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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 and . James proved that ''c''0 and ℓ1 are not distortable. Milman showed that if ''X'' is a Banach space that does not contain an isomorphic copy of ''c''0 or ℓ''p'' for some (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 . 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 that on ℓ1 there are Lipschitz functions which do not stabilize, although this space is not distortable by . 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 ℓp-spaces, 1 < p < ∞, the distortion problem was solved affirmatively by , who showed that ℓ2 is arbitrarily distortable, using the first known arbitrarily distortable space constructed by . ==See also== * Tsirelson space *Banach space 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distortion problem」の詳細全文を読む スポンサード リンク
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