Functional Analysis and Operator Theory de Volodymyr Brayman

Functional Analysis and Operator Theory: Problem Books in Mathematics

Autor Volodymyr Brayman, Andrii Chaikovskyi, Oleksii Konstantinov, Alexander Kukush, Yuliya Mishura, Oleksii Nesterenko

en Limba Engleză Hardback – 19 iun 2024

În procesul de învățare a matematicii avansate, interfața dintre analiza matematică, algebra liniară și ecuațiile diferențiale este locul unde se cristalizează înțelegerea profundă. Găsim în lucrarea Functional Analysis and Operator Theory o resursă pedagogică riguroasă, care pune accent pe interdisciplinaritate, solicitând cititorului cunoștințe prealabile de calcul, teoria măsurii și geometrie analitică pentru a naviga prin complexitatea operatorilor liniari.

Această ediție 2024, apărută sub egida Springer, se distinge prin abordarea pragmatică a celor peste 800 de probleme. Structura este concepută pentru a facilita tranziția de la teorie la practică: fiecare capitol debutează cu o sinteză teoretică necesară înțelegerii enunțurilor, urmată de exemple rezolvate care servesc drept model metodologic. Apreciem în mod deosebit includerea unor teme precum funcțiile generalizate și ecuațiile integrale, care oferă o perspectivă aplicată asupra disciplinei. Comparativ cu A Concise Introduction to Functional Analysis de César R. de Oliveira, care este structurat pentru un curs de un semestru, volumul de față servește ca o alternativă robustă pentru cursurile de analiză funcțională axate pe rezolvarea de probleme, având avantajul unui volum mult mai mare de exerciții aplicate și soluții detaliate.

Poziționată în contextul operei colectivului de autori, care include și Undergraduate Mathematics Competitions (1995-2016), această lucrare păstrează spiritul riguros al competițiilor matematice, dar îl adaptează nevoilor curriculare universitare. Progresia conținutului, de la spații Banach la teoria spectrală a operatorilor compacți, este logică și liniară, oferind instrumentele necesare atât pentru studiul individual, cât și pentru pregătirea seminarelor de specialitate.

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Specificații

ISBN-13: 9783031564260
ISBN-10: 303156426X
Pagini: 364
Ilustrații: XVI, 346 p.
Dimensiuni: 160 x 241 x 24 mm
Greutate: 0.78 kg
Ediția:2024
Editura: Springer
Colecția Problem Books in Mathematics
Seria Problem Books in Mathematics

Locul publicării:Cham, Switzerland

De ce să citești această carte

Recomandăm această carte studenților și cercetătorilor care doresc să treacă dincolo de demonstrațiile teoretice abstracte. Prin cele 800 de exerciții, cititorul câștigă o dexteritate tehnică esențială în manipularea operatorilor pe spații Hilbert și Banach. Este un instrument de lucru indispensabil pentru pregătirea examenelor și pentru aprofundarea fundamentelor analizei funcționale moderne, oferind soluții clare acolo unde intuiția poate eșua.

Despre autor

Echipa de autori, formată din matematicieni recunoscuți precum Volodymyr Brayman, Alexander Kukush și Yuliya Mishura, aduce o expertiză vastă în pedagogia matematică și cercetarea academică. Experiența lor colectivă este reflectată în calitatea selecției problemelor, mulți dintre aceștia fiind implicați activ în organizarea competițiilor internaționale de matematică, așa cum demonstrează lucrarea lor anterioară despre olimpiadele studențești. Expertiza lor acoperă domenii diverse, de la analiză stocastică la teoria operatorilor, asigurând o tratare riguroasă și exhaustivă a subiectelor abordate în cadrul seriei Problem Books in Mathematics.

Descriere scurtă

The book contains a collection of more than 800 problems from all main chapters of functional analysis, with theoretical background and solutions. It is mostly intended for undergraduate students who are starting to study the course of functional analysis. The book will also be useful for graduate and post- graduate students and researchers who wish to refresh their knowledge and deepen their understanding of the subject, as well as for teachers of functional analysis and related disciplines. It can be used for independent study as well. It is assumed that the reader has mastered standard courses of calculus and measure theory and has basic knowledge of linear algebra, analytic geometry, and differential equations.
This collection of problems can help students of different levels of training and different areas of specialization to learn how to solve problems in functional analysis. Each chapter of the book has similar structure and consists of the following sections: Theoretical Background, Examples of Problems with Solutions, and Problems to Solve. The book contains theoretical preliminaries to ensure that the reader understands the statements of problems and is able to successfully solve them. Then examples of typical problems with detailed solutions are included, and this is relevant not only for those students who have significant difficulties in studying this subject, but also for other students who due to various circumstances сcould be deprived of communication with a teacher. There are problems for independent solving, and the corresponding selection of problems reflects all the main plot lines that relate to a given topic.
The number of problems is sufficient both for a teacher to give practical lessons, to set homework, to prepare tasks for various forms of control, and for those students who want to study the discipline more deeply. Problems of a computational nature are provided with answers, while theoretical problems, the solutions ofwhich require non-trivial ideas or new techniques, are provided with detailed hints or solutions to introduce the reader to the corresponding ideas or techniques.

Cuprins

Preface.- Banach Spaces.- Hilbert Spaces.- Continuous Linear Functionals.- Hahn - Banach Theorem.- Weak and Weak* Convergence.- Bounded Linear Operators.- Uniform, Strong and Weak Operator Convergences.- Inverse Operators.- Classes of Linear Operators in Hilbert Space.- Compact Sets and Operators.- Spectrum of Linear Operators.- Spectral Theory of Compact Operators.- Integral Equations.- Generalized Functions.- Answers, hints and solutions.- List of notations.- References.

Notă biografică

Dr. Volodymyr Brayman received his PhD in probability and statistics from the Institute of Mathematics of National Academy of Sciences of Ukraine in 2007. He is currently an Assistant Professor at the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is an expert in stochastic processes, a jury member in various mathematical competitions, and an author of numerous problems proposed at the competitions. He co-authored with Prof. Alexander Kukush a problem book titled «Undergraduate Mathematics   Competitions (1995–2016)», Taras Shevchenko National University of Kyiv (ISBN 978-3-319-58672-4), published with Springer.
Prof. Andrii Chaikovskyi received his PhD in differential equations from Taras Shevchenko National University of Kyiv in 2001 and completed his postdoctoral degree in differential equations (Habilitation) in 2012. He is currently the Head of the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 60 research papers. His research interests include abstract differential and difference equations and approximation theory.
Dr. Oleksii Konstantinov is an Associate Professor  of the Department of Mathematical Analysis   at Taras Shevchenko National University of Kyiv. His research interests include Operator Theory, Mathematical Scattering Theory and Differential Operators.
Prof. Alexander Kukush received his PhD in probability and statistics from Kyiv University in 1982 and completed his postdoctoral degree in probability and statistics (Habilitation) in 1995. He is a Professor of the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 150 research papers and 4 books. His research interests include mathematical and applied statistics, actuarial and financial mathematics.
Prof. Yuliya Mishura received her PhD in probability and statistics from Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a Professor   of the Department of Probability, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. She is the author/coauthor of more than 300 research papers and 12 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, fractional calculus and fractional processes, stochastic analysis.Dr. Oleksii Nesterenko received his PhD in mathematical analysis from Taras Shevchenko National University of Kyiv in 2007. He is currently an Associate Professor of the Department of Mathematical Analysis of Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 10 research papers. His research interests include approximation theory.

Caracteristici

The book contains problems related to main topics of the standard undergraduate courses of functional analysis Vast spectrum of problems of varying difficulty is presented for students to work on independently Answers and detailed hints or solutions are provided to almost all the problems