Norm Derivatives And Characterizations Of Inner Product Spaces
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Norm Derivatives and Characterizations of Inner Product Spaces
1. Introduction. 1.1. Historical notes. 1.2. Normed linear spaces. 1.3. Strictly convex normed linear spaces. 1.4. Inner product spaces. 1.5. Orthogonalities in normed linear spaces -- 2. Norm derivatives. 2.1. Norm derivatives : Definition and basic properties. 2.2. Orthogonality relations based on norm derivatives. 2.3. p'[symbol]-orthogonal transformations. 2.4. On the equivalence of two norm derivatives. 2.5. Norm derivatives and projections in normed linear spaces. 2.6. Norm derivatives and Lagrange's identity in normed linear spaces. 2.7. On some extensions of the norm derivatives. 2.8. p-orthogonal additivity -- 3. Norm derivatives and heights. 3.1. Definition and basic properties. 3.2. Characterizations of inner product spaces involving geometrical properties of a height in a triangle. 3.3. Height functions and classical orthogonalities. 3.4. A new orthogonality relation. 3.5. Orthocenters. 3.6. A characterization of inner product spaces involving an isosceles trapezoid property. 3.7. Functional equations of the height transform -- 4. Perpendicular bisectors in Normed spaces. 4.1. Definitions and basic properties. 4.2. A new orthogonality relation. 4.3. Relations between perpendicular bisectors and classical orthogonalities. 4.4. On the radius of the circumscribed circumference of a triangle. 4.5. Circumcenters in a triangle. 4.6. Euler line in real normed space. 4.7. Functional equation of the perpendicular bisector transform -- 5. Bisectrices in real Normed spaces. 5.1. Bisectrices in real normed spaces. 5.2. A new orthogonality relation. 5.3. Functional equation of the bisectrix transform. 5.4. Generalized bisectrices in strictly convex real normed spaces. 5.5. Incenters and generalized bisectrices -- 6. Areas of triangles in Normed spaces. 6.1. Definition of four areas of triangles. 6.2. Classical properties of the areas and characterizations of inner product spaces. 6.3. Equalities between different area functions. 6.4. The area orthogonality.
Norm Derivatives and Characterizations of Inner Product Spaces
The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordanvon Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications).
Characterizations of Inner Product Spaces
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 =