Notation for the Dual of a Vector Space

Click For Summary
The notation ##V^\vee## is commonly used to denote the dual space of a vector space ##V##, primarily to avoid confusion with the group of units of a ring, which can be denoted as ##R^*##. The discussion raises questions about the reasoning behind this specific notation and its implications for defining the dual of a ring. It suggests that defining a ring as a module over itself could clarify the concept of duality in this context. The focus remains on understanding the rationale for using the "\vee" symbol in vector space notation. Ultimately, the conversation seeks to clarify the distinctions and definitions related to dual spaces and units in algebraic structures.
Mandelbroth
Messages
610
Reaction score
24
I've been reading about algebraic geometry lately. I see that a lot of authors use ##V^\vee## to denote the dual space of a vector space ##V##. Is there any particular reason for this?

The only reason I could think of is that this notation leaves us free to use ##R^*## to denote the units of ##R##. However, this still doesn't make sense of the reasoning behind the notation.

Thank you!
 
Physics news on Phys.org
What do you mean by the units of R, how do you define the dual of a ring? Do you define the ring as a module over itself and then define its dual?
 
Bacle2 said:
What do you mean by the units of R, how do you define the dual of a ring? Do you define the ring as a module over itself and then define its dual?
Sorry. I'm saying that the notation prevents confusion between the dual of a vector space and the group of units of a ring. The question is why the "\vee" is used.
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Undergrad Terminologies used to describe tensor product of vector spaces
  • · Replies 7 ·
Replies
7
Views
3K
MHB Dual Vector Space and Dual Basis - another question - Winitzki Section 1-6
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Understanding the concepts of isometric basis and musical isomorphism
  • · Replies 3 ·
Replies
3
Views
2K
High School Understanding Bases of a Vector Space
  • · Replies 3 ·
Replies
3
Views
3K
MHB Showing the Dual Basis is a basis
  • · Replies 1 ·
Replies
1
Views
3K
Why the bra vector is said to belong in the dual space?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Coefficients of a vector regarded as a function of a vector
  • · Replies 13 ·
Replies
13
Views
3K
MHB How Are Coefficients of a Vector Linear Functions of the Vector?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Inner product vs dot/scalar product
  • · Replies 4 ·
Replies
4
Views
3K
Null space of dual map of T = annihilator of range T
  • · Replies 1 ·
Replies
1
Views
2K