Quasi-derivative

In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative.

Let f : A → F be a continuous function from an open set A in a Banach space E to another Banach space F. Then the quasi-derivative of f at x₀ ∈ A is a linear transformation u : E → F with the following property: for every continuous function g : [0,1] → A with g(0)=x₀ such that g′(0) ∈ E exists,

\lim _{t\to 0^{+}}{\frac {f(g(t))-f(x_{0})}{t}}=u(g'(0)).

If such a linear map u exists, then f is said to be quasi-differentiable at x₀.

Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x₀, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x₀. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.

References

Dieudonné, J (1969). Foundations of modern analysis. Academic Press.

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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
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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