LB-space


In mathematics, an LB-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system of Banach 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 Banach space.
Some authors restrict the term LB-space to mean that V is a strict locally convex inductive limit of Banach spaces, which means that implies, each mapping is the natural inclusion, and strict means that each has 's subspace topology. To distinguish this type of space from the more general definition, we will call this a strict LB-space.
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

A strict LB-space is complete, barrelled,, and bornological.

Examples