Remark 9.5.3. The projective topology behaves well across the constraints of topological vector spaces and locally convex spaces: the preimage of a vector space/locally convex topology is also a vector space/locally convex topology.

On the inductive side, the story is not as simple: In principle, the locally convex inductive topology is smaller than the vector space inductive topology, which is smaller than the inductive topology. As such, the same construction must be performed three separate times, each time restricting to a smaller collection of sets.

In addition to the neighbourhood construction given above, the inductive topology may also be constructed as the weak topology generated by all topologies satisfying certain properties. While this more non-constructive method is simpler, it does not directly provide an explicit fundamental system of neighbourhoods at $0$.