I am trying to solve assignments of an institute of which I am not a student, and could not solve this problem. Can somebody please help?
Question is:
Let $X$ be a normed linear space and let $P: X \to Y$ be a projection. Then $P$ is closed map iff the subspaces $R(P)$ and $N(P)$ are closed in $X$.
Assuming $P$ is closed map, I am able to prove $R(P)$ and $ N(P)$ are closed by considering convergent sequences in these subspaces.
But can somebody please help me with the converse part?
Suppose $(x_n)$ is a sequence in $X$ converging to $x$ and $(Px_n)$ converges to $y$. Since $R(P)$ is closed, $y\in R(P)$. Now $x_n-Px_n\in N(P)$ and $$x_n-Px_n\to x-y \:\text{as} \:n\to\infty.$$ Since $N(P)$ is closed, $x-y\in N(P)$. So $$P(x-y)=0\implies Px=Py.$$ As $y\in R(P)$, we have $P(y)=y$. Hence $P(x)=y$.