Connecting a vector space to its dual - why?

Question

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the dual gives an non-degenerate bilinear form (inner product), which allows us to define length, angle, etc, while conversely the bilinear form has an embedding into the dual as a by-product", but I feel that there is some sort of understanding that is fluttering just over my head as I ponder these arguments. Why is this the right way, or the natural way, to go about putting a geometry on the vector space? Once we have fixed such a geometry, what do we gain from interpreting a vector as a functional and vice versa? It just seems very strange that once a correspondence with the dual has been fixed, then so should whatever geometrical properties that the space may have, or why the natural geometry of the space should arise from this connection.

Answer 1

Defining an isomorphism between $V$ and $V^\ast$ doesn't fix its geometry. It just defines a non-degenerate bilinear form which isn't quite an inner product.

However, if we have an inner product i.e. "common" geometry, we define an isomorphism.

So to answer your question, there is no reason.

Although, then again, a non-degenerate bilinear form probably fixes some notion of geometry, if we consider geometry to be defined as "a way to compare two vectors with each other".

A parting comment: A linear functional measures a vector, so if we associate each vector with a linear functional then we are giving each vector "a way to compare itself with another vector" i.e. we are giving any two vectors "a way to compare two vectors with each other".