Assume we have a directed set $L$, a Banach space $Z$ and a family of closed subspaces $Z_\gamma \subset Z$ with $\forall \gamma' \geq \gamma: Z_\gamma \subseteq Z_{\gamma'}$.
Why is in this case the space $\bigcup_{\gamma \in L} Z_\gamma$ in general no Banach space (but its completion is one)?
Consider $\mathcal{C}[0,1]$ and let $P_n$ be the set of all polynomials of degree $n$ or less.