The closure relation in infinite dimension is : ∫|x><x|dx =I (identity operator),but if we apply the limit definition of the integral the result is not logic or intuitive. The limit definition of the integral is a∫b f(x)dx=lim(n-->∞) [i=1]∑[i=∞]f(ci)Δxi, where Δxi=(b-a)/n (n--.>∞) and ci=a+((b-a)/n)Δxi. Apply the limit definition of the integral in the case of the closure relation above, it seems not logic and intuitive that ∫|x><x|dx=I(identity operator). ???? so how it comes ∫|x><x|dx=I(identity operator). I should take it for granted or there is a prove for it ?