2

I'm working on the following problem in Royden:

As a consequence of the Baire Category Theorem we showed that a mapping that is the pointwise limit of a sequence of continuous mappings on a complete metric space must be continuous at some point. Use this to prove that the pointwise limit of a sequence of linear operators on a Banach space has a limit that is continuous at some point and hence, by linearity, is continuous.

I'm stuck on how to start, linear operators on Banach Spaces need not be continuous. So I don't know how to get started. Even if I did know this, its not clear to me how linearity helps me go from pointwise continuity to full continuity.

yoshi
  • 3,509
  • 2
    I suspect the sequence of linear operators are each required to be continuous - if not just take the constant sequence of some discontinuous operator. – B. Mehta Apr 16 '18 at 03:48
  • could you elaborate on your comment about a constant sequence of a discontinuous operator? do you mean a sequence of such operators or such an operator acting on a sequence? – yoshi Apr 16 '18 at 04:04
  • 1
    For some discontinuous operator $T$, the sequence $(T,T,T,T,T,\dotsc)$ is what I mean. – B. Mehta Apr 16 '18 at 04:06

0 Answers0