James Tauber's Blog 2008/03/23
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.
by James Tauber : Created on March 23, 2008 : Last modified March 23, 2008 : Categories poincare_project : (permalink)