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This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course focuses on three major topics: the elementary theory of Schauder bases, an introduction to Lp spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, our primary interest is in the postwar renaissance of Banach space theory brought about by James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their elegant and insightful results are useful in many contemporary research endeavors and deserve greater publicity. By way of prerequisites, the reader will need an elementary understanding of functional analysis and at least a passing familiarity with abstract measure theory. An introductory course in topology would also be helpful; however, the text includes a brief appendix on the topology needed for the course.

Banach spaces. --- Banach, Espaces de --- Functions of complex variables --- Generalized spaces --- Topology

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This elementary introduction was developed from lectures by the authors on business mathematics and the lecture "Analysis and Linear Algebra" for Bachelor's degree programmes. It is designed for courses in business administration and business informatics at universities, universities of applied sciences and cooperative universities. With the 5th edition, the title was changed to "Analysis and Linear Algebra". The treatment of sequences and series has been added and some exercises have been added to the introductory chapters. The focus is on teaching mathematical basics with regard to applications in business and financial mathematics. Contents from the upper secondary school are repeated in a compact form. Numerous examples and exercises make the book clear and promote understanding of interrelationships. The introduction is therefore also suitable for A-level students at business schools. The detailed solutions to the exercises are provided on the book's website. The book is therefore also very suitable for self-study. The contents Elementary basics Functions Differential calculus Integral calculus Linear Algebra Functions with several variables Financial mathematics The Authors Prof. Dr. Thomas Holey is head of the Business Information Systems programme at the Baden-Württemberg Cooperative State University Mannheim and represents the basic mathematical subjects in teaching. Prof. Dr. Armin Wiedemann teaches formal methods of computer science as well as the mathematical subjects at the Baden-Württemberg Cooperative State University Mannheim. He is retired now.

Algebras, Linear. --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology

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This book aims to give an encyclopedic overview of the state-of-the-art of Krylov subspace iterative methods for solving nonsymmetric systems of algebraic linear equations and to study their mathematical properties. Solving systems of algebraic linear equations is among the most frequent problems in scientific computing; it is used in many disciplines such as physics, engineering, chemistry, biology, and several others. Krylov methods have progressively emerged as the iterative methods with the highest efficiency while being very robust for solving large linear systems; they may be expected to remain so, independent of progress in modern computer-related fields such as parallel and high performance computing.The mathematical properties of the methods are described and analyzed along with their behavior in finite precision arithmetic. A number of numerical examples demonstrate the properties and the behavior of the described methods. Also considered are the methods’ implementations and coding as Matlab®-like functions. Methods which became popular recently are considered in the general framework of Q-OR (quasi-orthogonal )/Q-MR (quasi-minimum) residual methods. This book can be useful for both practitioners and for readers who are more interested in theory. Together with a review of the state-of-the-art, it presents a number of recent theoretical results of the authors, some of them unpublished, as well as a few original algorithms. Some of the derived formulas might be useful for the design of possible new methods or for future analysis. For the more applied user, the book gives an up-to-date overview of the majority of the available Krylov methods for nonsymmetric linear systems, including well-known convergence properties and, as we said above, template codes that can serve as the base for more individualized and elaborate implementations.

Numerical analysis. --- Numerical Analysis. --- Mathematical analysis --- Algebras, Linear. --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Calculus of operations --- Line geometry --- Topology

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Algebras, Linear. --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- Àlgebra lineal

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This book is based on a course for first-semester science students, held by the second author at the University of Zurich several times. Its goal is threefold: to have students learn a minimal working knowledge of linear algebra, acquire some computational skills, and familiarize them with mathematical language to make mathematical literature more accessible. Therefore, we give precise definitions, introduce helpful notations, and state any results carefully worded. We provide no proofs of these results but typically illustrate them with numerous examples. Additionally, for better understanding, we often give supporting arguments for why they are valid.

Algebras, Linear. --- Algebra. --- Linear Algebra. --- Mathematics --- Mathematical analysis --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Calculus of operations --- Line geometry --- Topology