  We investigate the existence of an H-space structure on the function space,
F_*(X,Y,*), of based maps in the component of the trivial map between two
pointed connected CW-complexes X and Y. For that, we introduce the notion of
H(n)-space and prove that we have an H-space structure on F_*(X,Y,*) if Y is an
H(n)-space and X is of Lusternik-Schnirelmann category less than or equal to n.
When we consider the rational homotopy type of nilpotent finite type
CW-complexes, the existence of an H(n)-space structure can be easily detected
on the minimal model and coincides with the differential length considered by
Y. Kotani. When X is finite, using the Haefliger model for function spaces, we
can prove that the rational cohomology of F_*(X,Y,*) is free commutative if the
rational cup length of X is strictly less than the differential length of Y,
generalizing a recent result of Y. Kotani.
