# Functional analysis convergence question

1. Apr 8, 2010

### iceblue

If $$X$$ is Banach space and $$F:X \rightarrow X$$ is a linear and bounded map and that $$F^n(x)\rightarrow0$$ pointwise .. How can I show that it converges to zero uniformly also?

Thanks

Last edited: Apr 8, 2010
2. Apr 9, 2010

### Eynstone

http: //en.wikipedia.org/wiki/Bounded_operator#Equivalence_of_boundedness_and_continuity