Suppose $X$ is a Banach space and $X = E \bigoplus F$ where $E,F$ are vector subspace of $X$. Define $P$ : $X$ $\rightarrow$ $X$ by $P(u\oplus v) = u$ for $u \in E$ and $v \in F$.
Prove that if $E$ and $F$ are closed in $X$, then $P$ is bounded.
I would like to know the idea of solving this problem. I was thinking two ways to prove the boundedness of P. Applying the Closed Graph Theorem or proving by definition. I failed either way since I don't know how to apply the assumption that $E$ and $F$ being closed.
Suppose $(x_nPx_n) \to (u,w)$. Write $x_n$ as $u_n+v_n$ with $u_n \in E, v_n \in F$. Then $Px_n=u_n$ so $u_n \to w$. [This implies $w \in E$ since $E$ is closed]. Also, $x_n \to u$ so $v_n=x_n-u_n \to u-w$. Since $F$ is closed we get $u-w \in F$. Thus, $u=w+(u-w)$ with $w \in E$ and $u-w \in F$. Hence, $Pu=w$ (by defintion of $P$). Now conclude by applying Closed Graph Theorem