Does Continuous projections imply complementation? (This is a possibly trivial question.) Let $B$ be a Banach space for which there exists a continuous projection from $B$ onto a closed subspace $X\subseteq B$. > **Question.** Does there exist a closed subspace $Y\subseteq B$ such that $$ X\cap Y=\\{0\\}\,\,\text{ and }\,\,X+Y=B? $$

Yes, take $Y=\\{x-Px:x \in B\\}$. Then any point $x \in B$ is $Px + (x-Px) \in X+Y$. If $x \in X \cap Y$ then $x=Pz$ for some $z$. Since $P^{2}=P$ we see that $P$ vanishes on $Y$. Hence $Pz=0$ so $x=0$. Finally we have to show that $Y$ is closed. Let $x_n-Px_n \to u$. Since $P$ is continuous we get $$0=Px_n-P^{2}x_n=P(x_n-Px_n) \to Pu.$$ Thus $Pu=0$ and $u=u-Pu \in Y$. Hence $Y$ is closed.