Tsirelson space

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A specific example of a reflexive Banach space (cf. Reflexive space) which does not contain an imbedded $l_p$-space or an imbedded $c_0$-space. On the other hand, the classical Banach spaces, such as the spaces $L_p(\mu)=L_p(\Omega,\Sigma,\mu)$ of equivalence classes of measurable functions whose $p$-th powers are integrable and the spaces $C(K)$ of continuous scalar-valued functions on $K$ with the supremum norm, all do contain a copy of $c_0$ or $l_p$, and so do all Orlicz spaces (cf. Orlicz space).

For a selection of results concerning Banach spaces which do contain $l_p$ or $c_0$ see [a3], Sect. 2e.


[a1] B.S. Tsirelson, "Not every Banach space contains an imbedding of $l_p$ or $c_0$" Funct. Anal. Appl. , 8 : 2 (1974) pp. 138–141 Funkts. Anal. Prilozhen. , 8 : 2 (1974) pp. 57–60
[a2] P.G. Casazza, Th.J. Shura, "Tsirelson's space" , Lect. notes in math. , 1363 , Springer (1989)
[a3] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977)
[a4] D. van Dulst, "Characterizations of Banach spaces not containing $L^1$" , CWI (1989)
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