Inductive limit of locally convex spaces - Imported Knowlpedia Content

Definition

Let $(E_i,\phi_{ji})$ be a directed system of locally convex spaces and continuous linear maps, and let $E=\varinjlim E_i$ be its algebraic direct limit with canonical linear maps $\iota_i:E_i\to E$ . The locally convex inductive-limit topology on $E$ is the finest locally convex vector topology for which every $\iota_i$ is continuous. The resulting locally convex space is denoted

\varinjlim E_i \quad\text{or}\quad \operatorname{ind\,lim}_i E_i.

This is a categorical final construction within locally convex spaces. It need not equal the ordinary final topology in topological spaces, because that topology need not make $E$ a locally convex topological vector space.

Universal property

For every locally convex space $F$ , a linear map $T:E\to F$ is continuous exactly when each composite

T\circ\iota_i:E_i\to F

is continuous. This property characterizes the inductive limit up to canonical topological isomorphism and is often the most efficient way to prove continuity of a linear map defined on the limit.

Increasing sequences

A common case is an increasing sequence

E_1\subseteq E_2\subseteq\cdots,\qquad E=\bigcup_{n\geq1}E_n,

with continuous inclusions. If every $E_n$ is Fréchet, the inductive limit is called an LF-space. Compactly supported smooth functions are obtained in this way by taking smooth functions supported in a fixed compact set and then enlarging the compact set.

Hausdorffness and regularity

Some authors require locally convex spaces to be Hausdorff. In that convention, a non-Hausdorff locally convex inductive limit must be divided by the closure of $\{0\}$ to obtain the categorical Hausdorff limit. Completeness, bounded-set behavior, and the compatibility of subspace topologies do not follow formally from the definition; adjectives such as strict, regular, and complete record extra hypotheses.

References

François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier book record. Relevant: Chapters 13–14.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter II.