One might have wondered why contains only linear bounded operators, and not linear operators of any and every kind. This has a very specific reason. Unless we fix , we cannot construct a cauchy sequence of linear operators. And we do not want to fix , as we want to define the limit for every .
Now if we allow ourselves to take all , knowing that , we coud convert any sequence of linear operators into a divergent one. Hence, we need to construct sequences of the kind . This requires that we use bounded linear operators.