I'm sorry that the title is very general indeed. I'm looking for a theorem/corollary that uses all of the following four theorems/concepts in its course. This may be rather ambitious, but any ideas? I am not looking for the proof (just yet).
BCT, Uniform boundedness principle, open-mapping theorem, closed graph theorem.
From my text, BCT implies Open Mapping theorem and Uniform Boundedness Principle, and we use Open Mapping theorem to prove the Closed Graph theorem. So I suppose if we use Closed Graph theorem and Uniform Boundedness Principle to prove a theorem/corollary, then in fact we use the four theorems.
Here is one corollary:
Let $X$ and $Y$ be Banach spaces. If $T:\ X\to Y$ is a linear map such that $f\circ T\in X^*$ for every $f\in Y^*$, then $T$ is bounded.
You can use either Closed Graph theorem or Uniform Boundedness Principle to prove it, and I prefer the latter.
