Valentin Ferenczi, Gilles Godefroy
We prove several dichotomies on linear embeddings between Banach spaces. Given an arbitrary Banach space X with a basis, we show that the relations of isomorphism and bi-embedding are meager or co-meager on the Polish set of block-subspaces of X. We relate this result with tightness and minimality of Banach spaces. Examples and open questions are provided.
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Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric space $K$, are defined and characterized. Some applications to Banach spaces are given.
Banach spaces with arbitrary finite Baire order
本文构造了可分Banach空间，解决了关于内在Baire阶的开放问题，并推广了Lindenstrauss的结果，表明任何具有基的Banach空间可作为对偶空间中特定Baire类的泛函空间。
Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire sigma-algebras on C(K) associated to the weak and pointwise convergence topologies.
If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of $x^{**}$ are quite often determined by the behaviour of $x^{**}$ on the set of extreme points of $B_{E^*}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove several results on relation between Baire classes and intrinsic Baire classes of $L_1$-preduals which were introduced by S.A. Argyros, G. G…
The Fréchet derivative is in the first Baire class
该数学研究证明了赋范线性空间间映射的Fréchet导数在其定义集上属于第一Baire类，完善了导数的拓扑分类相关理论。
Tight-minimal dichotomies in Banach spaces
本文推广Ferenczi-Rosendal方法，证明带容许块系统的Banach空间包含A-紧或A-极小的无穷维子空间，并由此得到基的子序列二分性及紧性的Baire范畴解释。
We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces, but contrary to the recently constructed space of type (4) also tight with constants, thus essentially extending the list of known examples in Gowers classification program. The space is defined on the base on a boundedly modified mixed Tsirelson space with use of a special coding function.
We collect several open questions in Banach spaces, mostly related to measure theoretic aspects of the theory. The problems are divided into five categories: miscellaneous problems in Banach spaces (non-separable $L^p$ spaces, compactness in Banach spaces, $w^*$-null sequences in dual spaces), measurability in Banach spaces (Baire and Borel $σ$-algebras, measurable selectors), vector integration (Riemann, Pettis and McShane integrals), vector measures (range and associated $L^1$ spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).
Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pełczy{ń}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.
Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $λ$ on $\{0,1\}^{\mathb…
A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a countable subset of $C$. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of $B_{X^*}$ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under $\mathrm{MA}+\neg \mathrm{CH}$ for the class of $\mathcal{C}(K)$-spaces. In this article we provide a partial extension of …
Let X and Y be separable metrizable spaces, and f:X-->Y be a function. We want to recover f from its values on a small set via a simple algorithm. We show that this is possible if f is Baire class one, and in fact we get a characterization. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.
In this paper, we introduce the cone normed spaces and cone bounded linear mappings. Among other things, we prove the Baire category theorem and the Banach--Steinhaus theorem in cone normed spaces.
On some geometrization of compact metric spaces: A solution to the Banach-Ulam conjecture
提出基于Banach和Mycielski思想的紧度量空间几何化方法，证明了Banach-Ulam猜想：每个紧度量空间存在有限可加、合同不变的测度，并解决了Talagrand问题。
On extremal nonexpansive mappings
本文研究赋范空间非空有界闭凸子集上非扩张映射的极值性，证明满射等距映射在具有Radon-Nikodym性质的巴拿赫空间及C(K)空间中为极值，并表明Baire范畴意义下典型非扩张映射接近极值。
The aim of this paper is to establish Pontryagin's principles in a dicrete-time infinite-horizon setting when the state variables and the control variables belong to infinite dimensional Banach spaces. In comparison with previous results on this question, we delete conditions of finiteness of codi-mension of subspaces. To realize this aim, the main idea is the introduction of new recursive assumptions and useful consequences of the Baire category theorem and of the Banach isomorphism theorem.
Using the Baire Category Theorem to Explore Lions Problem for Quasi-Banach Spaces
本文利用 Baire 纲定理建立拟 Banach 偶插值空间严格居中的条件，结合重述定理部分解决 Lions 问题，并据此区分由逼近数生成的算子理想。
In this paper, first we present a new useful way of formulating probabilistic normed spaces. Then by using this formulation and probabilistic normed space version of the Baire category theorem, we prove four important results of functional analysis, i.e. the open mapping, closed graph, principle of uniform boundedness and Banach-Steinhaus theorem in PN-spaces.
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…
The condition onto pair ($F,G$) of function Banach spaces under which there exists a integral operator $T:F\to G$ with analytic kernel such that the inverse mapping $T^{-1}:$im$T\to F$ does not belong to arbitrary a priori given Borel (or Baire) class is found.
Equivalence of D and D' properties in Banach spaces
本文证明了在弱*拓扑下对偶Banach空间的序列性质$D$和$D'$等价，并利用Baire $σ$-代数扩展了先前结果，同时为非可分空间中性质$D$的失效提供了拓扑学解释。
Characterizations and Properties of Graphs of Baire Functions
本文研究了Baire函数的图特征，通过证明Baire-1函数的图是可表示为一系列开集的交集，并讨论了更高Baire类函数的类似问题，以及Baire-2函数图的积累集特性。
A cardinal k is called a Kunen cardinal if the sigma-algebra on k x k generated by all products AxB, coincides with the power set of k x k. For any cardinal k, let C(2^k) be the Banach space of all continuous real-valued functions on the Cantor cube 2^k. We prove that k is a Kunen cardinal if and only if the Baire sigma-algebra on C(2^k) for the pointwise convergence topology coincides with the Borel sigma-algebra on C(2^k) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.
Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ a mapping. We prove that there exists a $σ$-directionally porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is Lipschitz at $x$, and $f$ is Gâteaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux differentiable at all points, then $f$ is Hadamard differentiable at all points except a set which is $σ$-directionally porous set (and so is Aronszajn null, Haar null and $Γ$-null). Consequently, an everywhere G\^ ateaux diff…
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。
Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a regularity property (e.g. continuity, differentiability, etc.) on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on $x$ systematically. Afterwards, we apply this theoretical framework to several diffe…
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.
We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. Most of the results are about affine functions defined on convex subsets of vector spaces, rather than linear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duality theorem) and also results that use Baire's theorem (as in the Ro…
Topologies related to (I)-envelopes
本文研究巴拿赫空间对偶中(I)-包络的拓扑描述，给出可行与不可行条件，并引入中间闭包算子及开放问题。
We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is an image of $L$ under a upper semicontinuous set-function having finite values. Moreover we show that $K$ has a $π$-base of sets which closures a continuous images of compact subspaces of $L$. Our results imply in particular that if $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness of Frechetness, pass to the space $K$. We show that some arbitrary isomorphic…
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…
We extend the framework by Kawamura and Cook for investigating computational complexity for operators occurring in analysis. This model is based on second-order complexity theory for functions on the Baire space, which is lifted to metric spaces by means of representations. Time is measured in terms of the length of the input encodings and the required output precision. We propose the notions of a complete representation and of a regular representation. We show that complete representations ensure that any computable function has a time bound. Regular representations generalize Kawamura and Co…
Hyperbolic Banach spaces
本文提出双曲Banach空间的概念，通过替代传统椭圆框架中的线性组合、三角不等式和Cauchy完备性，探讨非光滑洛伦兹几何中的定向完备性，并研究L^p空间等结构在该框架下的应用。
We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrenc…
Compact sets in Banach lattices over spaces of homogeneous type
本文证明了齐次型空间上Banach格的Kolmogorov-Riesz紧性定理，用有界性、紧性和平均算子逼近刻画紧性，无需Vitali覆盖等假设，并推广至重排不变空间和Morrey空间。
Asplund spaces and the finest locally convex topology
本文在局部凸空间框架下研究Asplund性质，证明自由局部凸空间非GDS且无无限维Baire子空间，并给出quojection的Asplund等价条件。
Metrizable and Fréchet--Urysohn subgroups and linear subspaces of topological vector spaces
本文研究拓扑向量空间中Fréchet--Urysohn及Baire子群和线性子空间的结构，证明其局部有限维性，并将结果推广至更广泛的$\kappa$-Fréchet--Urysohn子群族。
Some sets of first category in product Calderón-Lozanovski\uı spaces on hypergroups
本文在局部紧超群的Calderón-Lozanovskiı空间框架下，受已有研究启发给出两类卷积相关集合为Baire第一纲集的充分条件，结果对超群上的Orlicz(-Lorentz)空间也具创新性。
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