Contents

# Contents

## Idea

Linear operators on normed spaces are continuous precisely iff they are bounded. A bornological space retains this property by definition.

The discussion below is about bornological CVSes, but there is a more general notion of bornological space.

## Definition

A locally convex topological vector space $E$ is bornological if every circled, convex subset $A \subset E$ that absorbs every bounded set in $E$ is a neighbourhood of $0$ in $E$. Equivalently every seminorm that is bounded on bounded sets is continuous.

The bornology of a given TVS is the family of bounded subsets.

Given a locally convex TVS $E$ with initial topology $T_0$, there is a finest topology $T$ such that the family of bounded subsets of $T$ coincides with $T_0$. The space $E$ equipped with the topology $T$ is called the bornologification of $E$, or the bornological space associated with $(E, T_0)$

## Properties

### Maps on bornological spaces

###### Theorem

Let $U$ be a linear map from a bornological space $E$ to any locally convex TVS, then the following statements are equivalent:

• $U$ is continuous,

• $U$ is bounded on bounded sets,

• $U$ maps null sequences to null sequences.

### Relation to Banach spaces

Every inductive limit of Banach spaces is a bornological vector space. (Alpay-Salomon 13, prop. 2.3)

Conversely, every bornological vector space is an inductive limit of normed spaces, and of Banach spaces if it is quasi-complete (Schaefer-Wolff 99)

## Examples

Every metrizible locally convex space is bornological, that is every Fréchet space and thus every Banach space.