LF-space

Summary

An LF-space is a locally convex space obtained as the inductive limit of an increasing sequence of Fréchet spaces. The letters stand for "limit of Fréchet spaces". The main example is the space $C_c^\infty(\Omega)$ of test functions on an open set $\Omega \subseteq \R^d$. It is the union, over compact sets $K \subseteq \Omega$, of the Fréchet spaces of smooth functions supported in $K$. Its dual is the space of distributions. LF-spaces are barrelled and bornological, although they are usually not metrizable.