前辅文
 I Complex Numbers
  I.1 The real number system
  I.2 Definition of the complex number field
  I.3 Elementary complex geometry
  I.4 Alternative definitions of the complex numbers
  I.5 Completeness
  I.6 Convergence for power series
  I.7 Trigonometry
  I.8 Roots of unity
  I.9 Summary
 II Complex Euclidean Spaces and Hilbert Spaces
  II.1 Hermitian inner products
  II.2 Orthogonality, projections and closed subspaces
  II.3 Orthonormal expansion
  II.4 The polarization identity
  II.5 Generating functions and orthonormal systems
 III Complex Analysis in Several Variables
  III.1 Holomorphic functions
  III.2 Some calculus
  III.3 The Bergman kernel function
 IV Linear Transformations and Positivity Conditions
  IV.1 Adjoints and Hermitian forms
  IV.2 Solving linear equations
  IV.3 Linearization
  IV.4 Eigenvalues and the spectral theorem in finite dimensions
  IV.5 Positive definite linear transformations in finite dimensions
  IV.6 Hilbert’s inequality
  IV.7 Additional inequalities from Fourier analysis
 V Compact and Integral Operators
  V.1 Convergence properties for bounded linear transformations
  V.2 Compact operators on Hilbert space
  V.3 The spectral theorem for compact Hermitian operators
  V.4 Integral operators
  V.5 A glimpse at singular integral operators
 VI Positivity Conditions for Real-valued Functions
  VI.1 Real variables analogues
  VI.2 Real-valued polynomials on Cn
  VI.3 Squared norms and quotients of squared norms
  VI.4 Plurisubharmonic functions
  VI.5 Positivity conditions for polynomials
 VII Stabilization and Applications
  VII.1 Stabilization for positive bihomogeneous polynomials
  VII.2 Positivity everywhere
  VII.3 Positivity on the unit sphere
  VII.4 Applications to proper holomorphic mappings between balls
  VII.5 Positivity on zero sets
  VII.6 Proof of stabilization
 VIII Afterword
 Appendix A
  A.1 Algebra
  A.2 Analysis
 Bibliography
 Index