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# Harmonic Approximation by Stephen J. Gardiner

By Stephen J. Gardiner

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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).