This book studies algebras and linear transformations acting on finite -dimensional vector spaces over arbitrary fields. It is written for students who have prior knowledge of algebra and linear algebra. The goal is to present a balance of theory and example in order for students to gain a firm understanding of the basic theory of finite- dimensional algebras and to provide a foundation for subsequent advanced study in a number of areas of mathematics. As such, the level of exposition is suitable for senior undergraduate students.
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Lower undergraduate
Illustrationen
Maße
Höhe: 241 mm
Breite: 160 mm
Dicke: 19 mm
Gewicht
ISBN-13
978-0-387-95062-4 (9780387950624)
DOI
10.1007/978-1-4613-0097-7
Schweitzer Klassifikation
1. Linear Algebra.- 1.1 Vector Spaces and Duality.- 1.2 Direct Sums and Quotients.- 1.3 Inner-Product Spaces.- 1.4 The Spectral Theorem.- 1.5 Fields and Field Extensions.- 1.6 Existence of Bases for Infinite-Dimensional Spaces.- 1.7 Notes.- 1.8 Exercises.- 2. Algebras.- 2.1 Algebrai c Structures.- 2.2 Algebras with a Prescribed Basis.- 2.3 Algebras of Linear Transformations.- 2.4 Inversion and Spectra.- 2.5 Division Algebras and Other Simple Algebras.- 2.6 Notes.- 2.7 Exercises.- 3. Invariant Subspaces.- 3.1 The Invariant-Subspace Lattice.- 3.2 Idempotents and Projections.- 3.3 Existence of Invariant Subspaces.- 3.4 Representations and Left Ideals.- 3.5 Functional Calculus and Polar Decomposition.- 3.6 Notes.- 3.7 Exercises.- 4. Semisimple Algebras.- 4.1 Nilpotent Algebras and the Nil Radical.- 4.2 Structure of Semisimple Algebras.- 4.3 Structure of Simple Algebras.- 4.4 Isomorphism Classes of Semisimple Algebras.- 4.5 Notes.- 4.6 Exercises.- 5. Operator Algebras.- 5.1 Von Neumann Algebras.- 5.2 Real and Complex Involutive Algebras.- 5.3 Representation of Operator Algebras.- 5.4 Wedderburn Theorems for Operator Algebras.- 5.5 C*-Algebras.- 5.5 Notes.- 5.7 Exercises.- 6. Tensor Products.- 6.1 Free Vector Spaces.- 6.2 Tensor Products of Vector Spaces.- 6.3 Tensor Products of Algebras.- 6.4 Tensor Products of Operator Algebras.- 6.5 Notes.- 6.6 Exercises.- References.
