Year: 2014
Type: article
Abstract: Let X X be a separable Banach space, Y Y a Banach space and f: X \to Y f : X → Y an arbitrary mapping. Then the following implication holds at each point x\in X x ∈ X except a \sigma σ -directionally porous set: If the one-sided Hadamard directional derivative f'_{H+}(x,u) f H + ′ ( x , u ) exists in all directions u u from a set S_x \subset X S x ⊂ X whose linear span is dense in X X , then f f is Hadamard differentiable at x x . This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of S_x S x equals X X and Y = \mathbb{R} Y = R . An analogous theorem, in which f f is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gâteaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which f
Source: Journal of convex analysis
DOI: 10.68381/jca21037
Author: Luděk Zajíček
Institution: Charles University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2280508283