This book evolved from notes originally developed for a graduate course, "Best Approximation in Normed Linear Spaces," that I began giving at Penn State Uni­ versity more than 25 years ago. It soon became evident. that many of the students who wanted to take the course (including engineers, computer scientists, and statis­ ticians, as well as mathematicians) did not have the necessary prerequisites such as a working knowledge of Lp-spaces and some basic functional analysis. (Today such material is typically contained in the first-year graduate course in analysis. ) To accommodate these students, I usually ended up spending nearly half the course on these prerequisites, and the last half was devoted to the "best approximation" part. I did this a few times and determined that it was not satisfactory: Too much time was being spent on the presumed prerequisites. To be able to devote most of the course to "best approximation," I decided to concentrate on the simplest of the normed linear spaces-the inner product spaces-since the theory in inner product spaces can be taught from first principles in much less time, and also since one can give a convincing argument that inner product spaces are the most important of all the normed linear spaces anyway. The success of this approach turned out to be even better than I had originally anticipated: One can develop a fairly complete theory of best approximation in inner product spaces from first principles, and such was my purpose in writing this book.

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.