Gateaux differential,
n. the directional derivative at x with increment h, of a given function f defined on an open domain, given by
δƒ(x;h) = limt→0(ƒ(x+th)-ƒ(x)) / t
If this limit exists for all h, then the function is said to be Gateaux differentiable at x, and, if it is linear in h, the mapping
T = δƒ(x; .)
is said to be (linear) Gateaux derivative of ƒ at x, and is often referred to as the gradient of ƒ, written ∇ ƒ(x). If a mapping of one finite-dimensional vector space into another is a Lipschitz function, then any linear Gateaux derivative is automatically a Frechet derivative.
δƒ(x;h) = limt→0(ƒ(x+th)-ƒ(x)) / t
If this limit exists for all h, then the function is said to be Gateaux differentiable at x, and, if it is linear in h, the mapping
T = δƒ(x; .)
is said to be (linear) Gateaux derivative of ƒ at x, and is often referred to as the gradient of ƒ, written ∇ ƒ(x). If a mapping of one finite-dimensional vector space into another is a Lipschitz function, then any linear Gateaux derivative is automatically a Frechet derivative.