Quasi-bounded sets - Jan Kucera (International Journal of Mathematics and Mathematical Sciences)
It is proved in [1] & [2] that a set bounded in an inductive limit E=indlim En of Fréchet spaces is also bounded in some En iff E is fast complete. In the case of arbitrary locally convex spaces En every bounded set in a fast complete indlim En is quasi-bounded in some En, though it may not be bounded or even contained in any En. Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.