  Kempf et al. in Ref. [1] have formulated a Hilbert space representation of
quantum mechanics with a minimal measurable length. Recently it has been
revealed, in the context of doubly special relativity, that a test particles'
momentum cannot be arbitrarily imprecise and therefore there is an upper bound
for momentum fluctuations. Taking this achievement into account, we generalize
the seminal work of Kempf {\it et al.} to the case that there is also a maximal
particles' momentum. Existence of an upper bound for the test particles'
momentum provides several novel and interesting features, some of which are
studied in this paper.
