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I'm having a bit of trouble understanding the dual space $V^*$ to a vector space $V$ over field $\mathbb F$.

So far I understand that a linear form/functional $f$ is a linear map from $V$ to its field of scalars such that $f:V\rightarrow \mathbb F$. However I have also read that the set of all linear forms/functionals is the dual space $V^*$.

My question is how can a vector space be made from the set of scalar fields? I don't think I have the right idea do I ...

Another thing, I have read that a one-form is an element of $V^*$, yet is also a linear map from $V$ to $V^*$. How can it be both? Many thanks for your time.

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"Field of scalars" is not the same as "scalar field." If for example $V$ is a vector space over the reals then linear functionals on $V$ are linear maps from $V$ to $\mathbb R$. The linear functionals form a vector space known as $V^{\ast}$, the dual space. A 1-form in the context of vector spaces is a linear functional.

1-forms on vector spaces are related to the 1-forms in differential geometry, which can be described as follows. At each point of $\mathbb R^n$ there is an attached tangent space, which is spanned by the partial derivatives. There is also an attached cotangent space which is the dual of the tangent space. This is spanned by the differentials $dx,dy$ etc. A 1-form is a function from $\mathbb R^n$ into the cotangent bundle, mapping each point to a differential contained in the cotangent space at that point. Similarly, a vector field is a function from $\mathbb R^n$ into the tangent bundle, mapping each point of $\mathbb R^n$ to some directional derivative at that point.

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  • $\begingroup$ I think 1-form in linear algebra is exactly the same as a linear functional as presented by the OP. Don't confuse this with the stuff from differential geometry/topology of the same name. $\endgroup$
    – Timbuc
    Dec 23, 2014 at 20:15
  • $\begingroup$ @Timbuc edited, thanks. No way I'm going to remove that part of the answer though since I put so much effort into typing it in my phone. It's an addendum now. $\endgroup$ Dec 23, 2014 at 20:20
  • $\begingroup$ Hehe...fair enough. The reference to differential geometry seems way too far from what the OP apparently meant, but it's fine. $\endgroup$
    – Timbuc
    Dec 23, 2014 at 20:21
  • $\begingroup$ @MattSamuel Thank you both for your answer, I am currently brushing up on my linear algebra before delving into differential geometry for classical mechanics so this relation between the two is really useful! $\endgroup$ Dec 23, 2014 at 20:21
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    $\begingroup$ @JanetthePhysicist yes. This is standard notation in math, not sure about physics though. $\endgroup$ Dec 23, 2014 at 20:30
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The set $\;V^*\;$ is made of functions , not "the set of scalar fields", whatever you may have meant with this: every element in $\;V^*\;$ is a linear functional $\;f:V\to \Bbb F\;$ , and we can sum these functionals and multiply by them by scalars from $\;\Bbb F\;$ (and still get linear functionals, of course).

About "one- forms": this is just another name for linear functionals. It is used in particular when we one to distinguish between one-variable functionals and multilinear functionals $\;V\times V\times\ldots\times V\to\Bbb F\;$ , like for example volume functionals (determinant) and etc.

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