When is the sum of complemented subspaces complemented?
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition a formula for a projection onto the sum is given. We also show that the condition is sharp (in a certain sense). As applications, we get a sufficient condition for the complementability of the sum of marginal subspaces in $L^p$ and a quantitative result on stability of the complementability property of the sum of linearly independent subspaces.