  An extensively tacit understandings of equivalency between the deformed
Heisenberg-Weyl algebra in noncommutative space and the undeformed
Heisenberg-Weyl algebra in commutative space is elucidated. Equivalency
conditions between two algebras are clarified. It is explored that the deformed
algebra related to the undeformed one by a non-orthogonal similarity
transformation. Furthermore, non-existence of a unitary similarity
transformation which transforms the deformed algebra to the undeformed one is
demonstrated. The un-equivalency theorem between the deformed and the
undeformed algebras is fully proved. Elucidation of this un-equivalency theorem
has basic meaning both in theory and practice.
