Let X,Y be two banach spaces and be linear operators that converge punctually to Show that
uniformly in each compact subset of X
First, replacing by , we may assume that , so that for all in .
Next, it follows from the uniform boundedness principle that is bounded, say for all .
Now let . For each in , there exists such that whenever . Let denote the open set . Then for all in , whenever . Such sets form an open cover of a given compact subset of , and by looking at a finite subcover you should be able to see that the pointwise convergence is uniform on that set.