Let X be a Banach space. Given a subset A of the dual space X* denote by $A_{(1)}$ the weak* sequential closure of A, i.e., the set of all limits of weak*-convergent sequences in A.
The study of weak* sequential closures of linear subspaces of the duals of separable Banach spaces was initiated by S.Banach. The first results of this study were presented in the appendix to his book "Theorie des operations lineaires" (1932).
It is natural to suppose that the reason for studying weak* sequential closures by S. Banach and S. Mazurkiewicz was the lack of acquaintance of S. Banach and his school with concepts of general topology. Although the name "General topology" was introduced later, the subject did already existed. F. Hausdorff introduced topological spaces in his book published in 1914, Alexandroff-Urysohn (1924) studied compactness, and A.Tychonoff published his theorem on compactness of products in 1929. Also J.von Neumann introduced the notion of a weak topology in his paper published in 1929.
Using the notions of a topological space and the Tychonoff theorem, more elegant treatment of weak and weak* topologies, and the duality of Banach spaces was developed by L.Alaoglu, N.Bourbaki and S.Kakutani (1938-1940).
Nevertheless, an "old-fashioned" treatment of S.Banach still attracts attention. It happens because the "sequential" approach is very useful in several contexts.
The main purpose of the paper is to describe the history of this direction of research and to give an up-to-date (1999) survey of the results on weak* sequential closures and their applications.
The paper is devoted to the convex-set counterpart of the theory of weak$^*$ derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space $X$ and every countable successor ordinal $α$, there exists a convex subset $A$ in $X^*$ such that $α$ is the least ordinal for which the weak$^*$ derived set of order $α$ coincides with the weak$^*$ closure of $A$. This result extends the previously known results on weak$^*$ derived sets by Ostrovskii (2011) and Silber (2021).
The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and…
A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball, answering in this way a …
A well known result of Lozanovsky states that a Banach lattice is weakly sequentially complete if and only if it does not contain a copy of $c_{0}$. In the current paper we extend this result to the class of Banach $C(K)$ modules of finite multiplicity and, as a special case, to finitely generated Banach $C(K)$-modules. Moreover, we prove that such a module is weakly sequentially complete if and only if each cyclic subspace of the module is weakly sequentially complete.
A Banach space $X$ is reflexive if the Mackey topology $τ(X^*,X)$ on $X^*$ agrees with the norm topology on $X^*$. Borwein [B] calls a Banach space $X$ {\it sequentially reflexive\/} provided that every $τ(X^*,X)$ convergent {\it sequence\/} in $X^*$ is norm convergent. The main result in [B] is that $X$ is sequentially reflexive if every separable subspace of $X$ has separable dual, and Borwein asks for a characterization of sequentially reflexive spaces. Here we answer that question by proving \proclaim Theorem. {\sl A Banach space $X$ is sequentially reflexive if and only if $\ell_1$ is not…
In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies \[\|e_1+\ldots+e_k\|_{\overlineδ_Y}\lesssim\|e_1+\ldots+e_k\|_S,\] where $\overlineδ_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. u…
In this note we provide examples that show that a common notion of causality for linear operators on Banach spaces does not carry over to the closure of the respective operators. We provide an alternative definition for causality, which is equivalent to the usual definition for closed linear operators but does carry over to the closure.
If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $α$ the dual of $X$ contains a subspace whose weak$^*$ sequential closures of orders less than $α$ are not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$ sequential closure of order $α+1$ coincides with $X^*$
We provide sufficient conditions for a Banach space Y to be weakly sequentially complete. These conditions are expressed in terms of the existence of directional derivatives for cone convex mappings with values in Y .
Let $A$ be a Banach algebra. For $f\in A^{\ast}$, we inspect the weak sequential properties of the well-known map $T_f:A\to A^{\ast}$, $T_f(a) = fa$, where $fa\in A^{\ast}$ is defined by $fa(x) = f(ax)$ for all $x\in A$. We provide equivalent conditions for when $T_f$ is completely continuous for every $f\in A^{\ast}$, and for when $T_f$ maps weakly precompact sets onto L-sets for every $f\in A^{\ast}$. Our results have applications to the algebra of compact operators $K(X)$ on a Banach space $X$.
In this paper, we first study the concept of $ p $-sequentially Right property, which is the $ p$-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called $ p$-Right$ ^{\ast} $ set and obtain the relationship between p-Right subsets and p-Right$ ^{\ast} $ subsets of dual spaces. Furthermore, for $ 1\leq p<q\leq\infty, $ we introduce the concepts of properties $ (SR)_{p,q}$ and $ (SR^{\ast})_{p,q}$ in order to find a condition which every Dunford-Pettis $ q $-convergent operator is Dunford-Pettis $p$-convergent. Finally, we apply the…
In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
Universidad de Murcia（穆尔西亚大学）；ETH Zürich（苏黎世联邦理工学院）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科学研究理事会-马德里自治大学-马德里卡洛斯三世大学-马德里大学））
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.
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 …
We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.
Let $M$ be a compact subset of a superreflexive Banach space. We prove that the Lipschitz-free space $\mathcal{F}(M)$, the predual of the Banach space of Lipschitz functions on $M$, has the Pełczyński's property ($V^\ast$). As a consequence, the Lipschitz-free space $\mathcal{F}(M)$ is weakly sequentially complete.
A recent result of T.~Abrahamsen, P.~Hájek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes through a sequential version of being almost square which we call being \textit{sequentially almost square}. In this article we study these conditions in the nonseparable setting. On one hand, we show that a Banach space $X$ contains a copy of $c_0$ if and only if there exists an equivalent renorming $| \cdot |$ on $X$ for which there exists $h\in S_{X^{****}}$ suc…
We show that operators on a separable infinite dimensional Banach space $X$ of the form $I +S$, where $S$ is an operator with dense generalised kernel, must lie in the norm closure of the hypercyclic operators on $X$, in fact in the closure of the mixing operators.
A set of all symmetric Banach function spaces defined on [0,1] is equipped with the partial order by the relation of continuous inclusion. Properties of symmetric spaces, which do not depend of their position in the ordered structure, are studied. With the help of the J. Peetre's interpolation scheme it is shown that for any pair of symmetric spaces E, F such that F is absolutely continuously included in E there exists an intermediate (so called, Peetre's) space K(E,F;W), where W is a space with an unconditional basis, that is reflexive or, respectively, weakly sequentially complete provided W…
We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.
These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.
A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property ($\mathcal{E}$). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to an…
We survey the recent investigations on (bounded, sequential) approximate amenability/contractibility and pseudo-amenability/contractibility for Banach algebras. We will discuss the core problems concerning these notions and address the significance of any solutions to them to the development of the field. A few new results are also included.
In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive - is weak-weakly sequentially continuous.
Let $E$ be a Banach lattice. Its ideal center $Z(E)$ is embedded naturally in the ideal center $Z(E')$ of its dual. The embedding may be extended to a contractive algebra and lattice homomorphism of $Z(E)"$ into $Z(E')$. We show that the extension is onto $Z(E')$ if and only if $E$ has a topologically full center. (That is, for each $x\in E$, the closure of $Z(E)x$ is the closed ideal generated by $x$.) The result can be generalized to the ideal center of the order dual of an Archimedean Riesz space and in a modified form to the orthomorphisms on the order dual of an Archimedean Riesz space.
Let F(X) denote the norm closure of the space of all finite rank operators on a Banach space X. We show that there are Banach spaces X for which the product map $a\otimes b\to ab$ does not define a surjective map from the projective tensor product $F(X) \hat\otimes F(X)$ onto F(X).
Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space $E$ and its closed unit ball $B$, both endowed with the weak topology of $E$. We show that $B$ has the Pytkeev property if and only if $E$ in the norm topology contains no isomorphic copy of $\ell_1$, while $E$ has the Pytkeev property if and only if it is finite-dimensional. We extend Schlüchtermann and Wheeler's result by showing that $B$ is a (separable) metrizable space if and only if it has countable $cs^\ast$-character and is a $k$-space. As a corollary we obtain that $B$ is Polish if…
We introduce and study certain type of variable exponent \ell^p spaces. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. We obtain some interesting examples of Banach spaces with a 1-unconditional basis.
$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_{w}$. It is unknown whether $C_p(K)$ and $C(L)_{w}$ can be homeomorphic for infinite compact spaces $K$ and $L$ \cite{Krupski-1}, \cite{Krupski-2}. In this paper we deal with a more general question: what are the Banach spaces $E$ which admit certain continuous surjective mappings $T: C_p(X) \to E_{w}$ for an infinite Tychonoff space $X$? First, we prove that if $T$ is linear and sequentia…
Let $X$ be a Banach space and $Γ\subseteq X^*$ a total linear subspace. We study the concept of $Γ$-integrability for $X$-valued functions $f$ defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions $\langle x^*,f \rangle$ for $x^*\in Γ$. We show that $Γ$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property ($\mathcal{D}'$) (meaning that every $w^*$-sequentially closed subspace of $X^*$ is $w^*$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^*$-angelic dual…
Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford--Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), and the sequential w*-continuity of the lattice operations. It is proved that the spaces consisting of regularly versions of the above operators are all Banach spaces. The domination problem for …
In this paper, we study properties of ultramaximally monotone operators. We characterize the interior and the closure of the range of an ultramaximally monotone operator. We establish the Brezis--Haraux condition in the setting of a general Banach space. Moreover, we show that every ultramaximally monotone operator is of type (NA). We also provide some sufficient conditions for a Banach space to be reflexive by a linear continuous and ultramaximally monotone operator.
A subsequence principle is obtained, characterizing Banach spaces containing $c_0$, in the spirit of the author's 1974 characterization of Banach spaces containing $\ell^1$. Definition: A sequence $(b_j)$ in a Banach space is called {\it strongly summing\/} (s.s.) if $(b_j)$ is a weak-Cauchy basic sequence so that whenever scalars $(c_j)$ satisfy $\sup_n \|\sum_{j=1}^n c_j b_j\| <\infty$, then $\sum c_j$ converges. A simple permanence property: if $(b_j)$ is an (s.s.) basis for a Banach space $B$ and $(b_j^*)$ are its biorthogonal functionals in $B^*$, then $(\sum_{j=1}^n b_j^*)_{n=1}^ \infty$…
Suppose $(X,Σ,μ)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,μ)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,μ)$ is a $KB$-space or has the sequential Fatou property if and only if so is $E$. Among this, some results about Bochner integral convergence in $B(X,E,μ)$, using order structure of $E$, have been proved, as well.
We give, for general Banach spaces, a characterization of the sequential lower limit of maximal monotone operators of type (D) and prove its representability. As a consequence, using a recent extension of the Moreau-Yosida regularization for type (D) operators, we extend to general Banach spaces the definitions of the variational sum of monotone operators and the variational composition of monotone operators with continuous linear mappings, and we prove that both operators are representable.
The parabolic algebra was introduced by Katavolos and Power, in 1997, as the operator algebra acting on $L^2(R)$ that is weakly generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces of the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on $L^p(R)$, where $1<p<\infty$. It is also shown that the reflexive closures of the Fourier binests on $L^p(R)$…
The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a \RM(real or complex\RM) Banach space has either a strongly summing sequence or a convex block basis equivalent to the summing basis. (A weak-Cauchy sequence is called {\it non-trivial} if it is {\it non-weakly convergent}.) The following characterization of spaces containing $c_0$ is thus obtained, in the sp…
In this paper, we investigate more about relationship between $uaw$ -convergence (resp. $un$-convergence) and the weak convergence. More precisely, we characterize Banach lattices on which every weak null sequence is $uaw$-null. Also, we characterize order continuous Banach lattices under which every norm bounded $un$-null net (resp. sequence) is weakly null. As a consequence, we study relationship between sequentially $uaw$-compact operators and weakly compact operators. Also, it is proved that every continuous operator, from a Banach lattice $E$ into a non-zero Banach space $X$, is unbounded…
A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $\aleph$. We perform a study of $\aleph$-injective and universally $\aleph$-injective Banach spaces which extends the basic case where $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding "isomet…
We prove the closure for the sequential weak $L^p$-topology of the class of vectorfields on $B^3$ having integer flux through almost every sphere. We show how this problem is connected to the study of the minimization problem for the Yang-Mills functional in dimension higher than critical, in the abelian case.
In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and Šmulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $ϕ(x,y)$, and show that $ϕ$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-Šmulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some …
It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.
It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf--convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to …
We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of $\ell_1$ holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property but fails the quantitative form of the Schur property.
For $0\leqslant ξ\leqslant ω_1$, we define the notion of $ξ$-weakly precompact and $ξ$-weakly compact sets in Banach spaces and prove that a set is $ξ$-weakly precompact if and only if its weak closure is $ξ$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant ξ\leqslant ω_1$, we prove that a Banach space $X$ has the $ξ$-Schur property if and only if every $ξ$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.
