S. K. Mercourakis, G. Vassiliadis
We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $α$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.
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Equilateral dimension of the planar Banach--Mazur compactum
本文证明了在巴拿赫-马祖尔距离下，平面对称凸体中存在任意大的等边集，并展示了构造的最优性。
An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.
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 $λ>0$ and an infinite sequence $(x_i)_{i=1}^\infty\subset X$ such that $\|x_i-x_j\|=λ$ for all $i\neq j$.
A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.
The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form $C(K)$ has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form $C(K)$ which has no unco…
We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spac…
We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity is at most 3/2, then there exist n+1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e^{c sqrt(log n)} equidistant points, where c>0 is an absolute constant. We also show that there exist n equidistant points in spaces sufficiently close to n-dimensional ell p (1 < p < infinity).
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equ…
A subset of a normed space $X$ is called equilateral if the distance between any two points is the same. Let $m(X)$ be the smallest possible size of an equilateral subset of $X$ maximal with respect to inclusion. We first observe that Petty's construction of a $d$-dimensional $X$ of any finite dimension $d\geq 4$ with $m(X)=4$ can be generalised to give $m(X\oplus_1\mathbb{R})=4$ for any $X$ of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $Γ$, $m(\ell_p(Γ))$ is finite and bounded above by a functio…
For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space of dimension $\log n$. The main result of the paper is a new example of a famil…
We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.
The following strengthening of the Elton-Odell theorem on the existence of a $(1+ε)-$separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subseteq S_X$ and a constant $d>1$, satisfying the property that for every $x,y\in S$ with $x\neq y$ there exists $f\in B_{X^*}$ such that $d\leq f(x)-f(y)$ and $f(y)\leq f(z)\leq f(x)$, for all $z\in S$.
Non-separable combinatorial Banach spaces
本文研究不可数密度的组合Banach空间，刻画其包含长$c_0$与长$\u2113_1$副本的条件，讨论固定基数无条件基范畴中的万有空间，并证明Suslin树生成的空间不含不可数等边集。
The paper concerns the problem whether a nonseparable $\C(K)$ space must contain a set of unit vectors whose cardinality equals to the density of $\C(K)$ such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of $\C(K)$ spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.
We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite dia…
We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric.
We give examples of real Banach spaces with exactly infinite countably many complex structures and with $ω_1$ many complex structures.
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincaré-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.
We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.
A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and Tomczak-Jaegermann, this gives a positive answer to Banach's problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.
Exact Banach-Mazur distances of certain $\ell_p$-sums and cones
本文研究了有限维实normed空间的ℓ_p直接和与相关凸体锥构造的Banach-Mazur距离，通过最近的最优Banach-Mazur位置特性，推导出从X_1⊕_p…⊕_pX_k到欧几里得空间的距离公式，并探讨了单锥与双锥的几何性质及距离关系。
Norms and Non-Equivalence in Infinite-Dimensional Banach Spaces
本文研究无穷维巴拿赫空间中不同范数的相互作用，分析双射线性映射诱导的范数、非等价范数的存在性以及哈默尔基的作用。
We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincaré-type polynomials. It turns out that the fulfillment of most of types of decoupling inequalities may depend on the geometry of Banach space.
A distortion lower bound of $Ω(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(β)$ of Rolewicz with modulus of power type $p\in(1,\infty)$ (in short property ($β_p$)). Also it is shown that a distortion lower bound of $Ω(\ell^{1/p})$ is incurred when embedding the parasol graph with $\ell$ levels into a Banach space with an equivalent norm with property ($β_p$). The tightness of the lower bound for trees is shown adjusting a construction of Matoušek to the case of infin…
Symmetry of Birkhoff-James orthogonality of operators defined between infinite dimensional Banach spaces
本文研究无穷维Banach空间之间有界线性算子在Birkhoff-James正交意义下的左右对称性，证明在严格凸且定义域自反并具有Kadets-Klee性质时左对称算子必为零算子，并给出非严格凸情形下的非零例子。
A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such that the norm of Tz equals the norm of T. The existence of an infinite dimensional banach space X, in which each bounded linear operator acting on X attains its norm, is still undetermined. This question was posed by M.I. Ostrovskii at St. John's University. In this paper we show that if an infinite dimensional banach space is considered over GF(2), then it i…
[REVISED VERSION] The aim of this paper is to state a sharp version of the König supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.
A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)
构造 Gowers-Maurey 型 Banach 空间 X，使有界算子代数 L(X) 为无限维自反空间与严格奇异算子理想的直和，从而商代数 L(X)/SS(X) 为无限维自反空间。
We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets o…
In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of asymptotic structures and asymptotic versions of a given Banach space. As an example of application of this approach, a class of asymptotic $l_p$-spaces is introduced and investigated in detail. Some properties of this class, as duality and complementation, are analogous to properties of classical $l_p$ spaces, although the latter is more ``regular'' than its classic…
Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces
本文证明对满足温和假设的任意Hausdorff函数对，可在Baire空间中构造紧集使其Hausdorff测度与packing测度均有限且正，并嵌入任意无限维Banach空间，肯定回答了Fan关于任意尺度度量空间存在性的问题。
The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace whic…
We present partial results to the following question: Does every infinite dimensional Banach space have an infinite dimensional subspace on which one can define an operator which is not a compact perturbation of a scalar multiplication?
In this note we study the endomorphisms of certain Banach algebras of infinitely differentiable functions on compact plane sets, associated with weight sequences M. These algebras were originally studied by Dales, Davie and McClure. In a previous paper this problem was solved in the case of the unit interval for many weights M. Here we investigate the extent to which the methods used previously apply to general compact plane sets, and introduce some new methods. In particular, we obtain many results for the case of the closed unit disc. This research was supported by EPSRC grant GR/M31132
We continue our investigation, from \cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a "Cuntz-like" Banach $*$-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algeb…
A New Shrinking projection Algorithm for an infinite family of Bregman weak relatively nonexpansive mappings in a Banach Space
提出一种结合广义预解式与广义投影的新型收缩投影算法，证明其在自反Banach空间中可强收敛到不动点、零点及平衡问题解的公共点。
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…
This article is a continuation of a paper of the first author \cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from \cite{F} and $C(K)$ examples due to Plebanek \cite{P}. We extend results of \cite{F} relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of \cite{F} about…
A Banach space $X$ is called subprojective if any of its infinite dimensional subspaces $Y$ contains a further infinite dimensional subspace complemented in $X$. This paper is devoted to systematic study of subprojectivity. We examine the stability of subprojectivity of Banach spaces under various operations, such us direct or twisted sums, tensor products, and forming spaces of operators. Along the way, we obtain new classes of subprojective spaces.
We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.
In this paper, we study different kinds of normal properties for infinite system of arbitrarily many convex sets in a Banach space and provide the dual characterization for the normal property in terms of the extended Jamenson property for arbitrarily many weak*-closed convex cones in the dual space. Then, we use the normal property and the extended Jamenson property to study CHIP, strong CHIP and linear regularity for the infinite case of arbitrarily many convex sets and establish equivalent relationship among these properties. In particular, we extend main results in [3] on normal property, …
In this note a large class of primary Banach spaces is characterized. Namely, it will be demonstrated that under the Continuum Hypothesis the ultrapower of any infinite dimensional nonsuperreflexive Banach space is always primary. Consequently, any infinite dimensional nonsuperreflexive Banach space can be isometrically embedded into its primary ultrapowers.
A short review on infinite-dimensional Grassmann-Banach algebras (IDGBA) is presented. Starting with the simplest IDGBA over $K = {\bf R}$ with $l_1$-norm (suggested by A. Rogers), we define a more general IDGBA over complete normed field $K$ with $l_1$-norm and set of generators of arbitrary power. Any $l_1$-type IDGBA may be obtained by action of Grassmann-Banach functor of projective type on certain $l_1$-space. In non-Archimedean case there exists another possibility for constructing of IDGBA using the Grassmann-Banach functor of injective type.
The purpose of this paper is to lay the foundations for the study of the problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$, including the first example of a separable Banach space $\mathscr K$ so that $\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to comp…
We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous differential equation $x'=f(x)$ has no solution at any point.
Motivated by the study of dynamics of interacting spins for infinite particle systems, we consider an infinite family of first order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite life-time, and prove that they live in an incre…
Infinite dimensional spaces consisting of sequences that do not converge to zero
本文研究在巴拿赫空间或巴拿赫格中构造无穷维子空间，使其中的有界序列经映射后在给定向量拓扑下不收敛于零，并由此改进和拓展了关于非经典性质空间及非标准线性/非线性映射的若干结果。
Banach spaces of sequences arising from infinite matrices
本文研究由无限矩阵生成的序列空间 $\ell_M^p$，证明其在广泛矩阵类下对 $p>1$ 具有可分性与严格/一致凸性，并识别其对偶空间，推广经典分解结果。
We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini--Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler--Lagrange condition and the maximum principle.
Mixing rates for linear operators under infinitely divisible measures on Banach spaces
本文在 Banach 空间线性动力学中，利用余差泛函与控制测度研究无限可分测度下算子的混合速率，并给出了复合 Poisson、α-稳定及调和 α-稳定测度下加权移位算子的显式收敛速率。
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.
We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.
The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $\ell_p$ for $p\in[1,\infty)$. We show that there is no infinite dimensional Banach space that coarsely embeds into eve…
A quojection (projective limit of Banach spaces with surjective linking mappings) without infinite dimensional Banach subspaces is constructed. This results answers a question posed by G.Metafune and V.B.Moscatelli.
The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are studied. It is shown that the infinite dimensional generalization of the Flaschka map is a momentum map. Action-angle variables for the Toda system are constructed.
A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X/Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive finite-dimensional decomposition on the basis of $X$. A reflexive space with the restricted QHI property is in particular HI, has HI dual, and is saturated with subspaces which are HI and have HI dual. The following dichotomy theorem is proved: any infinite dimensional Banach space contains a quotient of subspace which either has an unconditional basis, or has the…
Decomposition of one-layer neural networks via the infinite sum of reproducing kernel Banach spaces
本文利用RKBS特征定理定义RKBS的和并证明其与特征空间直和相容，将积分RKBS分解为p-范数RKBS之和，为单层神经网络及积分RKBS类的结构理解提供应用。
It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(λ)$, $1<p<\infty $, is a quotient of a H.I Banach space.
We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on l_p (1 \le p < \infty) or c_0.
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