  We consider the inclusion of the space of algebraic (regular) maps between
real algebraic varieties in the space of all continuous maps. For a certain
class of real algebraic varieties, which include real projective spaces, it is
well known that the space of real algebraic maps is a dense subset of the space
of all continuous maps. Our first result shows that, for this class of
varieties, the inclusion is also a homotopy equivalence. After proving this, we
restrict the class of varieties to real projective spaces. In this case, the
space of algebraic maps has a ` minimum degree\rq filtration by finite
dimensional subspaces and it is natural to expect that the homotopy types of
the terms of the filtration approximate closer and closer the homotopy type of
the space of continuous mappings as the degree increases. We prove this and
compute the lower bounds of this approximation for ` even\rq components of
these spaces (more precisely, we prove a very similar and closely related
result, and state this one as a conjecture).
