Part of the glacial Poinaré Project.
We've introduced the notion of a linear space and seen that, in the context of a manifold, there are at least two distinct types of linear spaces:
Arrow-type linear spaces are about position on the manifold and rates of change of position. Stack-type linear spaces, on the other hand, are about rates of change of some other quantity defined on the manifold as position changes.
These two types of linear spaces have a special relationship to one another. An arrow-type vector and a stack-type vector can be multiplied together to give a quantity which has no reference to distance or direction (what is called a scalar) and which is immune to transformations that maintain the topology of the manifold. Geometrically, you can calculate this quantity by counting how many "stacks" of the stack-type vector the arrow-type vector passes through.
Because this operation of multiplying arrow-type and stack-type vectors doesn't require any additional structures and is preserved under a homeomorphism, is it more fundamental than, say, the inner (or dot) product. As we will see in the next couple of entries, though, it has a relationship to the inner product, mediated through the metric.
We can go one step further and, algebraically, think of a stack-type vector as a function that turns arrow-type vectors into scalars. In other words if V is an arrow-type linear space then a particular stack-type vector w can be thought of as a function w: V -> R. Because the stack-type vectors that apply to the arrow-type vectors in V follow the axioms of linear spaces, the following rules fall out:
Or put more succinctly, w is a linear, real-valued function on V.
Because the linear space of stack-vectors that apply to V has a special relationship to V, it is said to be the dual of V.
It is worth noting that everything above still works if you swap arrows and stacks. In other words, you can view arrow-type vectors as linear, real-valued functions on the linear space of stack-type vectors. The dual relationship is symmetrical. In fact, the only thing that makes one "arrow-type" and the other "stack-type" is their relationship to a manifold.
You can talk about a linear space and its dual without reference to an underlying manifold on which the vectors live. For example, the n-tuple space has a dual as well. For a given linear space of n-tuples, this dual space is the space of all linear, real-valued functions on those n-tuples.
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.