This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra. Throughout the book, examples and applications have been interspersed with the theory. Each chapter concludes with numerous exercises and a section in which the author puts the results of that chapter into a historical perspective.

1. Inner Product Spaces.- Five Basic Problems.- Inner Product Spaces.- Orthogonality.- Topological Notions.- Hilbert Space.- Exercises.- Historical Notes.- 2. Best Approximation.- Best Approximation.- Convex Sets.- Five Basic Problems Revisited.- Exercises.- Historical Notes.- 3. Existence and Uniqueness of Best Approximations.- Existence of Best Approximations.- Uniqueness of Best Approximations.- Compactness Concepts.- Exercises.- Historical Notes.- 4. Characterization of Best Approximations.- Characterizing Best Approximations.- Dual Cones.- Characterizing Best Approximations from Subspaces.- Gram-Schmidt Orthonormalization.- Fourier Analysis.- Solutions to the First Three Basic Problems.- Exercises.- Historical Notes.- 5. The Metric Projection.- Metric Projections onto Convex Sets.- Linear Metric Projections.- The Reduction Principle.- Exercises.- Historical Notes.- 6. Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-Spaces.- Bounded Linear Functionals.- Representation of Bounded Linear Functionals.- Best Approximation from Hyperplanes.- Strong Separation Theorem.- Best Approximation from Half-Spaces.- Best Approximation from Polyhedra.- Exercises.- Historical Notes.- 7. Error of Approximation.- Distance to Convex Sets.- Distance to Finite-Dimensional Subspaces.- Finite-Codimensional Subspaces.- The Weierstrass Approximation Theorem.- Müntz's Theorem.- Exercises.- Historical Notes.- 8. Generalized Solutions of Linear Equations.- Linear Operator Equations.- The Uniform Boundedness and Open Mapping Theorems.- The Closed Range and Bounded Inverse Theorems.- The Closed Graph Theorem.- Adjoint of a Linear Operator.- Generalized Solutions to Operator Equations.- Generalized Inverse.- Exercises.- Historical Notes.- 9. The Method of AlternatingProjections.- The Case of Two Subspaces.- Angle Between Two Subspaces.- Rate of Convergence for Alternating Projections (two subspaces).- Weak Convergence.- Dykstra's Algorithm.- The Case of Affine Sets.- Rate of Convergence for Alternating Projections.- Examples.- Exercises.- Historical Notes.- 10. Constrained Interpolation from a Convex Set.- Shape-Preserving Interpolation.- Strong Conical Hull Intersection Property (Strong CHIP).- Affine Sets.- Relative Interiors and a Separation Theorem.- Extremal Subsets of C.- Constrained Interpolation by Positive Functions.- Exercises.- Historical Notes.- 11. Interpolation and Approximation.- Interpolation.- Simultaneous Approximation and Interpolation.- Simultaneous Approximation, Interpolation, and Norm-preservation.- Exercises.- Historical Notes.- 12. Convexity of Chebyshev Sets.- Is Every Chebyshev Set Convex?.- Chebyshev Suns.- Convexity of Boundedly Compact Chebyshev Sets.- Exercises.- Historical Notes.- Appendix 1. Zorn's Lemma.- References.