The unconditional basic sequence problem

无条件与对称序列空间的$2$-圆范数

$2$-rotund norms for unconditional and symmetric sequence spaces

该研究证明带无条件基的自反Banach空间有等价1-无条件2R范数且可嵌入带1-对称2R范数的自反空间，还得到对称基空间1-对称2R重赋范的部分结果。

Unconditional structures of weakly null sequences

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of c, the Banach space of null sequences under the supremum norm, or there exists a subsequence which is boundedly convexly complete. This result generalizes J. Elton's dichotomy on weakly null sequences.

Uniqueness of unconditional bases in Banach spaces

We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have the unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis.

On Schreier unconditional sequences

Let $(x_n)$ be a normalized weakly null sequence in a Banach space and let $\varep>0$. We show that there exists a subsequence $(y_n)$ with the following property: $$\hbox{ if }\ (a_i)\subseteq \IR\ \hbox{ and }\ F\subseteq \nat$$ satisfies $\min F\le |F|$ then $$\big\|\sum_{i\in F} a_i y_i\big\| \le (2+\varep) \big\| \sum a_iy_i\big\|\ . $$

Erratum to: ``Banach spaces without local unconditional structure''

This note contains a corrected proof of the main result (which remains unchanged) from [K-T]. It was recently observed that an argument in a basic technical criterium has a gap.

Unconditional and bimonotone structures in high density Banach spaces

It is shown that every normalized weakly null sequence of length $κ_λ$ in a Banach space has a subsequence of length $λ$ which is an unconditional basic sequence; here $κ_λ$ is a large cardinal depending on a given infinite cardinal $λ$. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Mark…

Unconditionally saturated Banach space with the scalar-plus-compact property

We construct a Bourgain-Delbaen $\mathscr{L}_\infty$-space $\mathfrak{X}_{Kus}$ with strongly heterogenous structure: any bounded operator on $\mathfrak{X}_{Kus}$ is a compact perturbation of a multiple of the identity, whereas the space $\mathfrak{X}_{Kus}$ is saturated with unconditional basic sequences.

An Indecomposable and unconditionally saturated Banach space

We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T\in \mathcal{B}(X_{ius})$ is of the form $λI+S$ with $S$ a strictly singular operator.

On a Loomis-Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies

For a permutationally invariant unconditional convex body K in R^n we define a finite sequence (K_j), j = 1, ..., n of projections of the body K to the space spanned by first j vectors of the standard basis of R^n. We prove that the sequence of volumes (|K_1|, ..., |K_n|) is log-concave.

Unconditional structures of translates for $L_p(R^d)$

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 $2<p<\infty$, $d\in N$ and unbounded sequence $(λ_n)_{n\in N}\subset R^d$ we establish the existence of a function $f\in L_p(R^d)$ and sequence $(g^*_n)_{n\in N}\subset L_p^*(R^d)$ such that $(T_{λ_n} f, g^*_n)_{n\in N}$ forms an unconditional Schauder frame for $L_p(R^d)$. In particular, there exists a Schauder frame of integer translates for $L_p(R)$ if (and only if) $2<p<\infty$.

Unconditionality of orthogonal spline systems in $L^p$

Given any natural number $k$ and any dense point sequence $(t_n)$, we prove that the corresponding orthonormal spline system of order $k$ is an unconditional basis in reflexive $L^p$.

On quotients of Banach spaces having shrinking unconditional bases

It is proved that if a Banach space $Y$ is a quotient of a Banach space having a shrinking unconditional basis, then every normalized weakly null sequence in $Y$ has an unconditional subsequence. The proof yields the corollary that every quotient of Schreier's space is $c_o$-saturated.

Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits us to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from [F. Albiac and C. Leránoz, Uniqueness of unconditional bases in nonlocally convex $\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2, 394--401] we show that if $X$ is quasi-Banach space with a str…

Unconditionality of Periodic Orthonormal Spline Systems in $L^p$

Given any natural number $k$ and any dense point sequence $(t_n)$ on the torus $\mathbb T$, we prove that the corresponding periodic orthonormal spline system of order $k$ is an unconditional basis in $L^p$ for $1<p<\infty$.

The Unconditional Constants for Hilbert Space Frame Expansions

The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by $\sqrt{\frac{B}{A}}$, where $A,B$ are the frame bounds of the frame. Tight frames thu…

Examples of asymptotically \ell_^1 Banach spaces

Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers.

Unconditional convergence and invertibility of multipliers

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when…

Unconditional Frames of Translates in $L_p(\mathbb{R}^d)$

We show that, for $1<p \le 2$, the space $L_p(\mathbb{R}^d)$ does not admit unconditional Schauder frames $\left\lbrace f_i,f_i'\right\rbrace_{i\in\mathbb{N}}$ where $\left\lbrace f_i\right\rbrace$ is a sequence of translates of finitely many functions and $\left\lbrace f_i'\right\rbrace$ is seminormalized. In fact, the only subspaces of $L_p(\mathbb{R}^d)$ admitting such Banach frames are those isomorphic to $\ell_p$. On the other hand, if $2<p<+\infty$ and $\left\lbrace λ_{i}\right\rbrace_{i\in\mathbb{N}}\subseteq \mathbb{R}^d$ is an unbounded sequence, there is a subsequence $\left\lbrace λ…

1-complemented subspaces of spaces with 1-unconditional bases

We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or …

A study of conditional spreading sequences

It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models. Among other things, it is shown that the space $C(ω^ω)$ is universal for all spreading models, i.e., it admits all spreading sequences, both conditional and unconditional, as spreading models…

Asymptotic Unconditionality

We show that a separable real Banach space embeds almost isometrically in a space $Y$ with a shrinking 1-unconditional basis if and only if $\lim_{n \to \infty} \|x^* + x_n^*\| = \lim_{n \to \infty} \|x^* - x_n^*\|$ whenever $x^* \in X^*$, $(x_n^*)$ is a weak$^*$-null sequence and both limits exist. If $X$ is reflexive then $Y$ can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.

A universal Banach space with a $K$-unconditional basis

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $…

A $c\_0$-saturated Banach space with no long unconditional basic sequences

We present a Banach space $\mathfrak X$ with a Schauder basis of length $ω\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak X$ have ``few operators'' in the sense that every bounded operator $T:X \to \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $ω\_1$-sin…

Banach actions preserving unconditional convergence

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\inω}$ in $A$ and unconditionally convergent series $\sum_{n\inω}x_n$ in $X$ the series $\sum_{n\inω}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining …

Lower estimates of random unconditional constants of Walsh-Paley martingales with values in banach spaces

For a Banach space X we define RUMD_n(X) to be the infimum of all c>0 such that (AVE_{ε_k =\pm 1} || \sum_1^n epsilon_k (M_k - M_{k-1} )||_{L_2^X}^2 )^{1/2} <= c || M_n ||_{L_2^X} holds for all Walsh-Paley martingales {M_k}_0^n subset L_2^X with M_0 =0. We relate the asymptotic behaviour of the sequence {RUMD(X)}_{n=1}^{infinity} to geometrical properties of the Banach space X such as K-convexity and superreflexivity.

Complemented basic sequences in Frechet spaces with finite dimensional decomposition

Let $E$ be a Frechet-Montel space and $(E_n)_{n \in \mathbb{N}}$ be a finite dimensional unconditional decomposition of $E$ with $\dim(E_n)\leq k$ for some fixed $k \in \mathbb{N}$ and for all $n \in \mathbb{N}$. Consider a sequence $(x_n)_{n \in \mathbb{N}}$ formed by taking an element $x_n$ from each $E_n$ for all $n \in \mathbb{N}$. Then $(x_n)_{n \in \mathbb{N}}$ has a subsequence which is complemented in $E$

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $\om_n$ without unconditional subsequences. This together with a result of \cite{Do-Lo-To} shows that $\om_ω$ is the minimal cardinal $κ$ that could possibly have the property that every weakly null $κ$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $\ka$ which is smaller than the first $\om$-Erdös cardina…

Contractive projections and isometries in sequence spaces

We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.

具有无条件基的巴拿赫空间上全纯函数的单项式基

Monomial bases for holomorphic functions on Banach spaces with an unconditional basis

本文针对具有无条件基的复巴拿赫空间，证明特定排序的单项式构成其带紧开拓扑的整全纯函数空间的Schauder基，并将结果应用于多类巴拿赫序列格与向量值E-和空间。

On the number of non permutatively equivalent sequences in a Banach space

This paper contains results concerning the Borel reduction of the relation $E_0$ of eventual agreement between sequences of 0's and 1's, to the relation of permutative equivalence between basic sequences in a Banach space. For more clarity in this abstract, we state these results in terms of classification by real numbers. If $R$ is some (analytic) equivalence relation on a Polish space $X$, it is said that $R$ is classifiable (by real numbers) if there exists a Borel map $g$ from $X$ into the real line such that $x R x'$ if and only if $g(x)=g(x')$. If $R$ is not classifiable, there must be $…

关于具有无条件Schauder基的Banach空间中FPP的注记

Remarks on the FPP in Banach spaces with unconditional Schauder basis

本文研究带Schauder基的Banach空间中的FPP与弱FPP，证明含等价于c₀单位基的预单调基本序列的空间无FPP，给出弱FPP的距离不变性及基相关充分条件。

The basic sequence problem

We construct a quasi-Banach space $X$ which contains no basic sequence.

Systems formed by translates of one element in $L_p(\mathbb R)$

Let $1\le p <\infty$, $f\in L_p(\real)$ and $Λ\subseteq \real$. We consider the closed subspace of $L_p(\real)$, $X_p (f,Λ)$, generated by the set of translations $f_{(λ)}$ of $f$ by $λ\inΛ$. If $p=1$ and $\{f_{(λ)} :λ\inΛ\}$ is a bounded minimal system in $L_1(\real)$, we prove that $X_1 (f,Λ)$ embeds almost isometrically into $\ell_1$. If $\{f_{(λ)} :λ\inΛ\}$ is an unconditional basic sequence in $L_p(\real)$, then $\{f_{(λ)} : λ\inΛ\}$ is equivalent to the unit vector basis of $\ell_p$ for $1\le p\le 2$ and $X_p (f,Λ)$ embeds into $\ell_p$ if $2<p\le 4$. If $p>4$, there exists $f\in L_p(\re…

Some combinatorial principles for trees and applications to tree-families in Banach spaces

Suppose that $(x_s)_{s\in S}$ is a normalized family in a Banach space indexed by the dyadic tree $S$. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain $β$ of $S$ the sequence $(x_s)_{s\inβ}$ is weakly null, we prove that there exists a subtree $T$ of $S$ such that for any infinite chain $β$ of $T$ the sequence $(x_s)_{s\inβ}$ is nearly (resp., convexly) unconditional. In the case where $(f_s)_{s\in S}$ is a family of continuous functions, under some additional assumptions, we …

Isometries of cross products of sequence spaces

Let $X_0, X_1, ..., X_k$ with $k \in \IN\cup\{\infty\}$ be sequence spaces $($finite or infinite dimensional$)$ over $\IC$ or $\IR$ with absolute norms $N_i$ for $i = 0, ..., k$, $($i.e., with 1-unconditional bases$)$ such that $\dim X_0 = k$. Define an absolute norm on the cross product space $($also known as the $X_0$ 1-unconditional sum$)$ $X_1 \times ... \times X_k$ by $$ N(x_1, ..., x_k) = N_0(N_1(x_1), ..., N_k(x_k)) \quad \hbox{for all} \quad (x_1, ..., x_k) \in X_1 \times ... \times X_k. $$ We show that every sequence space with an absolute norm has an intrinsic cross product structure…

超交替达戈维特性质、无条件基与 SCD 几何

The super Alternative Daugavet property, unconditional bases and SCD geometry

回答关于具 1 - 无条件基且满足超交替达戈维特性质的无限维巴拿赫空间存在性问题，处理相关弱拓扑结构等问题，证明具特定基的巴拿赫空间子集性质及存在单位球非切片可数确定的巴拿赫空间。

Summing inclusion maps between symmetric sequence spaces

We prove a substantial extension of a well-known result due to Bennett and Carl: The inclusion of a 2-concave symmetric Banach sequence space E into l_2 is (E,1)-summing, i.e. for every unconditionally summable sequence (x_n) in E the scalar sequence (||x_n||_2) is contained in E. Various applications are given, e.g. to the theory of eigenvalue distribution of compact operators and approximation theory.

Banach spaces with many boundedly complete basic sequences failing PCP

We prove that there exist Banach spaces not containing $\ell_1$, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of the Remark 2 in H. P. Rosenthal. "Boundedly complete weak-Cauchy sequences in Banach spaces with PCP." J. Funct. Anal. 253 (2007) 772-781.

每个半规范无条件Schauder框架在Hilbert空间中都包含一个框架

Every semi-normalized unconditional Schauder frame in Hilbert spaces contains a frame

本文证明了每个半规范无条件Schauder框架在Hilbert空间中都包含一个框架，并探讨了无条件Schauder框架的存在性问题。

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that t…

A note on asymptotically monotone basic sequences and well-separated sets

We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unit sphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.

具有许多好块基本序列的 Schauder 基

Schauder Bases Having Many Good Block Basic Sequences

本文研究 Schauder 基中好块基本序列的丰富性，刻画块树分支皆好与空间为 $1$-渐近 $\ell_p$ 的等价性，并借助 Hindman-Milliken-Taylor 定理证明常系数块基本序列的稳定化结果。

Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space

If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional ba…

Extracting long basic sequences from systems of dispersed vectors

We study Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces. These properties are much weaker than WCG and closely related to Corson's property (C). Given a transfinite sequence of normalized vectors, which is dispersed or null in some sense, we extract a subsequence which is a biorthogonal sequence, or even a weakly null monotone basic sequence, depending on the setting. The Separable Complementation Property is established for spaces with an M-basis under rather weak geometric properties. We also consider a…

关于变指数勒贝格空间基本序列结构的研究

On the basic sequence structure of variable exponent Lebesgue spaces

本文研究了变指数勒贝格空间中子对称基本序列的结构，证明了在特定条件下此类序列与$\ell_r$标准基等价。

Tirilman 空间的对偶具有唯一的次对称基序列

Duals of Tirilman spaces have unique subsymmetric basic sequences

本文研究了 Tirilman 空间 $Ti(p,γ)$ 的对偶空间，证明了其中所有次对称基序列均等价于其典范的次对称但非对称基。

Macphail's Theorem revisited

In 1947, M. S. Macphail constructed a series in $\ell_{1}$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach Space Theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky--Rogers Theorem asserts that in every infinite-dimensional Banach space $E$ there exists an unconditionally convergent series ${\textstyle\sum}x^{(j)}$ such that ${…

Sequence space representations for spaces of smooth functions and distributions via Wilson bases

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.

Two characterizations of the standard unit vector basis of $l_1$

We show that for a sequence in a Banach space, the property of being stable under large perturbations characterizes the property of being equivalent to the unit vector basis of $l_1$. We show that a normalized unconditional basic sequence in $l_1$ that is semi-normalized in $l_\infty$ is equivalent to the standard unit vector basis of~$l_1$.

Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's $(A_p)$ condition

We describe the complete interpolating sequences for the Paley-Wiener spaces $L^p_π$ ($1<p<\infty$) in terms of Muckenhoupt's $(A_p)$ condition. For $p=2$, this description coincides with those given by Pavlov (1979), Nikol'skii (1980), and Minkin (1992) of the unconditional bases of complex exponentials in $L^2(-π,π)$. While the techniques of these authors are linked to the Hilbert space geometry of $L^2_π$, our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted $L^p$ spaces of functions and sequences.

Diagonals of injective tensor products of Banach lattices with bases

In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.

Quantifying shrinking and boundedly complete bases

We investigate possible quantifications of R. C. James' classical work on bases and reflexivity of Banach spaces. By introducing new quantities measuring how far a basic sequence is from being shrinking and/or boundedly complete, we prove quantitative versions of James' famous characterisations of reflexivity in terms of bases. Furthermore, we establish quantitative versions of James' characterisations of reflexivity of Banach spaces with unconditional bases.

Matrix Subspaces of $L_1$

If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r<p\le 2$, then the space of matrices $\{a_{i,j}\}$ with norm $\|\{a_{i,j}\}\|_{E(F)}=\big\|\sum_k \|\sum_l a_{k,l}f_l\|e_k\big\|$ embeds into $L_1$. This generalizes a recent result of Prochno and Schütt.

Basic sequences and spaceability in $\ell_p$ spaces

Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p \in [1,\infty]$) $Z(\ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(\ell_\infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X \setminus Z(X)$ and its algebraic genericity.