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I am currently working through 'Schutz-First course in General Relativity' problem sets. Question 2 of Chapter 3, asks me to prove the set of one forms is a vector space.

Earlier in the chapter, hedefines:

[tex] \tilde{s}=\tilde{p}+\tilde{q} [/tex]

[tex] \tilde{r}=\alpha \tilde{p} [/tex]

To be the one forms whose values on a vector [tex]\vec{A}[/tex] are:

[tex] \tilde{s} (\vec{A})=\tilde{p}(\vec{A})+\tilde{q}(\vec{A}) [/tex]

[tex] \tilde{r}(\vec{A})=\alpha \tilde{p}(\vec{A}) [/tex]

Given this definition, surely some of the axioms to be a vector space, are simply satisfied by defintion (namely closure under addittion, and closure under scalar multiplication), so how does one prove these things, if they have been definied so? Or should I be looking to prove other vector space axioms, like existance of additive inverse etc? If so still not sure where to start.

Any help, much appreciated.

Thanks

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# Proof set of one-forms is a vector space

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