Recent zbMATH articles in MSC 46Ahttps://www.zbmath.org/atom/cc/46A2021-02-27T13:50:00+00:00WerkzeugOn various functional representations of the space of Schwartz operators.https://www.zbmath.org/1453.810052021-02-27T13:50:00+00:00"Amosov, G. G."https://www.zbmath.org/authors/?q=ai:amosov.grigori-gSummary: In this paper, we discuss various representations in which the space \(S\) of Schwartz operators turns into the space of test functions, whereas the dual space \(S'\) turns into the space of generalized functions.A course on topological vector spaces.https://www.zbmath.org/1453.460012021-02-27T13:50:00+00:00"Voigt, Jürgen"https://www.zbmath.org/authors/?q=ai:voigt.jurgenThis booklet of just about 150 pages provides a concise yet very readable treatment of the basics of the theory of topological vector spaces which was developed from the 1940's till the 1970's. Many classical treatises of that period such as the books of Bourbaki or Köthe were written in a strict Bourbaki style, always presenting the most general and structural viewpoint, and are not easy to read. Compared to this style, the present author's course provides an introduction in a streamlined and reader-friendly way -- the somewhat baroque typography of the classics is cleaned up and the typesetting, e.g., of the theorems in coloured boxes makes orientation in this book very comfortable.
The main topics covered by the book are the principles of topological vector spaces which serve as tools, e.g., for partial differential equations or semigroup theory on Banach spaces. The course treats
\begin{itemize}
\item basic facts and constructions for topological vector spaces like initial and final locally convex topologies,
\item implications of the Hahn-Banach theorem for duality theory (presented for general dual pairs) with fortunately only a few of the huge zoo of locally convex properties (such as barrelled or bornological spaces),
\item completeness properties of topological vector spaces including Grothendieck's construction of the completion, and
\item some basic facts about Fréchet spaces and the dual concept of DF-spaces.
\end{itemize}
A relatively large part of the book is devoted to (weak) compactness, including standard characterizations of reflexivity and compactness criteria of Eberlein, Grothendieck, Krein, and Šmulian.
Almost entirely avoided are systematic results about the \emph{morphisms} of the category of topological vector spaces, i.e., continuous linear operators. Therefore, important topics such as closed graph theorems, adjoints, or tensor products are not covered -- even the uniform boundedness principle of Banach and Steinhaus is only stated for families of functionals instead of operators.
The book is thus not a panorama of the theory, but confines itself strictly to the tools serving for other disciplines of functional analysis.
Within this goal, the author presents a compact, precise, and very well written introduction to the theory of topological vector spaces, including many examples and, due to the restricted length, only a few applications (such as Bernstein's characterization of completely monotone functions via the Krein-Milman theorem). The text is practically free of typos and errors (the only exceptions the referee noticed are the definition of the absolutely convex hull, to make the theorem of bipolars correct the hull of \(\emptyset\) should not be empty but \(\{0\}\), and a missing reference that Theorem~7.11 is V.\,Klee's solution to a problem posed by Banach).
The book may be highly recommended to all students and researchers with some knowledge of Banach or Hilbert space oriented functional analysis who want to learn its general abstract foundations.
Reviewer: Jochen Wengenroth (Trier)On bornologicalness in locally convex algebras.https://www.zbmath.org/1453.460442021-02-27T13:50:00+00:00"Haralampidou, Marina"https://www.zbmath.org/authors/?q=ai:haralampidou.marina"Oudadess, Mohamed"https://www.zbmath.org/authors/?q=ai:oudadess.mohamed"Palacios, Lourdes"https://www.zbmath.org/authors/?q=ai:palacios.lourdes"Signoret, Carlos"https://www.zbmath.org/authors/?q=ai:signoret.carlos-j-eThe authors define six subclasses of bornological algebras within the class of locally convex algebras and study the properties of these subclasses of topological algebras. They show that some subclasses are more general than others and provide characterizations of four subclasses of bornological algebras within the class of Hausdorff locally convex algebras. The characterizations use some properties of continuity of vector space seminorms and linear mappings to any locally convex space.
Quite a long part of the paper is dedicated to examples of topological algebras which belong to some of the subclasses of bornological algebras. The authors provide examples showing that these subclasses of bornological algebras are really different from each other.
In the last two chapters, the case of locally \(A\)-convex algebras and the case of pseudo-Banach algebras are considered and some bornology-related results for these cases are obtained.
Reviewer: Mart Abel (Tartu)Direct limits of adèle rings and their completions.https://www.zbmath.org/1453.111512021-02-27T13:50:00+00:00"Kelly, James P."https://www.zbmath.org/authors/?q=ai:kelly.james-pierre"Samuels, Charles L."https://www.zbmath.org/authors/?q=ai:samuels.charles-lFor a Galois extension \(E/F\), with F a global field, the paper defines a topological ring, denoted by \(\overline{\mathbb{V}}_E\), called the generalized adèle ring of \(E\).
Denote by \(\mathcal{J}_E\) the set \(\{K \subseteq E : K/F\text{ finite Galois}\}\), by \(\mathbb{A}_K\) the adèle ring of \(K\) and by \(\overline{\mathbb{A}}_K\) its completion with respect to some (any) invariant metric on \(\mathbb{A}_K\).
Main theorems are now stated in a short form.
Theorem 1. If \(E/F\) is a Galois extension, then the following hold:
\begin{itemize}
\item[i)] \(\overline{\mathbb{V}}_E\) is a metrizable topological ring which is complete with respect to any invariant metric on \(\overline{\mathbb{V}}_E\).
\item[ii)] If \(\mathbb{V}_E = \bigcup_{K\in\mathcal J_E} \mathbb{V}_K\), then \(\overline{\mathbb{V}}_E\) equals the closure of \(\mathbb{V}_E\) in \(\overline{\mathbb{V}}_E\).
\item[iii)] There exists a topological ring isomorphism \(\phi: \overline{\mathbb{A}}_E \to \overline{\mathbb{V}}_E\) such that \(\phi(\mathbb{A}_E) = \mathbb{V}_E\).
\end{itemize}
Theorem 2.
If \(E/F\) is an infinite Galois extension, then \(\mathbb{A}_E\) has empty interior in \(\overline{\mathbb{A}}_E\).
Reviewer: Stelian Mihalas (Timişoara)Convex linear metric spaces are normable.https://www.zbmath.org/1453.460032021-02-27T13:50:00+00:00"Singh, Jitender"https://www.zbmath.org/authors/?q=ai:singh.jitender"Narang, T. D."https://www.zbmath.org/authors/?q=ai:narang.tulsi-dasThe authors call a linear metric space \((X,d)\) over the field \(\mathbb R\) convex if \(d(\lambda x+(1-\lambda)y,0)\le\lambda d(x,0)+(1-\lambda)d(y,0)\) holds for all \(x,y\in X\) and each \(\lambda\), \(0\le\lambda\le1\). They prove that each (real) convex linear metric space is normable (with the norm given by \(\|x\|=d(x,0)\). The same is true in the case of complex scalars if the metric \(d\) is rotation invariant (i.e., \(d(\alpha x,0)=d(|\alpha|x,0)\) for all complex scalars \(\alpha\) and all \(x\in X\)), but not in general.
Reviewer: Zoran Kadelburg (Beograd)On some double Cesàro sequence spaces.https://www.zbmath.org/1453.460062021-02-27T13:50:00+00:00"Sever, Yurdal"https://www.zbmath.org/authors/?q=ai:sever.yurdal"Altay, Bilal"https://www.zbmath.org/authors/?q=ai:altay.bilalSummary: In this study, we define the double Cesàro sequence spaces Ces$_p$, Ces$_{bp}$ and Ces$_{bp0}$ and examine some properties of those sequence spaces. Furthermore, we determine the $\beta(bp)$-duals of the spaces Ces$_{bp}$ and Ces$_p$.On absolute Riesz summability factors of infinite series and their application to Fourier series.https://www.zbmath.org/1453.260192021-02-27T13:50:00+00:00"Bor, Hüseyin"https://www.zbmath.org/authors/?q=ai:bor.huseyinSummary: In this paper, some known results on the absolute Riesz summability factors of infinite series and trigonometric Fourier series have been generalized for the \(|\bar{N}, p_n; \theta_n|_k\) summability method. Some new and known results are also obtained.\(C^*\)-algebra-valued modular spaces and fixed point theorems.https://www.zbmath.org/1453.470092021-02-27T13:50:00+00:00"Shateri, Tayebe Lal"https://www.zbmath.org/authors/?q=ai:shateri.tayebeh-lalSummary: In the present paper a concept of \(C^*\)-algebra-valued modular space is introduced which is a generalization of a modular space. Next, some fixed point theorems for self-maps with contractive or expansive conditions on such spaces are proved.Almost convergence and Euler totient matrix.https://www.zbmath.org/1453.460042021-02-27T13:50:00+00:00"Demiriz, Serkan"https://www.zbmath.org/authors/?q=ai:demiriz.serkan"İlkhan, Merve"https://www.zbmath.org/authors/?q=ai:ilkhan.merve"Kara, Emrah Evren"https://www.zbmath.org/authors/?q=ai:kara.emrah-evrenThe \(\Phi\)-transform of complex-valued sequences is defined by
\[
v_{n}=\sum_{k\mid n}\varphi( k) u_{k},\ n\in\mathbb{N},
\]
where \(\varphi\) denotes the Euler totient function. Let
\[
\widehat{c}\left(\Phi\right) :=\left\{(u_{k})\biggm\vert\exists\alpha\in\mathbb{C}:\lim_{m\rightarrow\infty}\sum_{j=0}^{m}\frac{v_{n+j}}{m+1}=\alpha\text{ uniformly in }n\right\}.
\]
From the authors' abstract: ``It is proved that the space \(\widehat{c}\left(\Phi\right)\) and the space of all almost convergent sequences are linearly isomorphic. Further, the \(\beta\)-dual of the space \(\widehat{c}\left(\Phi\right)\) is determined and Euler totient core of a complex-valued sequence has been defined. Finally, inclusion theorems related to this new
type of core are obtained.''
Reviewer: Toivo Leiger (Tartu)A quantitative version of Helly's selection principle in Banach spaces and its applications.https://www.zbmath.org/1453.460072021-02-27T13:50:00+00:00"García, Gonzalo"https://www.zbmath.org/authors/?q=ai:garcia.gonzalo-aHelly's selection principle states that every sequence of real valued functions, defined on a compact interval, of uniformly bounded variation, has a pointwise converging subsequence. In other words, a set of real and uniformly bounded variation functions defined on a compact interval is sequentially compact in the pointwise topology. The main result of this paper is a quantitative version of this principle. The proofs use the so-called degree of nondensifiability, which measures (in a specified sense) the distance of a given convex subset of a Banach space to the class of its Peano continua. As application of the general results, the author analyzes the solvability of certain Volterra integral equations, the integral being considered in the sense of Bochner.
Reviewer: Ioan Raşa (Cluj-Napoca)Spaces of strongly lacunary invariant summable sequences.https://www.zbmath.org/1453.460052021-02-27T13:50:00+00:00"Savaş, E."https://www.zbmath.org/authors/?q=ai:savas.ekremSummary: In this paper, we introduce and examine some properties of three sequence spaces defined using lacunary sequence and invariant mean which generalize several known sequence spaces.On modulated topological vector spaces and applications.https://www.zbmath.org/1453.460082021-02-27T13:50:00+00:00"Kozlowski, Wojciech M."https://www.zbmath.org/authors/?q=ai:kozlowski.wojciech-mIf \(X\) is a real vector space, an even, convex function \(\rho:X\to [0,\infty]\) vanishing at \(0\) is a \textit{convex modular} and the set \(X_\rho = \{x\in X: \rho(\lambda x)\to 0\ \text{as}\ \lambda\to 0\}\) is a \textit{modular space}. In [Fixed point theory in modular function spaces. Cham: Birkhäuser/Springer (2015; Zbl 1318.47002)], \textit{M.\,A.\thinspace Khamsi} and the author published a treatise on fixed point theory in modular function spaces.
In the article under review, the author introduces a class of vector spaces that includes Banach spaces, modular function spaces, and many others, and obtains fixed point results in this larger class of vector spaces. With \(\rho\) and \(X_\rho\) as above and \(\tau\) a linear Hausdorff topology on \(X\), the triple \((X_\rho,\rho,\tau)\) is a \textit{modulated topological vector space} if \(\rho\) is \(\tau\)-lower semicontinuous on \(X\) and, if whenever the sequence \((x_n)\) in \(X\) \(\rho\)-converges to \(x\) in \(X\), there exists a subsequence of \((x_n)\) that is \(\tau\)-convergent to \(x\). The author uses a modular analogue of normal structure to obtain an analogue of Kirk's well-known fixed point theorem in the setting of modulated topological vector spaces. In particular, let \((X_\rho, \rho, \tau)\) be a \(\rho\)-complete modulated topological vector space and let \(C\) be a nonempty, \(\rho\)-closed, \(\rho\)-bounded subset of \(X_\rho\). Then, if \(C\) is \(\tau\)-sequentially compact and satisfies a \(\rho\)-analogue of normal structure, then every \(\rho\)-nonexpansive mapping \(T:C\to C\) has a fixed point. The author notes that analogous results can be obtained if \(\rho\) is an \(s\)-convex modular for any \(s>0\).
Reviewer: Barry Turett (Rochester)