Let's say $V$ is a vector space over a ground field $\mathbb K$. Normally, the tensor algebra of $T(V)$ over $V$ is defined as: $$T(V)=\bigoplus_{i=0}^{\infty}V^{\otimes i}.$$ As far as I know, this seems fairly standard. However, in the context of differential geometry (for example), we often use the different definition $$T(V)=\bigoplus_{i,j=0}^{\infty}V^{\otimes i}\otimes(V^*)^{\otimes j}.$$
Is there a standard notation for differing between these two definitions of $T(V)$? I know $V^{\otimes i}$ is often denoted $T^iV$ and $V^{\otimes i}\otimes(V^*)^{\otimes j}$ is denoted as $T^i_jV$.