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.)

First of all: I think it is impossible to give a complete answer to your question. The reason is that the dual space is such a general and central notion that it is ubiquiteous in mathematics, physics and other sciences. I would only exaggerate a bit if I said you might as well have asked what applications of vector spaces are.
But to give some ideas:


*

*A neat way to define the tangent space of a manifold is as the dual space of the cotangent space. This is rather abstract but can easily be transfered to other settings, e.g. varieties. For example vector fields - which are commonly used in physics - rely on this.

*In quantum physics people often use the "bra"- and "ket"-vector notation. In the end these are just other names for a vector and its dual.

*A scalar product $V\times V\to k$ (also central in, say, physics) can be written as an isomorphism $V\to V^*$. Many applications actually only need non-degenerate forms, which in turn are injective maps $V\to V^*$.

*In constrained optimisation one often passes from the so called primal problem to the dual problem. A great example from computer siences are support vector machines. After some work it turns out that the "dual" vectors (in the dual problem) are indeed dual vectors (in the classical sense) of the data space. This trick makes a huge difference in computations.
