Gateaux and Fréchet Derivatives

Abstract:

Let W₁ be some topological vector space and f a (nonlinear) mapping f : W → W₁. We call f Gateaux differentiable in u ∈ W if there exists a mapping θ ∈ L(W,W₁) such that for all υ ∈ W

in the topology of W₁. The linear mapping θ ∈ L(W,W₁) is the called Gateaux derivative of f in u and is written as θ = f′(u). The Gateaux derivative is a linear operation…