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.

Table of Contents:
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.