What is an application of the dual space? Does somebody know any application of the dual space in physics, chemistry, biology, computer science or economics?
(I would like to add that to the german wikipedia article about the dual space.)

When you are dealing with a vector space $X$ its dual $X^*$ is there, whether you are willing to apply it or not. It is a fact that the presence of a scalar product in $X$ tends to obscure the difference between $X$ and $X^*$. This is the case when $X$ is our geometrical or physical space ${\mathbb R}^3$ with the "standard scalar product", i.e., is  a euclidean space where an orthonormal basis has been chosen. 
An example: When $f$ is a differentiable scalar function of ${\bf x}$ then for "small" increment vectors ${\bf X}$ one has
$$f({\bf x}+{\bf X})=f({\bf x})+ \omega.{\bf X}+ o\bigl(|{\bf X}|\bigr)\qquad ({\bf X}\to{\bf 0})\ ,$$
where $\omega.{\bf X}$ depends linearly on ${\bf X}$. That is to say: The symbol $\omega:=df({\bf x})$ denotes an element of $X^*$. Now the availability of a scalar product $\bullet$ in $X$ allows one to write the above equation in the form
$$f({\bf x}+{\bf X})=f({\bf x})+ \nabla f({\bf x})\bullet{\bf X}+ o\bigl(|{\bf X}|\bigr)\qquad ({\bf X}\to{\bf 0})\ ,$$
where $\nabla f({\bf x})$ denotes the gradient of $f$ at ${\bf x}$ and is a bona fide vector in $X$.
There are situations in physics or economy where the variables are not "geometric" variables $x_i$ but $p$, $V$, $T$ for pressure, volume, and temperature (or similar). In such a case it doesn't make sense to introduce a scalar product making the "norm" of a state $s$ equal to $\|s\|:=\sqrt{p^2+V^2+T^2}$. As a consequence a linear function of the state variables $p$, $V$, $T$ is a genuine element of $X^*$ and cannot be represented by some "gradient vector" belonging to the state space $X$.
