Completeness relation or closure relation
 
The completeness relation or also called closure relation is often used in the algebra in quantum mechanics:
 
Σ_i |s_i><s_i| = 1 = 1
 
Suppose we have a complete set of orthonormal functions (states) |s_i>
 
An arbitrary function Ψ can be written as a linear combination: |Ψ>=Σ_ic_i|s_i>
 
Multiplying both sides by <s_j| gives:
<s_j|Ψ> = <s_j|Σ_ic_i|s_i> <=>
<s_j|Ψ> = Σ_ic_i<s_j|s_i> = Σ_ic_iδ_ji = c_j <=>  
<s_j|Ψ> = c_j
 
Thus, |Ψ> = Σ_ic_i|s_i> = Σ_i<s_i|Ψ>|s_i> = Σ_i|s_i><s_i|Ψ>   <=>
|Ψ> = (Σ_i|s_i><s_i|) |Ψ>   thus, Σ_i|s_i><s_i| = 1 = 1