viii Contents Chapter 3. Hilbert Spaces 67 3.1. Inner products 67 3.2. Orthogonality 71 3.3. Adjoints 76 3.4. Some historical notes 85 Chapter 4. The Spectral Theorem 87 4.1. The Spectral Theorem 87 4.2. Quadratic forms and quadratic surfaces 94 4.3. Critical points 98 4.4. The Spectral Theorem for real linear spaces 103 4.5. The functional calculus 106 4.6. The commutant 111 4.7. Unitarily equivalent hermitian transformations 113 Chapter 5. Matrices and Topology 117 5.1. Euclidean topology 118 5.2. The topology on the space of matrices 120 5.3. The general linear group 126 5.4. A polynomial interlude 132 5.5. Sets defined by the adjoint operation 137 Chapter 6. Modules 143 6.1. Definitions and examples 143 6.2. Submodules 147 6.3. Simple rings 154 6.4. A ring theory interlude 162 6.5. Factorization in rings 167 6.6. Cyclic modules 172 6.7. Torsion modules over a principal ideal domain 179 6.8. Applications to linear algebra 188 6.9. Bases and free modules 197 6.10. Finitely generated modules over a principal ideal domain 201 6.11. Finitely generated abelian groups 204 Appendix 207 A.1. Groups 208 A.2. Rings 214 A.3. Vector spaces 218
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