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]

A map ${\displaystyle \varphi$ between Banach spaces $\mathbb {D}$ and $\mathbb {E}$ is Hadamard-directionally differentiable^[2] at $\theta \in \mathbb {D}$ in the direction $h\in \mathbb {D}$ if there exists a map $\varphi _{\theta }':\,\mathbb {D} \to \mathbb {E}$ such that ${\frac {\varphi (\theta +t_{n}h_{n})-\varphi (\theta )}{t_{n}}}\to \varphi _{\theta }'(h)$ for all sequences $h_{n}\to h$ and $t_{n}\to 0$ .

Note that this definition does not require continuity or linearity of the derivative with respect to the direction $h$ . 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 $X_{n}$ be a sequence of random elements in a Banach space $\mathbb {D}$ (equipped with Borel sigma-field) such that weak convergence $\tau _{n}(X_{n}-\mu )\to Z$ holds for some $\mu \in \mathbb {D}$ , some sequence of real numbers $\tau _{n}\to \infty$ and some random element $Z\in \mathbb {D}$ with values concentrated on a separable subset of $\mathbb {D}$ . Then for a measurable map ${\displaystyle \varphi$ that is Hadamard directionally differentiable at $\mu$ we have $\tau _{n}(\varphi (X_{n})-\varphi (\mu ))\to \varphi _{\mu }'(Z)$ (where the weak convergence is with respect to Borel sigma-field on the Banach space $\mathbb {E}$ ).

This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.^[3]

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

Hadamard derivative

Relation to other derivatives

Applications

See also

References

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