47 research outputs found
Weakly Lindelof determined Banach spaces not containing
The class of countably intersected families of sets is defined. For any such
family we define a Banach space not containing \ell^{1}(\NN ). Thus we obtain
counterexamples to certain questions related to the heredity problem for W.C.G.
Banach spaces. Among them we give a subspace of a W.C.G. Banach space not
containing \ell^{1}(\NN ) and not being itself a W.C.G. space
Examples of k-iterated spreading models
It is shown that for every and every spreading sequence
that generates a uniformly convex Banach space ,
there exists a uniformly convex Banach space admitting
as a -iterated spreading model, but not as a
-iterated one.Comment: 16 pages, no figure
Complexity of weakly null sequences
We introduce an ordinal index which measures the complexity of a weakly null
sequence, and show that a construction due to J. Schreier can be iterated to
produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in
C(omega^{omega^{alpha}})) with complexity alpha. As in the Schreier example
each of these is a sequence of indicator functions which is a suppression-1
unconditional basic sequence. These sequences are used to construct
Tsirelson-like spaces of large index. We also show that this new ordinal index
is related to the Lavrentiev index of a Baire-1 function and use the index to
sharpen some results of Alspach and Odell on averaging weakly null sequences
Banach Spaces Of The Type Of Tsirelson
To any pair ( M , theta ) where M is a family of finite subsets of N compact
in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type
Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is
greater than n and theta >{1/n} then T_M^theta is reflexive. Moreover, if the
Cantor-Bendixson index of M is greater than omega then T_M^theta does not
contain any l^p, while if the Cantor-Bendixson index of M is finite
thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N :
|A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p