Locally convex spaces which satisfy the uniform bounded principle, i.e., every pointwise bounded family of continuous linear maps (with values in any normed space) is equicontinuous, are called barrelled spaces.

Of course, the most prominent examples are those for which the uniform boundeness (or Banach-Steinhaus) theorem was proved originally, namely, Fréchet spaces. But compared to the class of Fréchet spaces, the class of barrelled spaces has much better permanence properties, for instance, it is (rather trivially) stable with respect to inductive limits in the category of locally convex spaces (in category theory, one would rather say colimits).

You can find much information (like the relation to completeness properties of the dual space) in the book Barrelled Locally Convex Spaces of Bonet and Pérez Carreras.

Locally convex spaces which satisfy the uniform bounded principle, i.e., every pointwise bounded family of continuous linear maps (with values in any normed space) is equicontinuous, are called barrelled spaces.

Of course, the most prominent examples are those for which the uniform boundeness (or Banach-Steinhaus) theorem was proved originally, namely, Fréchet spaces. But compared to the class of Fréchet spaces, the class of barrelled spaces has much better permanence properties, for instance, it is (rather trivially) stable with respect to inductive limits in the category of locally convex spaces (in category theory, one would rather say colimits).

You can find much information (like the relation to completeness properties of the dual space) in the book Barrelled Locally Convex Spaces of Bonet and Pérez Carreras. In particular, 4.3.2 contains examples of barrelled normed spaces which are not complete.