$\newcommand{\ot}{\otimes}\newcommand{\op}{\oplus}$Exercise from Kassel's Quantum Groups:
Show that the canonical isomorphisms $V^{\ot (n+m)}\cong V^{\ot n}\ot V^{\ot m}$ induce a coalgebra structure on $\hat TV=\prod_{n\geq 0}V^{\ot n}$.
I've seen that it is possible to define coalgebra structure on a subspace of $\hat TV$. Is it possible to define a coalgebra structure on $\hat TV$?