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

The author introduces a topological degree for a class of maps defined between real Banach spaces. These maps are Fredholm of index zero. The degree is based on a notion of orientation for any linear Fredholm operator of index zero between two real vector spaces (no topological structure is required). It possesses the most important properties usually associated with a degree theory and is invariant with respect to continuous homotopies of Fredholm maps.