I have a linear map $P$ on a Banach space, $X$, with $P^2 = P$ and I'm trying to show that $P$ is continuous if and only if $\ker(P)$ and $\ker(I-P)$ are closed. One direction is straight forward but I'm struggling with the other.
I noticed we can write $X = \ker(P) \bigoplus \ker(I-P)$ and now I'm trying to use the closed graph theorem to show $P$ is continuous.
So we take a sequence in the graph $(x_n, Px_n)$ with $x_n \to x$ and $Px_n \to y$ and I'd like to show $y \in im(P)$. We can write $x_n = a_n + b_n$ from the decomposition above and then $Px_n = b_n$ and since $ker(I-P)$ is closed then $y \in ker(I-P)$ and $x = \lim x_n = \lim a_n + y$ so $x = a + y$ with $a \in ker(P)$ (since it's closed).
I'm struggling to go any further with this and deduce that $y \in im(P)$
Thanks for any help