# 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.)

• In economics prices are in the dual of the commodity space assuming per-unit pricing. Using this duality pops up in proving some of the existence of equilibrium theorems. In fact, the existence of a market clearing price is (if I recall) just a statement of the separating hyperplane/ Hahn-Banach theorem (depending on dimension) in classical General Equilibrium theory. – Chris Janjigian Aug 29 '12 at 14:39

Dual space is used in the lemma's Lax- Milgran wich guarantees the existence of severals solutions of PDE wich this describe describe movements, minimal surfaces, heat distribution and other.

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 $X$.

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:

1. 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.

2. 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.

3. 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 $V\to V^*$.

4. 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.