Preface Note to the Reader List of Symbols 0 Mathematical Preliminaries 0.1 Sets 0.2 Maps 0.3 Metric Spaces 0.4 Cardinality 0.5 Mathematical Induction 0.6 Problems I Finite-Dimensional Vector Spaces 1 Vectors and Transformations 1.1 Vector Spaces 1.2 Inner Product 1.3 Linear Transformations 1.4 Algebras 1.5 Problems 2 Operator Algebra 2.1 Algebra of L (V) 2.2 Derivatives of Functions of Operators 2.3 Conjugation of Operators 2.4 Hermitian and Unitary Operators 2.5 Projection Operators 2.6 Operators in Numerical Analysis 2.7 Problems 3 Matrices: Operator Representations 3.1 Matrices 3.2 Operations on Matrices 3.3 Orthonormal Bases 3.4 Change of Basis and Similarity Transformation 3.5 The Determinant 3.6 The Trace 3.7 Problems 4 Spectral Decomposition 4.1 Direct Sums 4.2 Invariant Subspaces 4.3 Eigenvalues and Eigenvectors 4.4 Spectral Decomposition 4.5 Functions of Operators 4.6 Polar Decomposition 4.7 Real Vector Spaces 4.8 Problems II Infinite-Dimensional Vector Spaces 5 Hilbert Spaces 5.1 The Question of Convergence 5.2 The Space of Square-Integrable Functions 5.3 Problems 6 Generalized Functions 6.1 Continuous Index 6.2 Generalized Functions 6.3 Problems 7 Classical Orthogonal Polynomials 7.1 General Properties 7.2 Classification 7.3 Recurrence Relations 7.4 Examples of Classical Orthogonal Polynomials …… III Complex Analysis IV Differential Equations V Operators on Hilbert Spaces VI Green'S Functions VII Groups and Manifolds VIII Lie Croups and Their Applications Bibvliography Index
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