1. For every
there are linearly independent vectors
,
such
that
for some
and finite
.
2. For every
there are vectors
,
such
that
for some
and the vectors
are linearly independent.
Remark
The Caratheodory theorem does not state that
might serve as a fixed basis. Indeed, on the picture
(
Caratheodory theorem
remark
) if the set X is open then for any pair of vectors
and
from X a point
may be found outside of the area span by the positively linear combinations of
and
.
Caratheodory theorem
remark.
Proof
1. The definition of
provides that there are some vectors
such
that
If such vectors are linearly dependent then there are numbers
We take a linear combination of two
equalities
and note that for at least one
the
is positive. Hence, a
exists such that all
are non negative and
for at least one index
.
Hence, we decreased the number of terms in the sum. We continue this process
until
are linearly independent.
Proof
2. The definition of
provides that there are some vectors
such that
We consider
and restate the above conditions as
Therefore,
.
The first part of the theorem applies and the vectors
may be assumed linearly independent. Hence, no all non-zero
exist such
that
Equivalently,
We express the
from the second equation and substitute it into the first. We obtain the
following
consequence
We conclude that no all non-zero
exist such that the above is true. Hence, the
are linearly independent.
