“Distortion problem”的意思、由来-开放百科全书

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Distortion problem

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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 {{harvtxt|James|1964}} and {{harvtxt|Milman|1971}}.

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 ℓ^p for some {{nowrap|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 ℓ^p, all of which are separable and uniform convex, for {{nowrap|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)| < δ, for all y ∈ Y, with ||y|| = 1. But it follows from the result of {{harvtxt|Odell|Schlumprecht|1994}} that on ℓ¹ there are Lipschitz functions which do not stabilize, although this space is not distortable by {{harvtxt|James |1964}}. In a separable Hilbert space, the distortion problem is equivalent to the question of whether there exist subsets of the unit sphere separated by a positive distance and yet intersect every infinite-dimensional closed subspace. Unlike many properties of Banach spaces, the distortion problem seems to be as difficult on Hilbert spaces as on other Banach spaces. On a separable Hilbert space, and for the other ℓ^p-spaces, 1 < p < ∞, the distortion problem was solved affirmatively by {{harvtxt|Odell|Schlumprecht|1994}}, who showed that ℓ² is arbitrarily distortable, using the first known arbitrarily distortable space constructed by

{{harvtxt|Schlumprecht|1991}}.

References

{{citation |title=Uniformly nonsquare Banach spaces |first=R.C. |last=James |journal=Annals of Mathematics |year=1964 |volume=80 |issue=2 |pages=542–550 |doi=10.2307/1970663}}.
{{citation |last=Milman |title=Geometry of Banach spaces II, geometry of the unit sphere |journal=Russian Mathematical Surveys |year=1971 |volume=26 |pages=79–163|bibcode=1971RuMaS..26...79M |doi=10.1070/RM1971v026n06ABEH001273 }}.
{{citation |chapter=Distortion and asymptotic structure |first1=E |last1=Odell |first2=Th. |last2=Schlumprecht |title=Handbook of the geometry of Banach spaces, Volume 2 |editors=Johnson, Lindenstrauss |publisher=Elsevier |year=2003 |isbn=978-0-444-51305-2}}.
{{Citation | last1=Odell | first1=E. | last2=Schlumprecht | first2=Th. | title=The distortion problem of Hilbert space | mr=1209302 | year=1993 | journal= Geom. Funct. Anal. | issn=1016-443X | volume=3 | pages=201–207 | doi=10.1007/BF01896023}}.
{{Citation | last1=Odell | first1=E. | last2=Schlumprecht | first2=Th. | title=The distortion problem | mr=1301394 | year=1994 | journal= Acta Mathematica | issn=0001-5962 | volume=173 | pages=259–281 | doi=10.1007/BF02398436}}.
{{Citation | title= An arbitrary distortable Banach space | first= Th. |last= Schlumprecht| mr=1177333 | year=1991 | journal= Israel Journal of Mathematics | issn=0021-2172 | volume=76 | pages=81–95 | doi=10.1007/bf02782845| arxiv=math/9201225 }}.

1 : Functional analysis

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