 It is both useful and customary, though not strictly necessary, to employ a Hilbert space representation of a quantum theory. In such a representation, the operators corresponding to observables are represented as acting on elements of an appropriately constructed Hilbert space (see the entry on  quantum mechanics  for details). Usually, the Hilbert space representation is constructed in such a way that vectors in the space represent pure states; such a representation is called an irreducible representation. Irreducible representations, in which mixed states are also represented by vectors, are also possible.