Understanding Co-vectors to Dual Spaces and Linear Functionals

[JuanC97]

Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate if you could give me some nice structured references to self-study some of this topics.

One of my main confussions is related to co-vectors:

Co-vectors are said to be one-forms, but also, linear functionals, which can be interpreted as linear maps or linear operators that can be useful to view the dual space as a homeomorphism of a vector space. There are also other definitions of the dual space in terms of the tangent (and cotangent) bundle(s) but none of these concepts is clear for me right now.

Also check this page: https://en.wikipedia.org/wiki/Dual_basis
It says that the biorthogonality condition (which I suppose is related to the homeomorphism / isomorphism) can be expresed as a dot product "If one denotes the evaluation of a covector on a vector as a pairing" but I don't get how is it possible to denote the operation between an element of the dual space and one of the original space as a dot product since, clearly, both elements belong to different spaces and dot product is defined for entries of the same space.

That said, you should be able to see the kind of doubts that I'm having.
So... as I said before, any good reference will be welcome... and thanks in advance.

 

[FactChecker]

You can go crazy looking at all the different approaches to the same thing. I recommend that you concentrate on the textbook for your class next semester and ask questions specifically about that approach.

 

[chiro]

Hey JuanC97.

Do you understand the concept of a Frenet frame?

https://en.wikipedia.org/wiki/Frenet–Serret_formulas

 

[JuanC97]

Do you understand the concept of a Frenet frame?

Sure Chiro, it is essentially a co-moving frame that keeps the velocity vector of a particle in the tangent direction.

 

[chiro]

Do you know about the bi-normal and tangent vectors in relation to co-vectors?

 

[JuanC97]

Do you know about the bi-normal and tangent vectors in relation to co-vectors?

Hmmmm, nope, sorry.
Does it have something to do with this? : https://en.wikipedia.org/wiki/Frenet–Serret_formulas#Formulas_in_n_dimensions but

 

[chiro]

I was getting at that so yes.