An example of a non closed linear map.
Just take any non-continuous linear map on a Banach space, an example of a non-continuous functional (so image $\Bbb R$) can be found here.
On a non-Banach space (for $X$ a Banach space we get that continuity and closedness are equivalent, by the Closed Graph Theorem, hence the previous example) you can consider $X=C_p([0,1])$ (the continuous real functions on $[0,1]$ in the pointwise topology, a locally convex topological vector space but not normable) and $T(f)=\int_0^1 f(x)dx \in \Bbb R$ as a "natural" example.