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.