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Norm Derivatives and Characterizations of Inner Product by Claudi Alsina

By Claudi Alsina

The e-book offers a accomplished evaluate of the characterizations of actual normed areas as internal product areas in accordance with norm derivatives and generalizations of the main simple geometrical homes of triangles in normed areas. because the visual appeal of Jordan-von Neumann's classical theorem (The Parallelogram legislation) in 1935, the sphere of characterizations of internal product areas has bought an important quantity of awareness in numerous literature texts. in addition, the concepts coming up within the concept of practical equations have proven to be super worthy in fixing key difficulties within the characterizations of Banach areas as Hilbert areas. This publication provides, in a transparent and specified sort, cutting-edge equipment of characterizing internal product areas by way of norm derivatives. It brings jointly effects which were scattered in a number of guides during the last 20 years and contains extra new fabric and strategies for fixing sensible equations in normed areas. hence the booklet can function a sophisticated undergraduate or graduate textual content in addition to a source booklet for researchers operating in geometry of Banach (Hilbert) areas or within the idea of useful equations (and their applications).

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From the other side − ρ′+ (x0 , y + h − h) − ρ′+ (x0 , y + h) ≥ −ρ′+ (x0 , −h) = ρ′− (x0 , h) ≥ − x0 h . This gives ρ′+ (x0 , y + h) − ρ′+ (x0 , y) ≤ x0 h , which proves the continuity of ρ′+ in the second variable. 7), after dividing it by 2t for t < 0 and taking the limit while t → 0− . 24 Norm Derivatives and Characterizations of Inner Product Spaces The continuity in the second variable of ρ′− is obtained immediately from the condition ρ′− (x0 , y) = −ρ′+ (−x0 , y). 4 [Amir (1986)] Let (X, · ) be a real normed linear space.

This gives ρ′+ (x0 , y + h) − ρ′+ (x0 , y) ≤ x0 h , which proves the continuity of ρ′+ in the second variable. 7), after dividing it by 2t for t < 0 and taking the limit while t → 0− . 24 Norm Derivatives and Characterizations of Inner Product Spaces The continuity in the second variable of ρ′− is obtained immediately from the condition ρ′− (x0 , y) = −ρ′+ (−x0 , y). 4 [Amir (1986)] Let (X, · ) be a real normed linear space. Then there exists a set F ⊂ X of Lebesgue measure zero such that for all x in X\F and y in X we have ρ′+ (x, y) = ρ′− (x, y), and X\F is dense in X.

4). Next we will show that T (λx) = λT (x) for all λ ∈ R and x in X\F , x = 0. Take x in X\F . 1 let us choose y independent of x and such that |ρ′+ (x, y)| < x y . Since T (x) and T (y) are also independent, for every λ in R there exist α, β in R (depending on λ) such that T (λx) = αT (x) + βT (y). 2) and the general properties of ρ′± , we obtain the equalities (in the case λ = 0 we assume ρ′sgn(λ) (·, ·) = 0): λg(x) x 2 = g(x)ρ′+ (x, λx) = ρ′+ (T (x), T (λx)) = ρ′+ (T (x), αT (x) + βT (y)) = α T (x) = αg(x) x 2 + βg(x)ρ′sgn(β) (x, y) 2 + βρ′sgn(β) (T (x), T (y)) and, analogously, λg(y)ρ′sgn(λ) (y, x) = g(y)ρ′+ (y, λx) = ρ′+ (T (y), T (λx)) = ρ′+ (T (y), αT (x) + βT (y)) = β T (y) 2 + αρ′sgn(α) (T (y), T (x)) 33 Norm Derivatives = βg(y) y 2 + αg(y)ρ′sgn(α) (y, x).

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