The concept of a cotangent space is built upon the structures of a manifold and an associated tangent space, so it would be helpful to read related nodes first. The three-sentence version follows:

In a manifold M (an n-dimensional space), we want to be able to define "vectors" as we do in ordinary euclidean space. There are many ways to do so, and mathematically we categorize them with this object we call the tangent space of vectors at a point p in M, denoted TpM, which is essentially the space of all vectors defined at p. This construction is useful because the tangent space is a manifold in itself.

Now, it is a mathematical fact of linear algebra that every vector space V is naturally associated to a dual space, V*. For example, in the case of column vectors in euclidean space, the natural dual space is the space of row vectors. A more interesting example is in a hilbert space, where the linear space of bras < ψ | is associated with the dual space of kets | ψ >. The concept of a dual space is indeed covered with detail in its respective node, so we give a brief explanation before refining our attention to the special case of the dual of a tangent vector space TpM, which is what we call the cotangent space, Tp*M.

In general, given any n-dimensional+ vector space V, we look at the space of all real-valued linear maps on V. This space of linear maps forms a vector space in its own right, which we call the dual space, V*. This would appear to be an extremely abstract space, but in fact, it is not very different from the space V itself. Think of V* as the evil twin of V. Its initial definition depends on V, but once the computational machinery is put in place, it is entirely possible to redefine V as the space of linear maps on V*, a la Freaky Friday.

A dual vector space is often defined by its basis. Given a set of basis vectors {eα} of a vector space V, we define the dual space V* to be the vector space spanned by the basis vectors {e*β}, where each e*β is a linear map on the set of eα's. Specifically,

e*β[ eα ] = δαβ.

Where δαβ is simply the Kronecker delta, equal to one when the indices agree, and zero when they don't. This gives us a basis for linear maps, in that any linear map ω: V → R can be written as ω = ωβe*β, where the ωβ's are just real coefficients. Indeed, this is definitely a linear map on V, and it is fairly easy to show that any linear map on V can be written in this form. However, our time is best spent studying the special case where V = TpM, V* = Tp*M.

As was described thoroughly in the tangent space node, TpM is the space of directional derivative operators acting on functions defined on M. The basis for this space is the set of partial derivative operators {eα = ∂/∂xα}. We will now introduce the dual basis exactly as we did before, but using very suggestive notation: {eβ = "dxβ "}.

By our definition, dxβ[ ∂/∂xα ] = δαβ.

Noting that any vp in TpM can be written vp = vα ∂/∂xα,

and any ωp in Tp*M can be written ωp = ωβdxβ, we find a general formula for ω acting on v.

ωp[ vp ] = ωβdxβ[ vα ∂/∂xα ] = ωβ vα δαβ = ωα vα.

Dual vectors in Tp*M are known as one-forms. Notice that, although TpM and Tp*M have the same size and essentially the same structure, there is no natural map between the two spaces++. In other words, given an arbitrary vector vp, there is no natural way to associate it with a unique one-form, ωp. We could try identifying them component-wise, via vα ↔ ωα, but the components aren't fundamental; they change under coordinate transformations. It is possible to choose some particular identification between vectors and one-forms, and this additional structure is known as a metric. The bottom line is, you need to input some additional information before you can directly compare vectors and one-forms.

Now, we could just stop here, having defined Tp*M via the dual space, {dxβ}. However, the notation "dxβ " seems to cry out for some motivation. We've seen this notation before in calculus. To see the relationship, consider the following:

Given a function f: M → R, define a special element of Tp*M. Call it ωf. Define ωf by the following:

ωf = (∂f/∂xβ) dxβ. In other words, the coefficients ωβ = ∂f/∂xβ.

Now, let's see what happens if we act with ωf on a vector vp in TpM:

ωf[ vα∂/∂xα ] = (∂f/∂xβ) vβ = vp(f).

ωf[ v ] is the directional derivative of f in the direction of v. It exactly gives us the same result we would get if we acted on f with v as a directional derivative.

We give ωf a new name: ωf = df = (∂f/∂xβ) dxβ.

Now, df[ v ] = v(f) is the directional derivative of f in the v-direction.

Now, we can see that our notation for "df" connects with our notation for "dxβ ". We have used boldface to distinguish them thus far, but soon that will not be necessary. As a special case, let f be the coordinate function f = xβ. Then,

df = d(xβ) = (∂xβ/∂xα) dxα, where "dxα " is still our dual vector notation.

Now, ∂xβ/∂xα = δαβ is just our friendly Kronecker Delta again, which is fairly easy to see, since our coordinates are independent of each other. Therefore, we have the relation

d(xβ) = dxβ. Our notation is consistent.

So, in a fairly unorthodox manner, the notation has lead us to a map d from functions into forms. d is known as the exterior derivative.

You might ask, "What about our notion of dx as a small change in x?" Well, one day, you're going to have to throw that picture out of the window, because the notation "dx" does not actually mean a small change in x. "dx" isn't even really a number. It's a linear map from vectors to numbers. It can act on small vectors to produce small numbers, but it isn't a small number in itself; it's not even an element of R. It's an element of Tp*M. So what does it really mean when we see "dx" in an integral? That is a question for another node.

+In this writeup, we work with finite-dimensional vector spaces. Infinite-dimensional dual spaces are very interesting (like the example of bras and kets in hilbert space), but not applicable to the study of tangent spaces of n-dimensional manifolds.
++As unperson put it, there are an infinite number of ways to connect the two spaces, but we have to choose one. That is what I mean when I say that there is no "natural" choice.

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