According to the Fredholm alternative theorem, either the adjoint of a continuous linear operator with a closed range has a non-trivial kernel or the operator is of this type.
A corollary of the Lyusternik theorem is that, at a certain point, the Frechet derivative of the map between Banach spaces has this property. According to the Fredholm alternative theorem, either the adjoint of a continuous linear operator with a closed range has a non-trivial kernel or the operator is of this type. More generally, a function is said to have this property if and only if the composition of it and its inverse is everywhere the identity function. Only applicable to functions whose domain spans the entire codomain, this is, FTP, what property, functions having which are bijective only if they are also injective?
A corollary of the Lyusternik theorem is that, at a certain point, the Frechet derivative of the map between Banach spaces has this property.
surjectiveness (or onto)