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 c0 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 (Figiel & Johnson (1974)) where they used the notation T for the dual of Tsirelson's example. Today, the letter T is the standard notation[1] 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 ℓ p space, 1 ≤ p < ∞, or to c0.

All classical Banach spaces known to Banach (1932), 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 c0. Also, new attempts in the early '70s[2] to promote a geometric theory of Banach spaces led to ask [3] whether or not every infinite-dimensional Banach space has a subspace isomorphic to some ℓ p or to c0. Moreover, it was shown by Baudier, Lancien, and Schlumprecht that ℓ p and c0 do not even coarsely embed into T*.

The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Thomas Schlumprecht (Schlumprecht (1991)), on which depend Gowers' solution to Banach's hyperplane problem[4] and the Odell–Schlumprecht solution to the distortion problem. Also, several results of Argyros et al.[5] are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.[6]

  1. ^ see for example Casazza & Shura (1989), p. 8; Lindenstrauss & Tzafriri (1977), p. 95; The Handbook of the Geometry of Banach Spaces, vol. 1, p. 276; vol. 2, p. 1060, 1649.
  2. ^ see Lindenstrauss (1970), Milman (1970).
  3. ^ The question is formulated explicitly in Lindenstrauss (1970), Milman (1970), Lindenstrauss (1971) on last page. Lindenstrauss & Tzafriri (1977), p. 95, say that this question was "a long standing open problem going back to Banach's book" (Banach (1932)), 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.