In mathematics, an LF-space is a topological vector spaceV that is a locally convex strict inductive limit of a countable inductive system of Fréchet spaces. This means that V is a direct limit of the system in the category of locally convex topological vector spaces and each is a Fréchet space. The word "strict" means that each of the bonding maps is an embedding of TVSs. Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on by is identical to the original topology on. The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if is an absolutely convex neighborhood of 0 in for every n.
Properties
Every LF-space is barrelled and bornological. Every LF-space is a meager subset of itself. The strict inductive limit of a sequence of complete locally convex spaces is necessarily complete. In particular, every LF-space is complete. An LF-space that is the inductive limit of a countable sequence of separable spaces is separable. If is the strict inductive limit of an increasing sequence of Fréchet space then a subset of is bounded in if and only if there exists some such that is a bounded subset of. A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous. A linear map from an LF-space into a Fréchet space is continuous if and only if its graph is closed in. Every bounded linear operator from an LF-space into another TVS is continuous. If is an LF-space defined by a sequence then the strong dual space of is a Fréchet space if and only if all are normable. Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.
A typical example of an LF-space is,, the space of all infinitely differentiable functions on with compact support. The LF-space structure is obtained by considering a sequence of compact sets with and for all i, is a subset of the interior of. Such a sequence could be the balls of radius i centered at the origin. The space of infinitely differentiable functions on with compact support contained in has a natural Fréchet space structure and inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets. With this LF-space structure, is known as the space of test functions, of fundamental importance in the theory of distributions.
Suppose that for every positive integer n, and for m < n, consider Xm as a vector subspace of Xn via the canonical embedding Xm → Xn defined by sending to. Denote the resulting LF-space by X. The continuous dual space of X is equal to the algebraic dual space of X and the weak topology on is equal to the strong topology on . Furthermore, the canonical map of X into the continuous dual space of is surjective.
