Quasi-bounded sets (Q1814024)

Summary: It is well-known that a set bounded in an inductive limit \(E=\mathrm{indlim} E_ n\) of Fréchet spaces is also bounded in some \(E_ n\) iff \(E\) is fast complete. In the case of arbitrary locally convex spaces \(E_ n\) every bounded set in a fast complete \(\mathrm{indlim} E_ n\) is quasi-bounded in some \(E_ n\), though it may not be bounded or even contained in ay \(E_ n\). Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.