I. Introduction.- II. Basic Properties of n-Widths.- 1. Properties of dn.- 2. Existence of Optimal Subspaces for dn.- 3. Properties of dn.- 4. Properties of ?n.- 5. Inequalities Between n-Widths.- 6. Duality Between dn and dn.- 7. n-Widths of Mappings of the Unit Ball.- 8. Some Relationships Between dn(T), dn(T) and ?n(T).- Notes and References.- III. Tchebycheff Systems and Total Positivity.- 1. Tchebycheff Systems.- 2. Matrices.- 3. Kernels.- 4. More on Kernels.- IV. n-Widths in Hilbert Spaces.- 1. Introduction.- 2. n-Widths of Compact Linear Operators.- 3. n-Widths, with Constraints.- 3.1 Restricted Approximating Subspaces.- 3.2 Restricting the Unit Ball and Optimal Recovery.- 3.3 n-Widths Under a Pair of Constraints.- 3.4 A Theorem of Ismagilov.- 4. n-Widths of Compact Periodic Convolution Operators.- 4.1 n-Widths as Fourier Coefficients.- 4.2 A Return to Ismagilov's Theorem.- 4.3 Bounded mth Modulus of Continuity.- 5. n-Widths of Totally Positive Operators in L2.- 5.1 The Main Theorem.- 5.2 Restricted Approximating Subspaces.- 6. Certain Classes of Periodic Functions.- 6.1 n-Widths of Cyclic Variation Diminishing Operators.- 6.2 n-Widths for Kernels Satisfying Property B.- Notes and References.- V. Exact n-Widths of Integral Operators.- 1. Introduction.- 2. Exact n-Widths of K? in Lq and Kp in L1.- 3. Exact n-Widths of K?r in Lq and Kpr in L1.- 4. Exact n-Widths for Periodic Functions.- 5. n-Widths of Rank n + 1 Kernels.- Notes and References.- VI. Matrices and n-Widths.- 1. Introduction and General Remarks.- 2. n-Widths of Diagonal Matrices.- 2.1 The Exact Solution for q ? p and p = 1, q = 2.- 2.2 Various Estimates for p = 1, q = ?.- 3. n-Widths of Strictly Totally Positive Matrices.- Notes and References.- VII. Asymptotic Estimates for n-Widths of Sobolev Spaces.- 1. Introduction.- 2. Optimal Lower Bounds.- 3. Optimal Upper Bounds.- 4. Another Look at ?n(B1(r); L?).- Notes and References.- VIII. n-Widths of Analytic Functions.- 1. Introduction.- 2. n-Widths of Analytic Functions with Bounded mth Derivative.- 3. n-Widths of Analytic Functions in H2.- 4. n-Widths of Analytic Functions in H?.- 5. n-Widths of a Class of Entire Functions.- Notes and References.- Glossary of Selected Symbols.- Author Index.