(1) and (2) are clear. (3). The inclusion
-
is also clear. Let
, .
Then we can choose a
basis
of and extend it to a basis of . The linear form vanishes on , therefore, it belongs to . Because of
-
we have
.
(4). Let be a basis of , and let
-
denote the mapping where these linear forms are the components. Here, we have
-
Assume that the mapping is not surjective. Then is a strict linear subspace of and its dimension is at most . Let be a -dimensional linear subspace with
-
Due to
fact,
there is a
linear form
-
whose kernel is exactly . Write
.
Then
-
contradicting the linear independence of the . Moreover, is surjective and the statement follows from
fact.