LB-space

In mathematics, an LB-space, also written (LB)-space, is a topological vector space ${\displaystyle X}$ that is a locally convex inductive limit of a countable inductive system ${\displaystyle (X_{n},i_{nm})}$ of Banach spaces. This means that ${\displaystyle X}$ is a direct limit of a direct system ${\displaystyle \left(X_{n},i_{nm}\right)}$ in the category of locally convex topological vector spaces and each ${\displaystyle X_{n}}$ is a Banach space.

If each of the bonding maps ${\displaystyle i_{nm}}$ is an embedding of TVSs then the LB-space is called a strict LB-space. This means that the topology induced on ${\displaystyle X_{n}}$ by ${\displaystyle X_{n+1}}$ is identical to the original topology on ${\displaystyle X_{n}.}$ Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."

The topology on ${\displaystyle X}$ can be described by specifying that an absolutely convex subset ${\displaystyle U}$ is a neighborhood of ${\displaystyle 0}$ if and only if ${\displaystyle U\cap X_{n}}$ is an absolutely convex neighborhood of ${\displaystyle 0}$ in ${\displaystyle X_{n}}$ for every ${\displaystyle n.}$

A strict LB-space is complete, barrelled, and bornological (and thus ultrabornological).

If ${\displaystyle D}$ is a locally compact topological space that is countable at infinity (that is, it is equal to a countable union of compact subspaces) then the space ${\displaystyle C_{c}(D)}$ of all continuous, complex-valued functions on ${\displaystyle D}$ with compact support is a strict LB-space. For any compact subset ${\displaystyle K\subseteq D,}$ let ${\displaystyle C_{c}(K)}$ denote the Banach space of complex-valued functions that are supported by ${\displaystyle K}$ with the uniform norm and order the family of compact subsets of ${\displaystyle D}$ by inclusion.

Let

denote the space of finite sequences, where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}$ let ${\displaystyle \mathbb {R} ^{n}}$ denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}$ denote the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$ so that its image is

and consequently,