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. d F ( u ; ψ ) = lim τ → 0 F ( u + τ ψ ) − F ( u ) τ = d d τ F ( u + τ ψ ) | τ = 0 {displaystyle dF(u;psi )=lim _{ au ightarrow 0}{frac {F(u+ au psi )-F(u)}{ au }}=left.{frac {d}{d au }}F(u+ au psi ) ight|_{ au =0}}     (1) d n F ( u ; h ) = d n d τ n F ( u + τ h ) | τ = 0 . {displaystyle d^{n}F(u;h)=left.{frac {d^{n}}{d au ^{n}}}F(u+ au h) ight|_{ au =0}.}     (2) D 2 F ( u ) { h , k } = lim τ → 0 D F ( u + τ k ) h − D F ( u ) h τ = ∂ 2 ∂ τ ∂ σ F ( u + σ h + τ k ) | τ = σ = 0 {displaystyle D^{2}F(u){h,k}=lim _{ au o 0}{frac {DF(u+ au k)h-DF(u)h}{ au }}=left.{frac {partial ^{2}}{partial au ,partial sigma }}F(u+sigma h+ au k) ight|_{ au =sigma =0}}     (3) 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. Suppose X {displaystyle X} and Y {displaystyle Y} are locally convex topological vector spaces (for example, Banach spaces), U ⊂ X {displaystyle Usubset X} is open, and F : X → Y {displaystyle F:X o Y} . The Gateaux differential d F ( u ; ψ ) {displaystyle dF(u;psi )} of F {displaystyle F} at u ∈ U {displaystyle uin U} in the direction ψ ∈ X {displaystyle psi in X} is defined as If the limit exists for all ψ ∈ X {displaystyle psi in X} , then one says that F {displaystyle F} is Gateaux differentiable at u {displaystyle u} . The limit appearing in (1) is taken relative to the topology of Y {displaystyle Y} . If X {displaystyle X} and Y {displaystyle Y} are real topological vector spaces, then the limit is taken for real τ {displaystyle au } . On the other hand, if X {displaystyle X} and Y {displaystyle Y} are complex topological vector spaces, then the limit above is usually taken as τ → 0 {displaystyle au o 0} in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative. At each point u ∈ U {displaystyle uin U} , the Gateaux differential defines a function This function is homogeneous in the sense that for all scalars α {displaystyle alpha } , However, this function need not be additive, so that the Gateaux differential may fail to be linear, unlike the Fréchet derivative. Even if linear, it may fail to depend continuously on ψ {displaystyle psi } if X {displaystyle X} and Y {displaystyle Y} are infinite dimensional. Furthermore, for Gateaux differentials that are linear and continuous in ψ {displaystyle psi } , there are several inequivalent ways to formulate their continuous differentiability. For example, consider the real-valued function F {displaystyle F} of two real variables defined by