We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which $\parallel {\sum }_{k=1}^n{\xi }_k{\parallel }_E\le C{n}^q$, where ${{\xi }_k}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $exp\left(L_p\right)$, p ≥ 1. We further apply our results to the study of Banach-Saks index sets in...

The Kolmogorov n-diameter of a bounded set B in a non-archimedean normed space, as defined by the first author in a previous paper, is studied in terms of the norms of orthogonal subsets of B with n + 1 points.

In this paper we describe the structure of surjective isometries of the spaces of all absolutely continuous, singular, or discrete probability distribution functions on R equipped with the Kolmogorov-Smirnov metric. We also study the structure of affine automorphisms of the space of all distribution functions.

In this paper, we show the representation of Köthe dual of Banach sequence spaces ${\ell }_p\left[X\right]$$(1\le p<\infty )...$

We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited${}^{*}$ and the $p$-(SR${}^{*}$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited${}^{*}$ and $p$-Right${}^{*}$ subsets. The $p$-$L$-limited${}^{*}$ property is studied in some spaces of operators.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum, while ${\ell }_{\infty }$...

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X=T\left[{(\theta ₙ,ₙ)}_{n=1}^{\infty }\right]$: (1) Every block subspace of X contains an $\ell ¹{-}_{\omega }$-spreading model, (2) The Bourgain ℓ¹-index $I_b\left(Y\right)=I\left(Y\right)>{\omega }^{\omega }$ for any block subspace Y of X, (3) $limₘlimsupₙ{\theta }_{m+n}/\theta ₙ>0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...