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 GreenS Functions
VII Groups and Manifolds
VIII Lie Croups and Their Applications
Bibvliography
Index
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