The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.

Question. Let $X$ be a Banach space, and let $V$ and $W$ be complemented subspaces of $X$. Is it true that $V \cap W$ is a complemented subspace? If not, is it true under (nontrivial) additional assumptions?

In the case of a Hilbert space $X$, where the answer is of course yes, the orthogonal projector onto $V \cap W$ may be found as a strong limit of operators $P_{V\cap W}=\lim_{n\to \infty}(P_V P_W)^n $ . Is there a similar procedure to obtain a linear projector onto $V\cap W$ in the general case of a Banach space $X$?