Bounded linear map which is not continuous

For every Hausdorff locally convex space $(X,\mathcal T)$ the weak topology $\sigma(X,X')$ has the same bounded sets as $\mathcal T$, hence the identity $id: (X,\sigma(X,X')) \to (X,\mathcal T)$ is bounded but discontinuous whenever, e.g., $(X,\mathcal T)$ has a continuous norm. For example, $(X,\mathcal T)$ could be an infinite deimensional normed space.

Locally convex spaces $X$ with the property that every bounded linear map on $X$ is continuous are called bornological. Besides metrizable spaces locally convex inductive limits (aka colimits) of metrizable spaces belong to this class.