Gâteaux Derivative

We have the the limit called the directional derivative of $\phi$ at $u_0$ in the direction $v$. We get the function $\mathcal{G}\phi (u_0):V\to W$, given as

The function $\phi :\mathrm{\Omega }\to W$ is Gâteaux differentiable at the point $u_0\in \mathrm{\Omega }$ if for all $v\in V$, there's a directional derivative of $\phi$ at $u_0$ in the direction $v$ and the function $\mathcal{G}\phi (u_0)$ defined above belongs to $\mathcal{B}(V,W)$.
$\phi$ is Gâteaux-differentiable on $\mathrm{\Omega }$ if it is differentiable at every point $u\in \mathrm{\Omega }$. The function

is called the Gâteaux derivative of $\phi$.

$\phi :\mathrm{\Omega }\to W$ is of class ${\mathcal{C}}^{\mathcal{1}}$ on $\mathrm{\Omega }$ iff $\phi$ is Gâteaux-differentiable on $\mathrm{\Omega }$ and its Gâteaux derivative $\mathcal{G}\phi :\mathrm{\Omega }\to \mathcal{B}(V,W)$ is continuous. In this case ${\phi }^{\prime }=\mathcal{G}\phi$.