Suppose $X$ is a Banach space, $T \in \mathscr{B}(X, Y)$ is a bounded linear operator, and $T$ is compact. If $\lambda \neq 0$, and $X = N(T-\lambda I) \oplus E$ (note:'$N$' indicates the null space, and the direct sum $\oplus$ is defined as the picture shows); $x = x_1 + x_2 \in X$, $x_1 \in N(T-\lambda I), x_2 \in E$, and $Px = x_1$. Prove that $P$ is a bounded linear operator. Click here to see the definition of direct sum, which is in Rudin's book
Hint:the basic idea of this problem is: Assume $x_n$ is a sequence in $X$, and $\lim_{n\to\infty} x_n = x$, and $lim_{n \to\infty }P(x_n) = y$. Then if we can prove $P(x) = y$, we are done. But I don't know how to prove $P(x) = y$. Can anybody help me out? Thanks.