- #1

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

thanks

- Thread starter quietrain
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- #1

- 654

- 2

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

thanks

- #2

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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

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so why is it called the dual space? whats the significance?

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

so is the crux is

? i don't really get this, can you elaborate? thanks!dual space of a given vector space contains all linear functions that can act on the aforementioned vector space

- #4

Fredrik

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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.

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

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ok thanks

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