It is shown in this paper that
total biorthogonal systems have the same cone if and only if they differ at most by
rearrangement and by positive scalar multiplication. A connection is demonstrated
between this result and work done by R. E. Fullerton in which he characterized the
existence of an unconditional basis in terms of the existence of certain type cones.
The paper is concluded by generalizing the first result to the situation in
which two biorthogonal systems have cones which induce order isomorphic
orderings.
