Dual spaces and gradients and subgradients Suppose we have some function $f:{\mathbb R}^n \rightarrow \mathbb{R}$. Its gradient is defined as the vector which gives the directional derivative via $(v,\nabla f )=D_{v}f$ for any direction $v$.
Could, or should, we think of $\nabla f$ as something belonging to the dual space of the domain of $f$? And if yes, what is the idea of going about this in this way? In particular are there some geometric ideas involved? 
I ran into this idea while learning about subgradients and generalised subgradients, which are defined as functionals on the space of the domain of $f$. 

You are right to mention that $\nabla f$ is vector information coded in the dual.
On the level curves $f(x)=a$ for a constant value, we perform a composition $f\circ C:I\to\Bbb R^n\to\Bbb R$ with $f\circ C(t)=f(C(t))$, so  you are going to get that for $x$ which are on the level curve $$f(C(t)=f(x)=a,$$ and $$\nabla f(C(t))\cdot C'(t)=0,$$ by the change rule. And this is the same for all $x\in f^{-1}(a)$.
So you can interpret that the gradient has the components of a vector, in each point in the level set, which is perpendicular to tangent $C'$ at each point $C(t)=x$, at each level curve.
