*is*, rather that the usual descriptions of what it

*does*. But first a couple of disclaimers. I am

*not*a physicist, so I may not be able to give much insight into applications. Second, I

*am*human; I make mistakes, and have possibly misunderstood some stuff - please correct me.

I will assume we all know what's meant by a

**vector space**. The particular properties of such spaces that I will need here, I will explain as we go. If you're not sure what a vector space is, you can think of arrows in the plane (not recommended), or you can accept that the set R of real numbers is a vector space

*provided only*that addition and multiplication are defined.

OK. Let V be a vector space over some scalar field, say F, and let v be an arbitrary vector in V. I define these vectors as

$$v=\sum_i \alpha^ie_i,$$, where the $$\alpha^i \in F$$, and where the set $$\{e_i\} \in V$$ is called a

**basis**for V. (We shan't need this equality for now, but we shall when we come to do some notational housework, so keep it in mind)

Let's just accept as a fact that the cardinality of the set $$\{e_i\}$$ defines the dimension of our space V, and for convenience we will say this is finite. Then we see straight away that the vector v depends only on the scalars $$\alpha^i$$, where i = 1,2,.....n.

I will say that, to any vector space V, I can associate linear maps of the form $$f :V \rightarrow W$$ and $$\varphi : V \rightarrow F$$. I'll explain linearity in a while, but note this:

The map $$f$$ above is technically known as an operator, or linear transformation (vector to vector) and the $$\varphi$$ is called a linear functional (vector to scalar); I will use the generic term "linear map", if that's OK with you. You will see the virtue of this in due course.

So consider the map $$\varphi : V \rightarrow F,\; \varphi(v) =\alpha$$. As there is no obvious way to specify the image of all v under $$\varphi$$, we must conclude there is a multiplicity of such maps, one for each v in V, in fact. We can gather these maps into a set, and easily show (Exercise!) that this set satisfies the axioms of a vector space (closed under map addition and scalar multiplication, etc)

Let's call this vector space the

**dual space**to V, and denote it by V*, the space of all maps from V to F, or, if you prefer, $$V^*: V \to F; \; \varphi, \; \psi, .... \in V^*$$.

OK, that's the foreplay done, let's get stuck in (whoops, sorry!). But later for that.