Abstract
We recall some immediate properties of the one-sided derivatives \( \varphi {'_ \pm }\left( {x,y} \right) \equiv \mathop {\lim }\limits_{t \to \pm 0} \frac{{\varphi \left( {x + ty} \right) - \varphi \left( x \right)}}{t} \) of a convex functional φ on a linear space E, at the point x in the direction of y (these limits exist by convexity).
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(i)
φ′+(x,y)= αφ′+(x,y) for for α≥0, φ′+(x,αy)=αφ′-(x,y) for α≤0
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(ii)
φ′+(x,y+z)≤φ′+(x,y)+φ′+(x,z).
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(iii)
φ′+(x,y)≥φ′-(x,y).
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(iv)
If φ′+(x,y) = φ′-(x,y) for all y ∈ E then the common value φ′ x (y) is a linear functional on y ∈ E (called: the derivative φ′ x of φ at x)
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© 1986 Springer Basel AG
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Amir, D. (1986). Norm Derivatives Characterizations. In: Characterizations of Inner Product Spaces. Operator Theory: Advances and Applications, vol 20. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5487-0_3
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DOI: https://doi.org/10.1007/978-3-0348-5487-0_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5489-4
Online ISBN: 978-3-0348-5487-0
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