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:

• w(kv) = 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.