An application of the closed graph theorem.

[Hjensen]

So I have to show that a projection P (i.e a linear operator with P=P²) on a Banach space X is bounded if and only if \ker (P) and P(X) are closed subspaces of X.My idea was to boil it down, using the closed graph theorem. What's left for me now is to show that the graph G(P):=\{(x,y)\in X\times X: y=Px\} is closed if \ker(P) and P(X) are closed. I don't quite know how this can be achieved though. Does anyone know how this could be done? Or am I simply taking the wrong approach by using the closed graph theorem?

 

[arkajad]

Edit: I wrote a nonsense. Thinking ...

 

[arkajad]

Assume ker(P) and P(X) closed. Let (x_n,y_n)\rightarrow (x,y),\, y_n=Px_n. Then (y_n-x_n)\in \ker(P) and so y=x+x',\, x'\in\ker (P). From (I-P)y_n=0 it follows (I-P)y=0, so y=Py=P(x+x')=Px.