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In functional analysis, a topological vector space (TVS) is called **ultrabornological** if every bounded linear operator from into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces.
Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck [1955, p. 17] "espace du type (β)").^{[1]}

Let be a topological vector space (TVS).

A **disk** is a convex and balanced set.
A disk in a TVS is called **bornivorous**^{[2]} if it absorbs every bounded subset of

A linear map between two TVSs is called **infrabounded**^{[2]} if it maps Banach disks to bounded disks.

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

- absorbs every Banach disks in

while if locally convex then we may add to this list:

- the gauge of is an infrabounded map;
^{[2]}

while if locally convex and Hausdorff then we may add to this list:

- absorbs all compact disks;
^{[2]}that is, is "compactivorious".

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

- every infrabornivorous disk in is a neighborhood of the origin;
^{[2]}

while if is a locally convex space then we may add to this list:

- every bounded linear operator from into a complete metrizable TVS is necessarily continuous;
- every infrabornivorous disk is a neighborhood of 0;
- be the inductive limit of the spaces as D varies over all compact disks in ;
- a seminorm on that is bounded on each Banach disk is necessarily continuous;
- for every locally convex space and every linear map if is bounded on each Banach disk then is continuous;
- 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:

- is an inductive limit of Banach spaces;
^{[2]}

Every locally convex ultrabornological space is barrelled,^{[2]} quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.

- Every ultrabornological space is the inductive limit of a family of nuclear Fréchet spaces, spanning
- Every ultrabornological space is the inductive limit of a family of nuclear DF-spaces, spanning

The finite product of locally convex ultrabornological spaces is ultrabornological.^{[2]} Inductive limits of ultrabornological spaces are ultrabornological.

Every Hausdorff sequentially complete bornological space is ultrabornological.^{[2]} Thus every complete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological.^{[2]}

The strong dual space of a complete Schwartz space is ultrabornological.

Every Hausdorff bornological space that is quasi-complete is ultrabornological.^{[citation needed]}

- Counter-examples

There exist ultrabarrelled spaces that are not ultrabornological. There exist ultrabornological spaces that are not ultrabarrelled.

- Bounded linear operator – Linear transformation between topological vector spaces
- Bounded set (topological vector space) – Generalization of boundedness
- Bornological space – Space where bounded operators are continuous
- Bornology – Mathematical generalization of boundedness
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Space of linear maps
- Topological vector space – Vector space with a notion of nearness
- Vector bornology

- Some characterizations of ultrabornological spaces

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