Bornological space - Imported Knowlpedia Content

Definition

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 Hogbe-Nlend, Chapters I–II.

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.