Understanding Co-vectors to Dual Spaces and Linear Functionals

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Discussion Overview

The discussion revolves around the concepts of co-vectors, dual spaces, and linear functionals within the context of mathematics. Participants express confusion regarding the terminology and definitions across different mathematical fields, particularly in relation to the dual space and its properties.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the terms co-vectors, one-forms, and linear functionals, noting their different interpretations across mathematical fields.
  • There is mention of the dual space being viewed as a homeomorphism of a vector space, but the participant seeks clarity on this concept.
  • A question is raised regarding the biorthogonality condition and its relation to the dot product, highlighting uncertainty about how operations between elements of different spaces can be defined.
  • Another participant suggests focusing on the textbook for the upcoming class to clarify these concepts.
  • Discussion includes references to the Frenet frame and its relation to co-vectors, with one participant confirming their understanding of the concept.
  • There is a query about the relationship between bi-normal and tangent vectors and co-vectors, indicating a gap in knowledge among participants.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions and relationships discussed, with multiple viewpoints and areas of confusion remaining evident throughout the conversation.

Contextual Notes

Participants express uncertainty regarding the definitions and relationships between co-vectors, dual spaces, and linear functionals, as well as the implications of the biorthogonality condition. There are references to specific mathematical concepts and frameworks that may require further exploration.

JuanC97
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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.
 
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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.
 
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chiro said:
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.
 
Do you know about the bi-normal and tangent vectors in relation to co-vectors?
 
I was getting at that so yes.
 

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