Dual Vector Space: Simple Explanation

[quietrain]

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

 

[FunkyDwarf]

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.

 

[quietrain]

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!

 

[Fredrik]

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 $$(f+g)(v)=f(v)+g(v)$$ 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 $$(kf)(v)=k(f(v))$$ 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.

 

[quietrain]

ok thanks