  We construct a Hilbert holomorphic function space $H$ on the unit disk such
that the polynomials are dense in $H$, but the odd polynomials are not dense in
the odd functions in $H$. As a consequence, there exists a function $f$ in $H$
that lies outside the closed linear span of its Taylor partial sums $s_n(f)$,
so it cannot be approximated by any triangular summability method applied to
the $s_n(f)$. We also show that there exists a function $f$ in $H$ that lies
outside the closed linear span of its radial dilates $f_r, ~r<1$.
