  Let $\mathcal{DZ}_k$ be the $\mathbb{Q}$-vector space spanned by double zeta
values with weight $k$, and $\mathcal{DM}_k$ be its quotient space divided by
the space $\mathcal{PZ}_k$ spanned by the zeta value $\zeta(k)$ and products of
two zeta values with total weight $k$. When $k$ is even, an upper bound for the
dimension of $\mathcal{DM}_k$ is known. By adding the dimensions of
$\mathcal{DM}_k$ and $\mathcal{PZ}_k$, an upper bound of $\mathcal{DZ}_k$ which
equals $k/2$ minus the dimension of the space of modular forms of weight $k$ on
the modular group is given. In this note, we obtain some specific sets of
generators for $\mathcal{DM}_k$ which represent the upper bound. These yield
the corresponding sets and the upper bound for $\mathcal{DZ}_k$.
