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In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces $X$ whose asymptotic structures are closely related to $c_0$ and do not contain a copy of $\ell_1$: i) Suppose $X$ has a normalized weakly null basis $(x_i)$ and every spreading model $(e_i)$ of a normalized weakly null block basis satisfies $\|e_1-e_2\|=1$. Then some subsequence of $(x_i)$ is equivalent to the unit vector basis of $c_0$. This generalizes a similar theorem of Odell and Schlumprecht, and yields a new proof of the Elton-Odell theorem on the existence of infinite $(1+\varepsilon)$-separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of $X$ generated by weakly null arrays are equivalent to the unit vector basis of $c_0$. Then $X^*$ is separable and $X$ is asymptotic-$c_0$ with respect to a shrinking basis $(y_i)$ of $Y\supseteq X$., Comment: 25 pages

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given $\vare>0$, so that the basis becomes $(1+\vare)$-democratic, and hence $(2+\vare)$-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is $(1+\vare)$-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1

Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $\lambda>0$ and an infinite sequence $(x_i)_{i=1}^\infty\subset X$ such that $\|x_i-x_j\|=\lambda$ for all $i\neq j$., Comment: 11 pages

We prove that a sequence $(f_i)_{i=1}^\infty$ of translates of a fixed $f\in L_p(R)$ cannot be an unconditional basis of $L_p(R)$ for any $1\le p<\infty$. In contrast to this, for every $2

We prove thatthe Banach space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq p \leq\infty$ and $1

Let $1\le p <\infty$, $f\in L_p(\real)$ and $\Lambda\subseteq \real$. We consider the closed subspace of $L_p(\real)$, $X_p (f,\Lambda)$, generated by the set of translations $f_{(\lambda)}$ of $f$ by $\lambda \in\Lambda$. If $p=1$ and $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is a bounded minimal system in $L_1(\real)$, we prove that $X_1 (f,\Lambda)$ embeds almost isometrically into $\ell_1$. If $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is an unconditional basic sequence in $L_p(\real)$, then $\{f_{(\lambda)} : \lambda\in\Lambda\}$ is equivalent to the unit vector basis of $\ell_p$ for $1\le p\le 2$ and $X_p (f,\Lambda)$ embeds into $\ell_p$ if $24$, there exists $f\in L_p(\real)$ and $\Lambda \subseteq \zed$ so that $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is unconditional basic and $L_p(\real)$ embeds isomorphically into $X_p (f,\Lambda)$.

We consider the $X$-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in $L_p[0,1]$ ($1 < p < \infty$) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $\omega^{\alpha\omega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.

We prove that if $X$ is a subspace of $L_p$ $(2

Let $(e_i)$ be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces., Comment: 33 pages

This paper deals with the following types of problems: Assume a Banach space $X$ has some property (P). Can it be embedded into some Banach space $Z$ with a finite dimensional decomposition having property (P), or more generally, having a property related to (P)? Secondly, given a class of Banach spaces, does there exist a Banach space in this class, or in a closely related one, which is universal for this class?, Comment: 26 pages

We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p., Comment: 12 pages, AMSLaTeX

Let (e_i) be a dictionary for a separable Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a `finite alphabet'. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c_0., Comment: LaTeX, 28 pages

We prove that if X is a separable, reflexive space which is asymptotic l_p, then X embeds into a reflexive space Z having an asymptotic l_p finite-dimensional decomposition. This result leads to an intrinsic characterization of subspaces of spaces with an asymptotic l_p FDD. More general results of this type are also obtained., Comment: 32 pages

J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional which is a weak form of unconditionality. The notion of near-unconditionality is quantified by a constant $K(\delta)$ depending on a parameter $\delta \in (0,1]$. It is unknown if $\sup_{\delta>0} K(\delta) < \infty$. This problem turns out to be closely related to the question whether every infinite-dimensional Banach space contains a quasi-greedy basic sequence. The notion of a quasi-greedy basic sequence was introduced recently by S. V. Konyagin and V. N. Temlyakov. We present an extension of Elton's result which includes Schreier unconditionality. The proof involves a basic framework which we show can be also employed to prove other partial unconditionality results including that of convex unconditionality due to Argyros, Mercourakis and Tsarpalias. Various constants of partial unconditionality are defined and we investigate the relationships between them. We also explore the combinatorial problem underlying the $\sup_{\delta>0} K(\delta) < \infty$ problem and show that $\sup_{\delta>0} K(\delta) > 5/4$., Comment: 50 pages

We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semi-lattices and countable lattices, and all finite lattices.

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c_0 then X is w.a.s.. We obtain an analogous result if c_0 is replaced by ell_1 and also show it is false if c_0 is replaced by ell_p, 1 < p < infinity. We prove that if 1 less than or equal p < infinity and the norm of the sum of (x_i)_1^n is of the order n^{1/p} for all (x_i)_1^n in the n^{th} asymptotic structure of $X$, then X contains an asymptotic ell_p, hence w.a.s. subspace., Comment: 22 pages, AMSLaTeX

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves a problem of J. Bourgain. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block $q$-Hilbertian and/or a block $p$-Besselian finite dimensional decomposition., Comment: v.2 (3 January 2007) 13 pages, amslatex

The hierarchy of the block bases of transfinite normalized averages of a normalized Schauder basic sequence is introduced and a criterion is given for a normalized weakly null sequence in C(K), the Banach space of scalar valued functions continuous on the compact metric space K, to admit a block basis of normalized averages equivalent to the unit vector basis of c_0, the Banach space of null scalar sequences. As an application of this criterion, it is shown that every normalized weakly null sequence in C(K), for countable K, admits a block basis of normalized averages equivalent to the unit vector basis of c_0., Comment: 36 pages

We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c_0. We call X elastic if for some K < infinity for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K < infinity every isomorph of X is K-elastic then X is finite dimensional., Comment: AMSLaTeX, 13 pages

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$ but none of them is isomorphic to $\ell_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha < \omega_1$, a spreading model $(\tilde x_i^\alpha)_i$ such that if $\alpha < \beta$ then $(\tilde x_i^\alpha)_i$ is dominated by (and not equivalent to) $(\tilde x_i^\beta)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.