Ultrabornological space


In functional analysis, a topological vector space X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous.

Definitions

Let be a topological vector space.

Preliminaries

A disk in a TVS is called infrabornivorous if it satisfies any of the following equivalent conditions:

  1. absorbs every Banach disks in.
while if locally convex then we may add to this list:

  1. the guage of is an infrabounded map;
while if locally convex and Hausdorff then we may add to this list:

  1. absorbs all compact disks.
    • i.e. is "compactivorious".

Ultrabornological space

A TVS is ultrabornological if it satisfies any of the following equivalent conditions:

  1. every infrabornivorous disk in is a neighborhood of the origin;
while if is a locally convex space then we may add to this list:

  1. every bounded linear operator from into a complete metrizable TVS is necessarily continuous;
  2. every infrabornivorous disk is a neighborhood of 0;
  3. be the inductive limit of the spaces as varies over all compact disks in ;
  4. a seminorm on that is bounded on each Banach disk is necessarily continuous;
  5. for every locally convex space and every linear map, if is bounded on each Banach disk then is continuous;
  6. for every Banach space and every linear map, if is bounded on each Banach disk then is continuous.
while if is a Hausdorff locally convex space then we may add to this list:

  1. is an inductive limit of Banach spaces;

Properties

Every ultrabornological space X is the inductive limit of a family of nuclear Fréchet spaces, spanning X.
Every ultrabornological space X is the inductive limit of a family of nuclear DF-spaces, spanning X.
Every ultrabornological space is a quasi-ultrabarrelled space.
Every locally convex ultrabornological space is a bornological space but there exist bornological spaces that are not ultrabornological.

Examples and sufficient conditions