Not every Banach space contains an imbedding ofl p or c0

Unconditionally converging polynomials on Banach spaces

“…; hence P ∈ P uc . ✷ It has been shown [1] that a Banach space E such that P( k E, K) ≡ P( k E) is reflexive for every k ∈ N has many of the properties of Tsirelson's space T * [17]. In fact, E must be reflexive, and the dual space E * cannot contain copies of ℓ p (1 < p < ∞).…”

Book Review: Classical sequences in Banach spaces

“…First,Ryszard Komorowski and Nicole Tomczak-Jaegermann [39] showed that a homogeneous Banach space which contains an unconditional basic sequence is isomorphic to a Hubert space. Next, W. T. Gowers [38] proved that a Banach space which contains no unconditional basic sequences must contain a hereditarily indecomposable subspace.…”

Some remarks on spreading models and mixed Tsirelson spaces

“…The first non-trivial example of an asymptotic 1 space was discovered by Tsirelson [17]. Recent results [6], [7], [15], have shown the necessity of studying the higher ordinal structure of an asymptotic 1 Banach space in order to obtain results on the global structure of its infinite dimensional subspaces.…”

“…The key point for the proof of this theorem is to produce for every normalized block sequence (x n ) of the basis, a vector in the linear span of (x n ), whose norm is arbitrarily small yet its support with respect to (x n ) belongs to S ω2 . The dual of the original Tsirelson's space [17] contains no ω 1 spreading model. This is due to the fact that every block sequence is equivalent to a subsequence of the basis.…”