Dual Vector Space: Simple Explanation

In summary, a dual vector space is a set of linear functions that can act on a given vector space. It is useful in applications such as quantum mechanics and differential geometry. The notation is also commonly used in these fields.
  • #1
quietrain
655
2
hi, anyone can provide a simple explanation of what is a dual vector space?

i have scoured the net and the explanations are all a tad too complicated for my understanding :(

thanks
 
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  • #2
My understanding, and others can feel free to correct me here, is that the dual space of a given vector space contains all linear functions that can act on the aforementioned vector space.

For example, if I have a displacement vector one can consider the gradient operator as an element of the dual space: the gradient is a linear operator that can act on a displacement vector to return a velocity vector.

I think.

Other examples are norms and inner products and things.
 
  • #3
by linear functions , do you mean linear operators?

so why is it called the dual space? what's the significance?

issn't the displacement velocity and gradient operators just acting in vector space?

so is the crux is
dual space of a given vector space contains all linear functions that can act on the aforementioned vector space
? i don't really get this, can you elaborate? thanks!
 
  • #4
Suppose that V is a normed vector space over ℂ. Let V* be the set of all bounded linear functions from V into ℂ. Now we define a function from V×V into V called addition by [tex](f+g)(v)=f(v)+g(v)[/tex] for all f,g in V* and all v in V. Then we define a function from ℂ×V into V called scalar multiplication by [tex](kf)(v)=k(f(v))[/tex] for all k in ℂ, all f in V*, and all v in V. These definitions give V* the structure of a vector space. It's called the dual space of V.

If V is a normed vector space over ℝ, replace every ℂ with ℝ in the definitions above.

Dual spaces aren't really significant for "elementary" applications. The concept is useful in QM, but it's mainly just to give us a notation (bra-ket notation) that's sometimes nicer than the alternatives. The only applications I know where dual spaces are needed are those that use differential geometry, in particular GR.
 
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  • #5
ok thanks
 

1. What is a dual vector space?

A dual vector space is a mathematical concept that refers to the set of all linear functionals on a given vector space. It is a vector space consisting of linear maps from the original vector space to the field of scalars.

2. What is the difference between a vector space and a dual vector space?

The main difference between a vector space and a dual vector space is that in a vector space, the elements are vectors, while in a dual vector space, the elements are linear functionals. Additionally, a vector space is defined over a field of scalars, while a dual vector space is defined over the same field of scalars.

3. How is a dual vector space related to the original vector space?

A dual vector space is related to the original vector space through the concept of duality. This means that for every vector in the original space, there is a corresponding dual vector in the dual space, and vice versa. The two spaces are also isomorphic, meaning they have the same dimension.

4. What is the importance of dual vector spaces in mathematics and science?

Dual vector spaces have many applications in mathematics and science, particularly in areas such as linear algebra, functional analysis, and physics. They allow for a deeper understanding of vector spaces and provide a powerful tool for solving problems in these fields.

5. How can I visualize a dual vector space?

While it can be difficult to visualize a dual vector space directly, one way to think about it is as a space of arrows pointing in different directions, with each arrow representing a different linear functional. Just as vectors can be added and scaled in a vector space, dual vectors can be added and scaled in a dual vector space.

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