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Every mathematician working in Banaeh spaee geometry or Approximation theory knows, from his own experienee, that most "natural" geometrie properties may faH to hold in a generalnormed spaee unless the spaee is an inner produet spaee. To reeall the weIl known definitions, this means IIx 11 = , where is an inner (or: scalar) product on E, Le. a function from ExE to the underlying (real or eomplex) field satisfying: (i) O for x ~ o. (ii) is linear in x. (iii) = (intherealease,thisisjust =
List of contents
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.
About the author
Amir is an Iranian-American human rights activist, journalist and documentary filmmaker. He has lived and worked in the U.S.A, Canada, Europe and Afghanistan.
