Orientation and degree for Fredholm maps of index zero between Banach spaces

We define a notion of topological degree for a class of maps (called orientable), defined between real Banach spaces, which are Fredholm of index zero. We introduce first a notion of orientation for any linear Fredholm operator of index zero between two real vector spaces. This notion (which does not require any topological structure) allows to define a concept of orientability for nonlinear Fredholm maps between real Banach spaces. The degree which we present verifies the most important properties, usually taken into account in other degree theories, and it is invariant with respect to continuous homotopies of Fredholm maps.