Part of the glacial PoinarĂ© 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:

*w*(k**v**) = k*w*(**v**)*w*(**u**+**v**) =*w*(**u**) +*w*(**v**)

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.

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.