The basic sequence problem

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。

函数空间中的无条件基序列及其在 Orlicz 空间中的应用

Unconditional basic sequences in function spaces with applications to Orlicz spaces

本文研究了函数空间 $\bf{L}$ 的无条件基序列，提出了一个二分法条件，使得其互补子空间的无条件基要么等价于 $\ell_2$ 的单位向量系，要么具有不相交支撑的子基，并据此分类了对称基序列，应用于 Orlicz 空间。

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 $E$

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.

Unconditional basic sequences in spaces of large density

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\aleph_ω$ contains an unconditional basic sequence.

On uniqueness and plentitude of subsymmetric sequences

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^*)$ of Tsirelson's original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a subsymmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to $Su(T^*)^*$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some $\ell_p$ or $c_0$…

On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by …

The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz bas…

Generalized function algebras as sequence space algebras

A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac's delta distribution induces discrete topology on the basic space. This result is in analogy to Schwartz' impossibility result concerning multiplication of distributions.

关于双重空间C(X)和Quojections中弱*基本序列的研究

On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections

研究无限紧致空间C(X)和Quojections的弱*基本序列问题，证明其双偶空间存在弱*基本序列并探讨相关开放问题。

Homogeneous families on trees and subsymmetric basic sequences

We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $κ$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $κ$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic struc…

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…

Rough Cauchy Sequences in a Cone Metric Space

Here we have introduced the idea of rough Cauchyness of sequences in a cone metric space. Also here we have discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu.

On strong $\mathcal{A}^{\mathcal{I}}$-statistical convergence of sequences in probabilistic metric spaces

In this paper using a non-negative regular summability matrix $\mathcal{A}$ and a non-trivial admissible ideal $\mathcal{I}$ in $\mathbb{N}$ we study some basic properties of strong $\mathcal{A}^{\mathcal{I}}$-statistical convergence and strong $\mathcal{A}^{\mathcal{I}}$-statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also introduce strong $\mathcal{A}^{\mathcal{I^*}}$-statistical Cauchyness in probabilistic metric space and study its relationship with strong A$\mathcal{A}^{\mathcal{I}}$-statistical Cauchyness there. Further, we study some basic propert…

On Strong A-statistical Convergence in Probabilistic Metric Spaces

In this paper we study some basic properties of strong A-statistical convergence and strong A-statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also study some basic properties of strong A-statistical limit points and strong A-statistical cluster points of a sequence in a probabilistic metric space. Further we also introduce the notion of strong statistically A-summable sequence in a probabilistic metric space and study its relationship with strong A-statistical convergence.

On strong λ-statistical convergence of sequences in probabilistic metric (pm) spaces

In this paper we study some basic properties of strong λ- statistical convergence of sequences in probabilistic metric (PM) spaces. We also introduce and study the notion of strong λ-statistically Cauchyness. Further introducing the notions of strong λ-statistical limit point and strong λ-statistical cluster point of a sequence in a probabilistic metric (PM) space we examine their interrelationship.

Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure $ν$ has no infinite set of complex exponentials which form an orthonormal set in $L^2(ν)$. We prove that any fractal measure $μ$ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(μ)$ such that the frequencies have positive Beurling dimension.

内积空间中子空间序列收敛的基于理想的广义概念

A generalized notion of convergence of sequences of subspaces in an inner product space via ideals

本文提出了内积空间中k维子空间序列基于理想的I-收敛广义概念，并研究了其基本性质。

Basic topological and geometric properties of Cesàro--Orlicz spaces

Necessary and sufficient conditions under which the Cesàro--Orlicz sequence space $\cfi$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro--Orlicz spaces $\cfi$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\cfi$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\cfi$ are given.

Partial Unconditionality

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 $K(δ)$ depending on a parameter $δ\in (0,1]$. It is unknown if $\sup_{δ>0} K(δ) < \infty$. 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 extens…

A characterization of moment sequences by extensions

The fibre theorem \cite{schm2003} for the moment problem on closed semi-algebraic subsets of $\R^d$ is generalized to finitely generated real unital algebras. A charaterization of moment functionals on the polynomial algebra $\R[x_1,\dots,x_d]$ in terms of extensions is given. These results are applied to reprove basic results on the complex moment problem due to Stochel and Szafraniec \cite{stochelsz} and Bisgaard \cite{bisgaard}.

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the Analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a s…

Strong $I$ AND $I^*$-statistically pre-Cauchy double sequences in Probabilistic Metric Spaces

In this paper we consider the notion of strong $I$-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et. al. [6] and introduce the new concept of strong $I^*$-statistically pre-Cauchy double sequences in real line as well as in probabilistic metric spaces. We mainly study inter relationship among strong $I$-statistical convergence, strong $I$-statistical pre-Cauchy condition and strong $I^*$-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions.

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 in…

Measures of Noncompactness in $\bar{N}(p,q)$ Summable Sequence Spaces

In this paper we first define the $\bar{N}(p,q)$ summable sequence spaces and obtain some basic results related to these spaces. The necessary and sufficient conditions for infinite matrices $A$ to map these spaces on $X~~,~~X=c_0, c \text{ or } \ell_{\infty}$ is obtained and Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of linear operators defined on these spaces.

Weakly null sequences in the Banach space C(K)

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 ve…

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…

A Conditional Quasi-greedy Basis of l_1

We show that the Lindenstrauss basic sequence in l_1 may be used to construct a conditional quasi-greedy basis of l_1, thus answering a question of Wojtaszczyk. We further show that the sequence of coefficient functionals for this basis is not quasi-greedy.

A note on rough $I$-convergence of double sequences

In this paper we study some basic properties of rough $I$-convergent double sequences in the line of D$\ddot{u}$ndar [8]. We also study the set of all rough $I$-limits of a double sequence and relation between boundedness and rough $I$-convergence of a double sequence.

Morrey Sequence Spaces: Pitt's Theorem and compact embeddings

Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0<p\leq u<\infty$, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical $\ell_p$ spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterize the compactness of embeddings of related finite-dimensional spaces.

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 $…

等距刚性和Orlicz序列空间的Fraïssé性质

Isometric rigidity and Fraïssé properties of Orlicz sequence spaces

论文研究了Orlicz序列空间的近等距刚性性质，证明了某些空间guarded Fraïssé但非ω-范畴，且不含ℓ₂的拷贝，回答了Cúth-de Rancourt-Doucha的问题。

广义Morrey序列空间

Generalised Morrey sequence spaces

本文针对广义Morrey函数空间的离散对应物展开研究，定义了广义Morrey序列空间$m_{φ,p}$，作为经典Morrey序列空间$m_{u,p}$的自然推广，并分析了其基本特征及嵌入性质。

序列空间间嵌入算子的最大非紧性

Maximal non-compactness of embeddings between sequence spaces

研究序列空间中嵌入算子的最大非紧性，分析洛伦兹空间的包含关系及身份算子的范数。

Baernstein 空间与 Schreier 空间上的严格奇异算子

Strictly singular operators on the Baernstein and Schreier spaces

该文证明在 Baernstein 与 Schreier 空间上，两个严格奇异算子复合为紧算子，并刻画了次对称基的等价性及非补块序列的存在。

Ramsey Property and Block Oscillation Stability on Normalized Sequences in Banach Spaces

A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using…

Syndetic Sets, Paving, and the Feichtinger Conjecture

We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen to be syndetic. We then apply this result to prove that if a Fourier frame for a measurable subset of a higher dimensional cube can be partitioned into Riesz basic sequences, then each subset can be chosen to be a syndetic subset of the corresponding higher dimensional integer lattice. Both of these results follow from a result about syndetic pavings of eleme…

An exact Ramsey principle for block sequences

We prove an exact, i.e., formulated without $Δ$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.

Banach spaces from a construction scheme

We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable $\varepsilon$-biorthogonal system but no uncountable $τ$-biorthogonal system for $τ<\varepsilon$. In particular the space have no uncountable biorthogonal system. We also construct a Banach space $\mathcal X_K$ with an uncountable $K$-basic sequence but no uncountable $K'$-basic sequence, for $1\leq K'<K$. A common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme.

Basic embeddings and Hilbert's 13th problem on superpositions (in Russian)

This note is purely expository. We show how in the course of the Kolmogorov-Arnold solution of Hilbert's 13-th problem on superpositions there appeared the notion of a basic embedding. A subset K of R^2 is {\it basic} if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y) = g(x) + h(y) for each point (x,y) in K. We present descriptions of basic subsets of the plane and graphs basically embeddable into the plane (solutions of Arnold's and Sternfeld's problems). We present some results and open problems on the smooth version of the property of being bas…

Basic embeddings and Hilbert's 13th problem

This note is purely expository. In the course of the Kolmogorov-Arnold solution of Hilbert's 13th problem on superpositions there appeared the notion of basic embedding. A subset K of R^2 is basic if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for each point (x,y) in K. We present descriptions of basic subsets of the plane (with a proof) and description of graphs basically embeddable into the plane (solutions of Arnold's and Sternfeld's problems). We present some results and open problems on the smooth version of the property of bein…