 If 

are finite, then the function "T"("x")"v" is integrable for each "v" in "E" with

The result can be proved by replacing sums by integrals in the previous proof or by using Riemann sums to approximate the integrals.Here is an example of an "orthogonal" family of operators.Consider the inifite-dimensional matrices

and also

Then
formula_29 for each formula_30,
hence the series formula_31
does not converge in the uniform operator topology.