The entry for Gâteaux gradient in the Encyclopedia of Mathematics wiki reads:
Gâteaux gradient of a functional $f$ at a point $x_0$ of a Hilbert space $H$. The vector in $H$ equal to the Gâteaux derivative $f_G'(x_0)$ of $f$ at $x_0$. In other words, the Gâteaux gradient is defined by the formula $$f(x_0+h) - f(x_0) = f(x_0) + (f_G'(x_0),h) + \epsilon(h),$$ where $\epsilon(th)\rightarrow 0$ as $t\rightarrow 0$. In an $n$-dimensional Euclidean space the Gâteaux gradient $f_G'(x0)$ is the vector with coordinates $$\left(\frac{\partial f(x_0)}{\partial x_1},\ldots,\frac{\partial f(x_0)}{\partial x_n}\right),$$ and is simply known as the gradient. The concept of the Gâteaux gradient may be extended to the case when $X$ is a Riemannian manifold (finite-dimensional) or an infinite-dimensional Hilbert manifold and $f$ is a smooth real function on $X$. The growth of $f$ in the direction of its Gâteaux gradient is larger than in any other direction passing through the point $x_0$.
I'm interested in learning more about the extension referenced at the end of the article to the case where $X$ is an infinite-dimensional Hilbert manifold. Unfortunately, the article doesn't provide a reference for this. Does anyone know where I could find more information about this generalization?