Why is the gradient of a function in the dual space?

Question

Suppose $f(x)$ where $x \in D \subset R^n$ is a real-valued function. Why is the gradient of $f(x)$, i.e. $\nabla f(x)$, or subgradient of $f(x)$, i.e. $g \in \partial f(x)$, in the dual space?

Answer 1

The gradient is a map $C^{1}\left(D,\,\mathbb{R}\right)\times D\to L\left(D,\,\mathbb{R}\right)$, where $L$ denotes the space of linear maps. (Strictly speaking we would only need differentiable and not $C^{1}$, but meh...) So if you fix a function $f$ and an evaluation point $x$ (where you take the gradient), you get an element of $L\left(D,\,\mathbb{R}\right)$ which is the dual space.

The trick is actually to forget all that you learned about derivatives in highschool. Derivatives are not numbers, partial derivatives are. The gradient gives you for each spot in the domain of a function a linear map describing the behaviour of the function at that point.