Let $V$ be an arbitrary $n$-dimensional inner product space over $\mathbb R$. Define the gradient of $f$ at $x$ as the vector such that for all $\mathbf v\in V$,
$${\displaystyle {\big \langle}\nabla f(x){\big )},\mathbf {v}\big\rangle =D_{\mathbf {v} }f(x),}$$
where $\langle,\rangle$ is the inner product on $V$ and $D_{\mathbf {v} }f(x)$ is the directional derivative of $f$ at $x$ along the tangent vector $\mathbf v\in \mathbb R^n$.
What do we need to assume to conclude that the gradient is the "steepest" direction in which to go? Does this hold for arbitrary inner products? Or just the Euclidean one?