A map between Banach spaces and is Hadamard-directionally differentiable[2] at in the direction if there exists a map such that
for all sequences and .
Note that this definition does not require continuity or linearity of the derivative with respect to the direction . Although continuity follows automatically from the definition, linearity does not.
Relation to other derivatives
If the Hadamard directional derivative exists, then the Gateaux derivative also exists and the two derivatives coincide.[2]
A version of functional delta method holds for Hadamard directionally differentiable maps. Namely, let be a sequence of random elements in a Banach space (equipped with Borel sigma-field) such that weak convergence holds for some , some sequence of real numbers and some random element with values concentrated on a separable subset of . Then for a measurable map that is Hadamard directionally differentiable at we have (where the weak convergence is with respect to Borel sigma-field on the Banach space ).