Hadamard derivative
In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications in stochastic programming and asymptotic statistics.[1]
Definition
A map between Banach spaces and is Hadamard-directionally differentiable[2] at in the direction if there exists a map such that
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]
- The Hadamard derivative is readily generalized for maps between Hausdorff topological vector spaces.
Applications
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
This result has applications in optimal inference for wide range of
See also
- Directional derivative – Instantaneous rate of change of the function
- Fréchet derivative – Derivative defined on normed spaces - generalization of the total derivative
- Gateaux derivative – Generalization of the concept of directional derivative
- Generalizations of the derivative – Fundamental construction of differential calculus
- Total derivative – Type of derivative in mathematics