One of the important takeaways from arrows and stacks as duals is that

Every linear space has a dual space whose elements are the linear, scalar-valued functions on the original space.

These linear, scalar-valued functions have a number of different names. They are variously called **linear functionals**, **linear forms** or, more specifically, and as we will call them, **one-forms**.

For our purposes, the one-forms are real-valued (because our linear spaces are real), although in quantum mechanics I think dual spaces are always made of complex-valued functions (and complex linear spaces).

Let's just quickly demonstrate that the dual space is itself a linear space:

Firstly, by virtue of the fact the one-forms are linear, we know:

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

We define addition of one-forms:

- (
*w*+*x*)(**v**) =*w*(**v**) +*x*(**v**)

and scaling of one-forms:

- (k
*w*)(**v**) = k*w*(**v**)

We can see that one-forms do form a linear space, just by the algebraic properties of real number addition and multiplication:

- ((
*w*+*x*) +*y*)(**v**) = (*w*(**v**) +*x*(**v**)) +*y*(**v**) =*w*(**v**) + (*x*(**v**) +*y*(**v**)) = (*w*+ (*x*+*y*))(**v**)

- (
*w*+*x*)(**v**) =*w*(**v**) +*x*(**v**) =*x*(**v**) +*w*(**v**) = (*x*+*w*)(**v**)

- there is an additive identity
*0*which simply maps every vector to 0.

- for every
*w*there is an additive inverse*-w*such that (*w*+*-w*)(**v**) =*w*(**v**) +*-w*(**v**) =*w*(**v**) -*w*(**v**) = 0 =*0*(**v**)

- a (
*w*+*x*)(**v**) = a (*w*(**v**) +*x*(**v**) ) = a*w*(**v**) + a*x*(**v**)

- (a + b)
*w*(**v**) = a*w*(**v**) + b*w*(**v**)

- a (b
*w*(**v**)) = (ab)*w*(**v**)

- 1
*w*(**v**) =*w*(**v**)

Remember that if we view vectors in the context of a manifold as arrows, the corresponding one-forms are stacks.

In matrix algebra, column vectors and row vectors are similarly duals of one another.

In the next entry, we'll look at the relationship between one-forms and coordinate systems.

Tweet

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