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 $$y=Py=P(x+x')=Px.$$