A. G. Smirnov
A convenient technique for calculating completed topological tensor products of functional Frechet and DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of spaces of smooth and entire analytic functions.
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Countable projective limits of countable inductive limits, so-called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet, who analyzed locally convex properties in terms of the defining double sequence of weights. We complement their results by considering a defining sequence which is the product of two single sequences. By associating these two sequences with a weighted Fréchet, resp. LB-space of continuous functions or with two weighted Fréchet spaces (by taking the reciprocal of one of the sequences) we derive a repre…
We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Frechet space. The investigations were stimulated by a recent metric approach to differentiability in Frechet spaces by Olaf Mueller. Our results also subsume generalizations of Mueller's Inverse Function Theorem for mappings between Frechet spaces. As an application, we prove exist…
A convenient technique for calculating completed topological tensor products of functional Frechet or DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of spaces of smooth and entire analytic functions.
Differentiation in Topological Vector Spaces
本文探讨拓扑向量空间中微分的定义与性质，基于F-半范数构建扩展的ε-δ语言，定义单值映射的连续性及广义Gâteaux和Frechet导数，并应用于非normed空间的优化问题。
The diametral dimension is an important topological invariant in the category of Frechet spaces which has been used, e.g., to distinguish types of Stein manifolds. We introduce variants of the classical definition in order to solve an old conjecture of Bessaga, Mityagin, Pelczynski, and Rolewicz at least for nuclear Frechet spaces. Moreover, we clarify the relation between an invariant recently introduced by Terzioglu and the by now classical condition $(\bar Ω)$ of Vogt and Wagner.
We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, this result implies that each closed convex subset of a Frechet space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals $0\le n\leω$, $0\le m\le 1$ and $k\ge 0$.
Baire-type properties of topological vector spaces
本文研究了拓扑向量空间的Baire型性质，引入了性质（MK），证明了局部完备的κ-弗雷歇-乌里索恩空间是Baire空间，并扩展了相关结果。
It is shown that every continuous homomorphism of Arens-Michael algebras can be obtained as the limit of a morphism of certain projective systems consisting of Fréchet algebras. Based on this we prove that a complemented subalgebra of an uncountable product of Fréchet algebras is topologically isomorphic to the product of Fréchet algebras. These results are used to characterize injective objects of the category of locally convex topological vector spaces. Dually, it is shown that a complemented subspace of an uncountable direct sum of Banach spaces is topologically isomorphic to the direct sum…
Let $(E,F)$ be a pair of Fréchet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_π F$. Specifically we focus on the two properties generalizing Köthe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fréchet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in…
We study the space of loops into a hypersurface complement, and show that the corresponding topological algebra of Laurent series with coefficients in $\mathcal{O}(L\mathbf{A}^{d}_{f})$ is a topological localisation of $\mathcal{O}(L\mathbf{A}^{d})$. This requires introducing a small amount of non-Archimedean functional analysis. In particular we work with topological algebras whose topology is generated by a family of sub-multiplicative, non-Archimedean semi-norms.
We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $ϕ: \X\to \Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(ϕ): H^n(\X) \to H^n(\Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E \hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ for nuclear F…
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。
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…
A topological space $X$ is $κ$-Fréchet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $κ$-Fréchet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $κ$-Fréchet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(κ)$.
We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor produc…
On automatic continuity of operators from ordered to topological vector spaces
本文研究了从有序到拓扑向量空间的算子连续性和有界性，探讨了Frechet空间到拓扑向量空间的算子自动连续性，特别研究了Levi和Lebesgue算子。
Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fréchet space or $E=ϕ$. We prove that the strong dual $E'_β$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fréchet--Montel space, so $E'_β$ is a sequential non-Fréchet--Urysohn space, or (ii) $E=ϕ$, so $E'_β= \mathbb{R}^ω$. Consequently, the space $\mathcal{D}(Ω)$ of test functions and the space of distribution…
Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fréchet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.
Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces
本文研究张量场高阶链式法则，利用嵌套集合划分与广义贝尔多项式推广 Faà di Bruno 公式，并给出 Sobolev-Slobodeckij、Musielak-Orlicz 范数及全变差半范数估计。
We give a survey on classical and recent results on dual spaces of topological tensor products as well as some examples where these are used.
We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups.
The work starts a series of papers on topological radicals and their applications. Among other results we present a theory of radicals related to the joint tensor radius.
The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $σ(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.
Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorp…
We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is concerned with properties of the~tensor rank that is a natural generalization of the matrix rank. The topological group structure of invertible matrices is involved in this study. The multilinear matrix multiplication is discussed from a viewpoint of transformation groups. We treat a low-rank tensor approximation in finite-dimensional tensor products. It is…
It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{y_{n}\right\}_{n=1}^{\infty}$ of linear continuous functionals in a Fréchet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n},x\right\rangle $ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fréchet space the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle $ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits hold…
A linear topological invariant for weighted spaces of holomorphic functions
本文研究由权函数系统定义的带状区域上急速衰减全纯函数Fréchet空间的线性拓扑不变量$(Ω)$，通过权函数系统的显式条件给出完全刻画，并应用于向量值光滑函数加权空间上Cauchy-Riemann算子的满射性研究。
Given a metrizable topological vector space, we can also use its von Neumann bornology or its bornology of precompact subsets to do analysis. We show that the bornological and topological approaches are equivalent for many problems. For instance, they yield the same concepts of convergence for sequences of points or linear operators, of continuity of functions, of completeness and completion. We also show that the bornological and topological versions of Grothendieck's approximation property are equivalent for Frechet spaces. These results are important for applications in noncommutative geome…
Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces
本文研究局部凸空间上算子的广义双曲性、拓扑稳定性与拓扑扩张性，建立其与有限跟踪、严格周期跟踪、混沌及传递性之间的关系，并刻画加权移位情形。
Linear topological invariants for kernels of differential operators by shifted fundamental solutions
本文利用满足特定性质的移位基本解，刻画了偏微分算子光滑核与分布核的线性拓扑条件$(Ω)$、$(PΩ)$及$(P\overline{\overlineΩ})$，并基于空间提升定理给出微分方程解参数依赖问题的特征与新证明。
A convenient technique for proving kernel theorems for (LF)-spaces (countable inductive limits of Frechet spaces)is developed. The proposed approach is based on introducing a suitable modification of the functor of the completed inductive topological tensor product. Using such modified tensor products makes it possible to prove kernel theorems without assuming the completeness of the considered (LF)-spaces. The general construction is applied to proving kernel theorems for a class of spaces of entire analytic functions arising in nonlocal quantum field theory.
Boundedness in Strict Deformation Quantization
本文通过复解析方法解释严格形变量子化中张量代数上的拓扑，证明其可视为有界Fréchet全纯函数代数，并推广了整函数阶的概念，同时给出全纯性与完备性的新结果。
We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\mathbb{R}^Γ$ for some $Γ$. Consequently, a Fréchet space $E$ is weakly Ascoli iff $E=\mathbb{R}^N$ for some $N\leqω$. If $X$ is a $μ$-space and a $k$-space (for example, metrizable),…
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