On ℓℓ-stable mappings with values in infinite-dimensional Banach spaces

The aim of the paper is to collect some results concerning ℓℓ-stability of mappings with values in possibly infinite-dimensional Banach spaces. We show that any ℓℓ-stable function from finite-dimensional space to an arbitrary Banach space is Lipschitz near the reference point. Further, we show that any ℓℓ-stable function from finite-dimensional space to a Banach space having Radon–Nikodým property is strictly differentiable at the reference point. As an application we present the second-order sufficient condition for the unconstrained optimization problem previously obtained for finite-dimensional range space.