Gateaux 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, 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 a nonlinear operator. 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

Suppose

X

{\displaystyle X}

and

Y

{\displaystyle Y}

are locally convex topological vector spaces (for example, Banach spaces),

U

⊆

X

{\displaystyle U\subseteq X}

is open, and

f

:

U

→

Y

.

{\displaystyle f:U\to Y.}

The Gateaux differential

d

f

(

x

,

v

)

{\displaystyle df(x,v)}

of

f

{\displaystyle f}

at

x

∈

U

{\displaystyle x\in U}

in the direction

v

∈

X

{\displaystyle v\in X}

is defined as

If the limit exists for all

v

∈

X

,

{\displaystyle v\in X,}

then one says that

F

{\displaystyle F}

is Gateaux differentiable at

x

.

{\displaystyle x.}

The limit appearing in (1) is taken relative to the topology of

Y

.