  The problem of approximating the infinite dimensional space of all continuous
maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite
dimensional spaces of algebraic maps arises in several areas of geometry and
mathematical physics. An often considered formulation of the problem (sometimes
called the Atiyah-Jones problem after \cite{AJ}) is to determine a (preferably
optimal) integer $n_D$ such that the inclusion from this finite dimensional
algebraic space into the corresponding infinite dimensional one induces
isomorphisms of homology (or homotopy) groups through dimension $n_D$, where
$D$ denotes a tuple of integers called the "degree" of the algebraic maps and
$n_D\to\infty$ as $D\to\infty$. In this paper we investigate this problem in
the case when $X$ is a real projective space and $Y$ is a smooth compact toric
variety.
