  For a nonempty compact set D of R we determine the maximal possible dimension
of a subspace X of polynomial functions of degree at most m which possesses a
positive bases (where positivity is understood on D). The exact value of this
maximal dimension depends on topological features of the base set D. We show
that at many cases dimension m can be achieved.Whereas only for low m or finite
sets D it is possible to have m+1 (the full dimension of the space of
polynomials of degree at most m) dimensional subspace X with positive basis.
Therefore, it turns out that for no D it is possible to have a positive basis
of the polynomial space of degree at most m for all m in N.
