I've just finished a course on bilinear forms and am now starting a cause on topological spaces and was just wondering; for a metric space which is made up of a set $M$ and a metric function $d$ such that $$d:M \times M \to \mathbb{R}$$ defines the distance between points in $M$ etc etc..
Is this 'distance function' a simply a bilinear form? And if not why not? What goes wrong?
Thanks!
Best Answer
This might be slightly more involved than what you are encountering right now, but there is a link between these two notions : first, a positive-definite bilinear form yields a norm on vector spaces, so this is a function on a vector space, but a distance function on a vector space is not necessarily of this form. Then, on a smooth manifold (a particularly well-behaved kind of topological space), there is the notion of a tangent space, hence tangent vector to a curve, and then, integrating the "norm" (i.e. a smoothly-varying bilinear form on the tangent bundle) of the tangent vector to a curve along the whole curve gives its length, and then, tanking the minimum of all lengths between two points give the distance between these two points.