  The aim of this work is to characterize linear maps of inner pro\-duct
infinite-dimensional vector spaces where the Moore-Penrose inverse exists. This
MP inverse generalizes the well-known Moore-Penrose inverse of a matrix $A\in
\text{Mat}_{n\times m} ({\mathbb C})$. Moreover, a method for the computation
of the MP inverse of some endomorphisms on infinite-dimensional vector spaces
is given. As an application, we study the least norm solution of an infinite
linear system from the Moore-Penrose inverse offered.
