pointwise limit of bounded operators is bounded
The following result is a corollary of the principle of uniform boundedness.
is linear and . Moreover, the sequence is bounded (http://planetmath.org/Bounded).
Proof : It is clear that the operator is linear.
For each we have since is . By the principle of uniform boundedness (http://planetmath.org/BanachSteinhausTheorem) we must also have .
Then for each we have
which means that is .
|Title||pointwise limit of bounded operators is bounded|
|Date of creation||2013-03-22 17:32:10|
|Last modified on||2013-03-22 17:32:10|
|Last modified by||asteroid (17536)|