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 ${\displaystyle (V_{n},i_{nm})}$ of Banach spaces. This means that V is a direct limit of the system ${\displaystyle \left(V_{n},i_{nm}\right)}$ in the category of locally convex topological vector spaces and each ${\displaystyle V_{n}}$ 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 ${\displaystyle m\leq n}$ implies ${\displaystyle V_{m}\subseteq C_{n}}$, each mapping ${\displaystyle i_{nm}:V_{m}\to V_{n}}$ is the natural inclusion, and strict means that each ${\displaystyle V_{m}}$ has ${\displaystyle V_{n}}$'s subspace topology (where ${\displaystyle m\leq n}$).[1] 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 ${\displaystyle U\cap V_{n}}$ is an absolutely convex neighborhood of 0 in ${\displaystyle V_{n}}$ for every n.