Cover -- Title -- Contents -- Preface -- 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.4.1 Using matrices -- I.4.2 Using polynomials -- 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 C^n -- 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.