Vector space/Bidual/Natural mapping/Fact/Proof

Proof

Let be fixed. First of all, we show that is a linear form on the dual space . Obviously, is a mapping from to . The additivity follows from

where we have used the definition of the addition on the dual space. The compatibility with the scalar multiplication follows similarly from

In order to prove the additivity of , let be given. We have to show the equality

This is an equality inside of , in particular, it is an equality of mappings. So let be given. Then, the additivity follows from

The scalar compatibility follows from

In order to prove injectivity, let with be given. this means that for all linear forms , we have . But then, due to fact, we have

By the criterion for injectiviy, is injective.

In the finite-dimensional case, the bijectivity follows from injectivity and from fact.