Understanding Co-vectors to Dual Spaces and Linear Functionals

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SUMMARY

This discussion centers on the concepts of co-vectors, dual spaces, and linear functionals in mathematics. Co-vectors, also known as one-forms, are linear functionals that serve as linear maps, providing a framework for understanding dual spaces as homeomorphisms of vector spaces. The biorthogonality condition, which relates to the isomorphism between dual spaces, is highlighted as a point of confusion, particularly regarding the use of dot products between elements of different spaces. The conversation emphasizes the importance of structured references and textbooks for deeper comprehension of these mathematical concepts.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically linear functionals and vector spaces.
  • Familiarity with dual spaces and their properties in mathematics.
  • Knowledge of the Frenet frame and its relation to tangent and bi-normal vectors.
  • Basic comprehension of homeomorphisms and isomorphisms in topology.
NEXT STEPS
  • Study the properties of co-vectors and their applications in linear algebra.
  • Explore the concept of dual spaces in depth, focusing on their mathematical definitions and implications.
  • Research the Frenet-Serret formulas and their significance in differential geometry.
  • Examine the biorthogonality condition and its role in establishing relationships between dual spaces and vector spaces.
USEFUL FOR

Mathematics students, particularly those studying linear algebra and differential geometry, as well as educators seeking structured references for teaching concepts related to co-vectors and dual spaces.

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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