Let
be a vector such that
is a subspace. One can choose vectors
such that
,
are linearly independent and the linear span of
is
.
All convex combinations of
belong to
and also belong to
.
Hence, the
is not empty. We construct
as claimed in (a) by taking
.
Consequently,
.
The statement (b) is evident from the picture
(
Relative interior
).
