Closed Complemented Subspaces of Banach Spaces and Existence of Bounded Quasi-linear Generalized Inverses

In this article, an equivalent condition for the existence of a bounded quasi-linear (BQL) generalized inverse of a closed linear operator with respect to projector between two Banach spaces is given. Using the BQL generalized inverse, we give a necessary and sucient condition for a closed linear subspace in a Banach space to be complemented. Finally, an application of the main results to the Saddle-Node bifurcation theorem from multiple eigenvalues in nonlinear analysis is given.