Geodesic
A generalized inductive limit topology for linear spaces
Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 105-110

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In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem. 
Iyahen, S. O.; Popoola, J. O. A generalized inductive limit topology for linear spaces. Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 105-110. doi: 10.1017/S001708950000183X
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