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.