Bornological space - Knowlpedia

Let $E$ be a Hausdorff locally convex space. A convex balanced set $D\subseteq E$ is bornivorous if it absorbs every bounded subset $B$ of $E$ : for each $B$ , some $r>0$ satisfies $B\subseteq tD$ whenever $\lvert t\rvert\ge r$ . The space $E$ is bornological if every bornivorous convex balanced set is a neighborhood of $0$ . Equivalently, the locally convex topology of $E$ is the finest locally convex topology having the same bounded subsets.

Mapping characterization

The defining condition is equivalent to a useful test: for every locally convex space $F$ , a linear map $T:E\to F$ is continuous whenever it maps bounded subsets of $E$ to bounded subsets of $F$ . Thus boundedness controls continuity on the source. The converse implication—continuous linear maps preserve bounded sets—holds for every topological vector space and does not require bornologicality.

Examples and permanence

Every metrizable locally convex space, hence every normed or Fréchet space, is bornological. Locally convex inductive limits of bornological spaces are bornological, so LF-spaces provide important nonmetrizable examples. Quotients and locally convex direct sums preserve bornologicality. Arbitrary subspaces need not: the bounded subsets inherited by a subspace may fail to determine its subspace topology.

Conventions and contrasts

“Bornological” here is a property of a locally convex topology, not merely the data of an abstract bornology. It differs from barreledness: bornologicality detects continuity of bounded linear maps, whereas barreledness supports uniform-boundedness principles for pointwise bounded families. Neither property implies the other without additional hypotheses. Some sources say bornologic, but bornological is the usual English form.

References

Henri Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland Mathematics Studies 26, 1977. Elsevier publisher record. Relevant: Chapters I–II on bounded structures and fundamental bornological constructions.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter IV on bornological spaces and dual topologies.