Projection operators

Armed with the bra-ket formalism we can construct any operator in Hilbert space. The projection operator P_a is defined as

 

Pa = | a><a|,

Note that and in general any projection operator P has the property P²=P. Consider operator X whose eigenstates are given by the set . If we define the projection operators P_an = | a_n><a_n|, show that operator X can be expressed as a sum of projection operators, i.e.