Characterizations of Inner Product Spaces

2-Dimensional Characterization.- The Parallelogram Equality and Derived Equalities.- Norm Derivatives Characterizations.- James' Isoceles Orthogonality (Midpoints of Chords).- Birkhoff Orthogolaity.- Best Approximation Characterizations.- Loewner Ellipsoids and Parallelogram Inequalities.- Pythagorean Type Orthogonalities.- Area Arguments and Area Orthogonalities.- Moduli of Convexity and Smoothness.- The Rectangular Constant and Orthogonality In S E .- Inversions and Four-Point Properties.- 3 - Dimensional Characterization.- Kakutani's conditions.- 3-Dimensional Approximation Properties.- Blaschke's Condition and Derived Characterizations.- Chebyshev Radius and Centers.- Combining the Garkavi-Klee Condition with the Hahn-Banach Theorem.- Best Coapproximation and Optimal Sets.- Symmetry of Orthogonality.- Symmetry of Orthogonality with Smoothness.- Subspace Homogeneity and Concluding Remarks.