GâteauxDerivative

The Gâteaux derivative in the context of the space $\mathcal{S}_0^\ast$ is defined for functions mapping from

$$\mathcal{S}_0^\ast \text{ to } \mathbb{C}$$

It is denoted by $D_{\xi} F(x)$ for a function $F : \mathcal{S}_0^\ast \rightarrow \mathbb{C}$, and is given by the formula:

$$ D_{\xi} F (x) = \frac{d}{dt} \Big|_{t = 0} F (x + t \xi) $$

where $\xi \in \mathcal{S}_0^\ast$, and $x$ is a point in $\mathcal{S}_0^\ast$. This definition captures the directional derivative of $F$ at $x$ in the direction of $\xi$, evaluating the rate at which $F$ changes as one moves from $x$ in the infinitesimal direction of $\xi$.

In measure-theoretic terms, this formulation is particularly relevant for functional analysis on spaces of distributions or functions, where the concept of differentiation needs to extend beyond finite-dimensional spaces. The Gâteaux derivative enables one to deal rigorously with the infinitesimal behavior of functionals, which is essential for the study of variational problems, differential equations, and functional equations in infinite-dimensional spaces such as those encountered in quantum field theory and the analysis of the Yang-Mills fields.