Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces

The purpose of this book is to give a comprehensive introduction to several inequalities in Inner Product Spaces that have important applications in various topics of Contemporary Mathematics such as: Linear Operators Theory, Partial Differential Equations, Non-linear Analysis, Approximation Theory, Optimisation Theory, Numerical Analysis, Probability Theory, Statistics and other fields. Preface......Page 5 1.1. Introduction......Page 7 1.2. Hermitian Forms, Fundamental Properties......Page 8 1.3. Superadditivity and Monotonicity......Page 14 1.4. Applications for General Inner Product Spaces......Page 21 1.5. Applications for Sequences of Vectors......Page 33 Bibliography......Page 40 2.1. Introduction......Page 43 2.2. Inequalities Related to Schwarz's One......Page 44 2.3. Kurepa Type Refinements for the Schwarz Inequality......Page 52 2.4. Refinements of Buzano's and Kurepa's Inequalities......Page 57 2.5. Inequalities for Orthornormal Families......Page 64 2.6. Generalizations of Precupanu 's Inequality......Page 71 2.7. Some New Refinements of the Schwarz Inequality......Page 80 2.8. More Schwarz Related Inequalities......Page 94 Bibliography......Page 110 3.1. Introduction......Page 113 3.2. Some Inequalities of Diaz-Metcalf Type......Page 114 3.3. Additive Reverses for the Triangle Inequality......Page 118 3.4. Further Additive Reverses......Page 122 3.5. Reverses of Schwarz Inequality......Page 127 3.6. Quadratic Reverses of the Triangle Inequality......Page 128 3.7. Further Quadratic Refinements......Page 135 3.8. Reverses for Complex Spaces......Page 141 3.9. Applications for Vector-Valued Integral Inequalities......Page 148 3.10. Applications for Complex Numbers......Page 151 Bibliography......Page 152 4.1. Introduction......Page 157 4.2. Multiplicative Reverses......Page 158 4.3. Some Additive Reverses......Page 166 4.4. Quadratic Reverses of the Triangle Inequality......Page 178 4.5. Refinements for Complex Spaces......Page 184 4.6. Applications for Complex-Valued Functions......Page 192 Bibliography......Page 199 5.1. Introduction......Page 203 5.2. Some Reverse Inequalities......Page 204 5.3. Other Reverses......Page 219 Bibliography......Page 228 6.1. Bounds for the Distance to Finite-Dimensional Subspaces......Page 231 6.2. Reversing the CBS Inequality for Sequences......Page 245 6.3. Other Reverses of the CBS Inequality......Page 265 Bibliography......Page 277 Index......Page 279 Inequalities for hermitian forms -- Schwarz related inequalities -- Reverses for the triangle inequality -- Reverses for the continous triangle inequality -- Reverses of the cbs and heisenberg inequalities -- Other inequalities in inner product spaces