Gâteaux derivative
Gâteaux derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
Definition
**(1)** |
Linearity and continuity
This is Gateaux differentiable at (0, 0), with its differential there being
- Relation with the Fréchet derivative
- Continuous differentiability
A stronger notion of continuous differentiability requires that
be a continuous mapping
Higher derivatives
**(2)** |
There is another candidate for the definition of the higher order derivative, the function
**(3)** |
is continuous in the product topology, and moreover that the second derivative defined by (3) is also continuous in the sense that
Properties
Suppose that is in the sense that the Gateaux derivative is a continuous function . Then for any and ,
Many of the other familiar properties of the derivative follow from this, such as multilinearity and commutativity of the higher-order derivatives. Further properties, also consequences of the fundamental theorem, include:
(The chain rule)
- for alland. (Note well that, as with simplepartial derivatives, the Gateaux derivative does not satisfy the chain rule if the derivative is permitted to be discontinuous.)
(Taylor's theorem with remainder)
- Suppose that the line segment betweenandlies entirely within. Ifisthen
- where the remainder term is given by
Example
See also
Hadamard derivative
Derivative (generalizations)
Differentiation in Fréchet spaces
Fractal derivative
Quasi-derivative
Quaternionic analysis