Weak convergence in infinite dimensional spaces

The Journal of Analysis

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1, 1,. .. , 1,. . .) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces.

European Journal of Mathematics

The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such spaces were named to honour Stefan Banach (1892-1945), one of the founders of Functional Analysis, who lived, worked and died in Lviv (now the largest city in western part of Ukraine). Of course, there are many important Banach spaces: spaces of sequences, functions, operators, etc. Yet, there exists one very concrete Banach space, called the Banach space. It includes numerous historical places in Lviv related to Stefan Banach: the houses where B Taras Banakh

arXiv (Cornell University), 2018

This book was intended to be Rafael Armario's Ph.D. thesis dissertation for obtaining his doctoral degree in Pure Mathematics at the University of Cádiz (Spain, EU). Unfortunately he passed away on January 28 th , 2013, right after he started writing this manuscript. His two Ph.D. advisors at the time of his death (the authors of this book) decided then to finish his work in book format and this is how this manuscript was given birth. 0.1 Rafael Armario's life Rafael Armario, better known as "Rafa" by his friends, was a joyful and warm person, a beloved son and friend, a very hard-working graduate student, an excellent highschool teacher, and an extremely brilliant mathematician. 0.1.1 Childhood and early life Rafa was born and raised in Cadiz, the oldest European city, within a humble middleclass family. He quickly developed a strong interest in the Cadiz culture and folklore, in particular, in the worldwide famous Cadiz carnival. Other local feasts such as the El Puerto fair also grew on him in a very profound way. 0.1.2 University of Cadiz undergraduate period Rafa decided to register as a math major in the university of Cadiz after graduating from his neighborhood high school. He met a lot of friends there among students and i ii PREFACE professors, including the one who left the strongest imprint on him, Prof. Antonio Aizpuru, who eventually became his Ph.D. advisor. He also met a bunch of friends a buddies like Fernandito, Mari, Sole, Marina, Marimar, el Gafas, etc. 0.2 Rafael Armario's work During his working period as a top-level mathematician, Rafa accomplished many objectives and obtained many different results. Indeed, all the theorems, definitions, examples, etc., which are anonymous in this book (including those of the framework) are due to his joint work with Prof. Aizpuru and the authors of this manuscript. 0.2.1 University of Cadiz graduate period Rafa's graduate period was undoubtedly conditioned by the death of his Ph.D. advisor at the time, Prof. Antonio Aizpuru, who passed away in March 2008 after Rafa and him published their first paper (see [5]), which was also co-authored by his Ph.D. co-advisor at the time, Prof. Francisco Javier Pérez-Fernández. In September 2010, Prof. García-Pacheco, better known as Paquito, was hired at the level of Assistant Professor by the Mathematics Department of the University of Cadiz after spending nine semesters (three American academic years) at Kent State University (Ohio, USA) and ten semesters at Texas A&M University (Texas, USA). One of the first assignments of Prof. García-Pacheco was to co-advice Rafael Armario together with Prof. Pérez-Fernández, continuing this way the work once started by Prof. Antonio Aizpuru. The fruit of this co-advising was the co-authorship of the papers [3, 4, 12, 13, 40]. The present book consists of those papers. Many anecdotes occurred during his graduate school period. For instance, in one occasion, Rafa and Paquito shared a hotel room, due to a lack of research funding, while attending a conference held in the university of Almeria (Spain, EU). The hotel workers will never forget that week.

2013

A sequence space is defined to be a linear space of real or complex sequences. Throughout the paper N , R and C denotes the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively. Let ω denote the space of all sequences (real or complex); l∞ and c respectively, denotes the space of all bounded sequences and the space of convergent sequences. Also, by cs we denote the space of all convergent series. Let X, Y be two sequence spaces and let A = (ank) be an infinite matrix of real or complex numbers ank, where n, k∈ N . Then, the matrix A defines the A−transformation from X into Y , if for every sequence x = (xk) ∈ X , the sequence Ax = {(Ax)n}, the A-transform of x exists and is in Y ; where (Ax)n = ∑ k ankxk.

Electronic Journal of Linear Algebra, 2022

Generalized locally Toeplitz (GLT) sequences of matrices originated from the spectral study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate either partial differential equations or fractional differential equations using any discretization by local methods (finite differences, finite elements, finite volumes, isogeometric analysis, etc.). The study of the asymptotic spectral behavior of GLT sequences is important in analyzing the solution of the corresponding partial differential equations and in finding fast and efficient methods for the corresponding large linear systems. Approximating classes of sequences (a.c.s.) and spectral symbols are important notions connected to GLT sequences. Recently, G. Barbarino obtained some results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudometric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various connections between the subalgebras of matrix-sequence spaces and the subalgebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. We also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. As an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.

Mathematische Nachrichten, 2011

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in L ∞ implies strong convergence in L p for all 1 ≤ p < ∞, weak convergence in L 1 v.s. strong convergence in L 1 and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on W 1,p .

Anais Da Academia Brasileira De Ciencias, 2003

In this paper, we prove that if a Nemytskii operator maps Lp( , E) into Lq( , F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p( ), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

Journal of Mathematical Analysis and Applications, 2015

We start the systematic study of Fréchet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ 0-space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space C c (X) of continuous realvalued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then C c (X) endowed with the weak topology is an ℵ 0-space if and only if X is countable. We extend Corson's result as follows: If the space E := C c (X) is a Fréchet lcs, then E endowed with its weak topology σ(E, E ′) is an ℵ-space if and only if (E, σ(E, E ′)) is an ℵ 0-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Fréchet lcs E in the weak topology σ(E, E ′) is an ℵ-space if and only if (E, σ(E, E ′)) is an ℵ 0-space if and only if E is separable. We show however that the nonseparable Banach space ℓ 1 (R) with the weak topology is an ℵ-space.

Journal of Function Spaces