Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

2013/08/11 by Zajicek, Ludek
#FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46G05 #Secondary: 26B05

paper · doi:10.48550/arxiv.1308.2415

Abstract

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions f: \R → \R was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on \R². This extension gives the strongest relation among upper and lower Hadamard directional derivatives f⁺_H (x,v), f^-_H (x,v) (v ∈ X) which holds almost everywhere for an arbitrary function f:\R²→ \R. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.

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