So I am revising Linear Algebra and have a question about finding isomorphisms between dual spaces. I am looking at this problem: Let $V$ be a finite dimensional vector space, over a field $F$ and let $V'$ be its dual space. Show that the map $(f,v) \mapsto f(v)$ from $V'\times V$ to $F$ induces an isomorphism $\theta : V\rightarrow (V')'$.
My first question is, is what is the isomorphism induced? Am I meant to be finding another map from the one given to me, and if so, how do I go about doing this?
If I am testing to see if something is an isomorphism, I think I want to check it is linear (so well-defined), surjective and injective. I can see the the map stated, if this is the map, is linear, but how would I go about proving its surjective and injective.
Any help understanding this problem and topic appreciated.Thanks.