One-Forms Form Linear Spaces

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:

We define addition of one-forms:

and scaling of one-forms:

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

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.

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