The theory of best approximation and functional analysis

Description

Title page Preface 1. Characterizations of elements of best approximation 2. Existence of elements of best approximation 2.1. Characterizations of proximinal linear subspaces 2.2. Some classes of proximinal linear subspaces 2.3 Normed linear spaces in which all closed linear subspaces are proximinal 2.4. Normed linear spaces which are proximinal in every superspace 2.5. Normed linear spaces which are, under the canonical embedding, proximinal in their second conjugate space 2.6. Transitivity of proximinality 2.7. Proximinality and quotient spaces 2.8. Very non-proximinal linear subspaces 3. Uniqueness of elements of best approximation 3.1. Characterizations of semi-Chebyshev and Chebyshev subspaces 3.2. Existence of semi-Chebyshev and Chebyshev subspaces 3.3. Normed linear spaces in which all linear (respectively, all closed linear) subspaces are semi-Chebyshev (respectively, Chebyshev subspaces 3.4. Semi-Chebyshev and Chebyshev subspaces and quotient spaces 3.5. Strongly unique elements of best approximation. Strongly Chebyshev subspaces. Interpolating subspaces 3.6. Almost Chebyshev subspaces. k-semi-Chebyshev and k-Chebyshev subspaces. Pseudo-Chebyshev subspaces 3.7. Very non-Chebyshev subspaces 4. Properties of metric projections 4.1. The mappings π_G Metric projections 4.2. Continuity of metric projections 4.3. Weak continuity of metric projections 4.4. Lipschitzian metric projections 4.5. Differentiability of metric projections 4.6. Linearity of metric projections 4.7. Semi-continuity and continuity of set-valued metric projections 4.8. Continuous selections and linear selections for set-valued metric projections 5. Best approximation by elements of nonlinear sets 5.1. Best approximation by elements of convex sets. Extensions to convex optimization in locally convex spaces 5.2. Best approximation by elements of N-parameter sets 5.3. Generalizations 5.4. Best approximation by elements of arbitrary sets References