I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex. 

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that the topology $\tau$  on $X_n$ is relatively  second countable and metrizable for every $n$.