Jarno Talponen
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.
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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 …
On the sequential monotone closure of $CD_ω(K)$ spaces
本文解决了Wickstead提出的问题，探讨了$CD_ω(K)$空间的序列单调闭包性质。
We consider several general conditions for integrability of two Roumieu ultradistributions and prove their equivalence. The discussed sequential conditions are based on two classes of approximate units and allow one to introduce sequential defnitions of the convolution in the space of Roumieu ultradistributions, analogous to the known defnitions in the space of distributions.
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\ell_\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such tha…
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…
On weak sequential completeness of spaces where weakly compact sets are super weakly compact
本文证明弱紧集为超弱紧集的 Banach 空间自动弱序列完备，回答了 Silber (2024) 的问题，并利用求和子列从非平凡弱 Cauchy 列构造非超弱紧的弱紧集。
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.
Total subspaces with long chains of nowhere norming weak$^*$ sequential closures
若可分巴拿赫空间$X$使得对于某个非拟自反巴拿赫空间$Y$存在满射严格奇异算子$T:X\to Y$，则对于每个可数序数$α$，$X$的对偶包含一个子空间，其弱$^*$序列闭包的阶小于$α$，在任何无穷-一个$X$的维数子空间，它的弱$^*$序列闭包为$α+1$，与$X^*$重合
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 $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
We consider several general sequential conditions for convolvability of two Roumieu ultradistributions and prove that they are equivalent to the convolvability of these ultradistributions in the sense of Pilipovic and Prangoski. The discussed conditions, based on two classes of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space of distributions and in the space of ultradistributions of Beurling type. Moreover, the useful property of the convolution and ultradiferential opera…
Cyclicity via weak$^\ast$ sequential cyclicity in Radially weighted Besov spaces
该研究针对径向加权Besov空间中循环性与弱*序列循环性的差异，给出无零点有界全纯函数循环性的对数条件，提出弱*序列循环乘子的比较原理，为非完全Pick空间下的循环性现象提供新认知。
We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, $1\le p<\infty$, and $(\bigoplus_{n=1}^\infty c_o)_{\ell_q}$, $1\le q<\infty$, do not have greedy bases. It follows from that and known results that a class of Besov spaces on $\R^n$ lack greedy bases as well.
Nonatomic bounded sequences in $\ell_1$, that is, those giving rise to nonatomic submeasures on $\mathbb N$ are introduced and shown to form a closed subspace nonat$(\ell_1)$ of $\ell_\infty(\ell_1)$, and some spaces of relevant operators in $\ell_1$ are considered.
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.
In this paper, we characterize when the $\ell^p$ uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for $p\in [1,\infty)$. Moreover, we show that the $\ell^p$ uniform Roe algebra is a (non-sequential) spatial $L^p$ AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero. We also consider the ordered $K_0$ groups of $\ell^p$ uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered $K_0$ group is trivial when the me…
We study equivalent descriptions of the vague, weak, setwise and total-variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent conditions on the vague convergence of sequences of measures following Kallenberg, and some equivalent conditions on the TV convergence of sequences of measures following Feinberg-Kasyanov-Zgurovsky. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give e…
Weak* sequential closures in Banach space theory and their applications
设X是巴拿赫空间。给定对偶空间X* 的子集A，用$A_{(1)}$表示A的弱 * 序列闭包，即，A中弱 *-收敛序列的所有极限的集合.可分巴拿赫空间的子空间的弱 * 序列闭包的研究是由S. Banach开创的。这项研究的第一个结果是在附录中提出的他的书“理论万业务线”（1932年）。可以很自然地认为，S. Banach和S. Mazurkiewicz是缺乏认识的S。巴拿赫及其学派与一般拓扑学的概念。虽然“一般拓扑学”这个名字是后来才被引入的，但这个主题确实已经存在了。F.豪斯多夫介绍了拓扑空间在他的书出版于1914年，亚历山德罗夫，Urysohn（1924年）研究的紧凑性，和A.吉洪诺夫发表了他的定理紧凑的产品于1929年。此外J.冯诺依曼介绍了概念的弱拓扑在他的论文发表于1929年。利用拓扑空间的概念和吉洪诺夫定理，更优雅地处理弱和弱 * 拓扑，以及巴拿赫空间的对偶性是由L.Alaoglu，N.Bourbaki和S.Kakutani（1938-1940）发展的。尽管如此，一个“老式”的治疗S.巴拿赫仍然引起注意。这是因为“顺序”方法在几种情况下非常有用。本文的主要目的是描述这个方向的研究历史，并给出一个最新的（1999年）调查结果弱 * 序列闭包及其应用。
In this paper we investigate the conditional expectation on the non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.
A class of preservers on Hilbert space effects including ortho-order automorphisms and sequential automorphisms
本文研究了所有希尔伯特空间效应集上的一类新的变换。这是由双射映射保持秩序和零产品在两个方向。本文的主要结果给出了这些变换的结构的完整描述。作为应用，我们得到了额外的新结果和一些以前的简单的推论。特别地，我们得到了正交序自同构和序列自同构的形式。在最后一段的文件中，我们表明，这两种自同构属于我们的类的变换，即使他们的域是一个一般的冯诺依曼代数的所有影响的集合。
A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $κ$-fan introduced by Banakh. In particular, we prove the following two results: (1) The space $S_{ω_{1}}\times S_{ω_{1}}$ is not a $k_{R}$-space. (2) The space $S_ω\times S_{ω_{1}}$ is a $k_{R}$-space if and only if $S_ω\times S_{ω_{1}}$ is a …
Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces.
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