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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Tsirelson space ： ウィキペディア英語版 Tsirelson space In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ''p'' space nor a ''c''0 space can be embedded. The Tsirelson space is reflexive. It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article () where they used the notation ''T'' for the ''dual'' of Tsirelson's example. Today, the letter ''T'' is the standard notation〔see for example , p. 8; , p. 95; the Handbook of the Geometry of Banach Spaces, vol. 1, p. 276; vol. 2, p. 1060, 1649.〕 for the dual of the original example, while the original Tsirelson example is denoted by ''T'' *. In ''T'' * or in ''T'', no subspace is isomorphic, as Banach space, to an ℓp space, 1 ≤ ''p'' < ∞, or to ''c''0. All classical Banach spaces known to , spaces of continuous functions, of differentiable functions or of integrable functions, and all the Banach spaces used in functional analysis for the next forty years, contain some ℓ''p'' or ''c''0. Also, new attempts in the early '70s〔see , .〕 to promote a geometric theory of Banach spaces led to ask 〔The question is formulated explicitly in , , on last page. , p. 95, say that this question was "''a long standing open problem going back to Banach's book''" (), but the question does not appear in Banach's book. However, Banach compares the ''linear dimension'' of ℓ''p'' to that of other classical spaces, a somewhat similar question.〕 whether or not ''every'' infinite-dimensional Banach space has a subspace isomorphic to some ℓ''p'' or to ''c''0. The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Schlumprecht (), on which depend Gowers' solution to Banach's hyperplane problem〔The question is whether every infinite-dimensional Banach space is isomorphic to its hyperplanes. The negative solution is in Gowers, "''A solution to Banach's hyperplane problem''". Bull. London Math. Soc. 26 (1994), 523-530.〕 and the Odell–Schlumprecht solution to the distortion problem. Also, several results of Argyros et al.〔for example, S. Argyros and V. Felouzis, "''Interpolating Hereditarily Indecomposable Banach spaces''", Journal Amer. Math. Soc., 13 (2000), 243–294; S. Argyros and A. Tolias, "''Methods in the theory of hereditarily indecomposable Banach spaces''", Mem. Amer. Math. Soc. 170 (2004), no. 806.〕 are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.〔S. Argyros and R. Haydon constructed a Banach space on which every bounded operator is a compact perturbation of a scalar multiple of the identity, in "''A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem''", Acta Mathematica (2011) 206: 1-54.〕 == Tsirelson's construction == On the vector space ℓ∞ of bounded scalar sequences , let ''P''''n'' denote the linear operator which sets to zero all coordinates ''x''''j'' of ''x'' for which ''j'' ≤ ''n''. A finite sequence $\backslash \_^N$ of vectors in ℓ∞ is called ''block-disjoint'' if there are natural numbers $\backslash textstyle\; \backslash \_^N$ so that , and so that $(x\_n)\_i=0$ when The unit ball  ''B''∞  of ℓ∞ is compact and metrizable for the topology of pointwise convergence (the product topology). The crucial step in the Tsirelson construction is to let ''K'' be the ''smallest'' pointwise closed subset of  ''B''∞  satisfying the following two properties:〔conditions b, c, d here are conditions (3), (2) and (4) respectively in , and a is a modified form of condition (1) from the same article.〕 :a. For every integer  ''j''  in N, the unit vector ''e''''j'' and all multiples $\backslash lambda\; e\_j$, for |λ| ≤ 1, belong to ''K''. :b. For any integer ''N'' ≥ 1, if $\backslash textstyle\; (x\_1,\backslash dots,x\_N)$ is a block-disjoint sequence in ''K'', then $\backslash textstyleP\_N(x\_1\; +\; \backslash cdots\; +\; x\_N)\}$ belongs to ''K''. This set ''K'' satisfies the following stability property: :c. Together with every element ''x'' of ''K'', the set ''K'' contains all vectors ''y'' in ℓ∞ such that |''y''| ≤ |''x''| (for the pointwise comparison). It is then shown that ''K'' is actually a subset of ''c''0, the Banach subspace of ℓ∞ consisting of scalar sequences tending to zero at infinity. This is done by proving that :d: for every element ''x'' in ''K'', there exists an integer ''n'' such that 2 ''P''''n''(''x'') belongs to ''K'', and iterating this fact. Since ''K'' is pointwise compact and contained in ''c''0, it is weakly compact in ''c''0. Let ''V'' be the closed convex hull of ''K'' in ''c''0. It is also a weakly compact set in ''c''0. It is shown that ''V'' satisfies b, c and d. The Tsirelson space ''T'' * is the Banach space whose unit ball is ''V''. The unit vector basis is an unconditional basis for ''T'' * and ''T'' * is reflexive. Therefore, ''T'' * does not contain an isomorphic copy of ''c''0. The other ℓp spaces, 1 ≤ ''p'' < ∞, are ruled out by condition b. 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「Tsirelson space」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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