Transport in metallic magnetic multilayers composed of two ferromagnetic films separated by a nonmagnetic material is analyzed theoretically within the real-space Kubo formalism. The electron system is approximated by the jellium model with a steplike and spin-dependent lattice potential. Considerations are restricted to the current-in-plane geometry. Particular attention is focused on the quantum size effect and its influence on the giant magnetoresistance (GMR). A simple analytical formula valid in the interference limit is also derived. It has been shown that the GMR effect can occur in structures with no spin asymmetry in relaxation times, but with spin-dependent electronic structure for conducting electrons.