本书细致且友好地讲述了一些相当有趣的数学内容。作者首先定义了复数域，并在前几章中对一些标准的数学分析内容做了新颖的介绍。作者从最近的研究文献所引入的一些成果，将本书推向高潮，对这些成果，本书给出易于理解的完整证明和一些令人惊讶的推论。一个统一主题是希尔伯特第十七问题的复变量类似。书中大量的例子、练习以及关于几何推理的讨论，对读者十分有益。本书可供数学专业的本科生和研究生阅读，也可供物理学家和工程师参考。 ---------------------------------------------------------------------------------------- 本书的文字优美、案例生动、练习明晰。 —Steven Krantz, Washington University, St. Louis

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

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