||
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