Bornological space

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In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

Bornological spaces were first studied by George Mackey.^{[citation needed ]} The name was coined by Bourbaki ^{[citation needed ]} after borné , the French word for "bounded".

A bornology on a set ${\displaystyle X}${\displaystyle X} is a collection ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} of subsets of ${\displaystyle X}${\displaystyle X} that satisfy all the following conditions:

1. ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} covers ${\displaystyle X;}${\displaystyle X;} that is, ${\displaystyle X=\cup {\mathcal {B}}}${\displaystyle X=\cup {\mathcal {B}}};
2. ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is stable under inclusions; that is, if ${\displaystyle B\in {\mathcal {B}}}${\displaystyle B\in {\mathcal {B}}} and ${\displaystyle A\subseteq B,}${\displaystyle A\subseteq B,} then ${\displaystyle A\in {\mathcal {B}}}${\displaystyle A\in {\mathcal {B}}};
3. ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is stable under finite unions; that is, if ${\displaystyle B_{1},\ldots ,B_{n}\in {\mathcal {B}}}${\displaystyle B_{1},\ldots ,B_{n}\in {\mathcal {B}}} then ${\displaystyle B_{1}\cup \cdots \cup B_{n}\in {\mathcal {B}}}${\displaystyle B_{1}\cup \cdots \cup B_{n}\in {\mathcal {B}}};

Elements of the collection ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} are called ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}-bounded or simply bounded sets if ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is understood.^[1] The pair ${\displaystyle (X,{\mathcal {B}})}${\displaystyle (X,{\mathcal {B}})} is called a bounded structure or a bornological set.^[1]

A base or fundamental system of a bornology ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is a subset ${\displaystyle {\mathcal {B}}_{0}}${\displaystyle {\mathcal {B}}_{0}} of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} such that each element of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is a subset of some element of ${\displaystyle {\mathcal {B}}_{0}.}${\displaystyle {\mathcal {B}}_{0}.} Given a collection ${\displaystyle {\mathcal {S}}}${\displaystyle {\mathcal {S}}} of subsets of ${\displaystyle X,}${\displaystyle X,} the smallest bornology containing ${\displaystyle {\mathcal {S}}}${\displaystyle {\mathcal {S}}} is called the bornology generated by ${\displaystyle {\mathcal {S}}.}${\displaystyle {\mathcal {S}}.}^[2]

If ${\displaystyle (X,{\mathcal {B}})}${\displaystyle (X,{\mathcal {B}})} and ${\displaystyle (Y,{\mathcal {C}})}${\displaystyle (Y,{\mathcal {C}})} are bornological sets then their product bornology on ${\displaystyle X\times Y}${\displaystyle X\times Y} is the bornology having as a base the collection of all sets of the form ${\displaystyle B\times C,}${\displaystyle B\times C,} where ${\displaystyle B\in {\mathcal {B}}}${\displaystyle B\in {\mathcal {B}}} and ${\displaystyle C\in {\mathcal {C}}.}${\displaystyle C\in {\mathcal {C}}.}^[2] A subset of ${\displaystyle X\times Y}${\displaystyle X\times Y} is bounded in the product bornology if and only if its image under the canonical projections onto ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} are both bounded.

If ${\displaystyle (X,{\mathcal {B}})}${\displaystyle (X,{\mathcal {B}})} and ${\displaystyle (Y,{\mathcal {C}})}${\displaystyle (Y,{\mathcal {C}})} are bornological sets then a function ${\displaystyle f:X\to Y}${\displaystyle f:X\to Y} is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}-bounded subsets of ${\displaystyle X}${\displaystyle X} to ${\displaystyle {\mathcal {C}}}${\displaystyle {\mathcal {C}}}-bounded subsets of ${\displaystyle Y;}${\displaystyle Y;} that is, if ${\displaystyle f({\mathcal {B}})\subseteq {\mathcal {C}}.}${\displaystyle f({\mathcal {B}})\subseteq {\mathcal {C}}.}^[2] If in addition ${\displaystyle f}${\displaystyle f} is a bijection and ${\displaystyle f^{-1}}${\displaystyle f^{-1}} is also bounded then ${\displaystyle f}${\displaystyle f} is called a bornological isomorphism.

Let ${\displaystyle X}${\displaystyle X} be a vector space over a field ${\displaystyle \mathbb {K} }${\displaystyle \mathbb {K} } where ${\displaystyle \mathbb {K} }${\displaystyle \mathbb {K} } has a bornology ${\displaystyle {\mathcal {B}}_{\mathbb {K} }.}${\displaystyle {\mathcal {B}}_{\mathbb {K} }.} A bornology ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} on ${\displaystyle X}${\displaystyle X} is called a vector bornology on ${\displaystyle X}${\displaystyle X} if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).

If ${\displaystyle X}${\displaystyle X} is a topological vector space (TVS) and ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is a bornology on ${\displaystyle X,}${\displaystyle X,} then the following are equivalent:

1. ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is a vector bornology;
2. Finite sums and balanced hulls of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}-bounded sets are ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}-bounded;^[2]
3. The scalar multiplication map ${\displaystyle \mathbb {K} \times X\to X}${\displaystyle \mathbb {K} \times X\to X} defined by ${\displaystyle (s,x)\mapsto sx}${\displaystyle (s,x)\mapsto sx} and the addition map ${\displaystyle X\times X\to X}${\displaystyle X\times X\to X} defined by ${\displaystyle (x,y)\mapsto x+y,}${\displaystyle (x,y)\mapsto x+y,} are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).^[2]

A vector bornology ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then ${\displaystyle {\mathcal {B}}.}${\displaystyle {\mathcal {B}}.} And a vector bornology ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is called separated if the only bounded vector subspace of ${\displaystyle X}${\displaystyle X} is the 0-dimensional trivial space ${\displaystyle \{0\}.}${\displaystyle \{0\}.}

Usually, ${\displaystyle \mathbb {K} }${\displaystyle \mathbb {K} } is either the real or complex numbers, in which case a vector bornology ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} on ${\displaystyle X}${\displaystyle X} will be called a convex vector bornology if ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} has a base consisting of convex sets.

A subset ${\displaystyle A}${\displaystyle A} of ${\displaystyle X}${\displaystyle X} is called bornivorous and a bornivore if it absorbs every bounded set.

In a vector bornology, ${\displaystyle A}${\displaystyle A} is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology ${\displaystyle A}${\displaystyle A} is bornivorous if it absorbs every bounded disk.

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.^[3]

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.^[4]

A sequence ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }} in a TVS ${\displaystyle X}${\displaystyle X} is said to be Mackey convergent to ${\displaystyle 0}${\displaystyle 0} if there exists a sequence of positive real numbers ${\displaystyle r_{\bullet }=(r_{i})_{i=1}^{\infty }}${\displaystyle r_{\bullet }=(r_{i})_{i=1}^{\infty }} diverging to ${\displaystyle \infty }${\displaystyle \infty } such that ${\displaystyle (r_{i}x_{i})_{i=1}^{\infty }}${\displaystyle (r_{i}x_{i})_{i=1}^{\infty }} converges to ${\displaystyle 0}${\displaystyle 0} in ${\displaystyle X.}${\displaystyle X.}^[5]

Every topological vector space ${\displaystyle X,}${\displaystyle X,} at least on a non discrete valued field gives a bornology on ${\displaystyle X}${\displaystyle X} by defining a subset ${\displaystyle B\subseteq X}${\displaystyle B\subseteq X} to be bounded (or von-Neumann bounded), if and only if for all open sets ${\displaystyle U\subseteq X}${\displaystyle U\subseteq X} containing zero there exists a ${\displaystyle r>0}${\displaystyle r>0} with ${\displaystyle B\subseteq rU.}${\displaystyle B\subseteq rU.} If ${\displaystyle X}${\displaystyle X} is a locally convex topological vector space then ${\displaystyle B\subseteq X}${\displaystyle B\subseteq X} is bounded if and only if all continuous semi-norms on ${\displaystyle X}${\displaystyle X} are bounded on ${\displaystyle B.}${\displaystyle B.}

The set of all bounded subsets of a topological vector space ${\displaystyle X}${\displaystyle X} is called the bornology or the von Neumann bornology of ${\displaystyle X.}${\displaystyle X.}

If ${\displaystyle X}${\displaystyle X} is a locally convex topological vector space, then an absorbing disk ${\displaystyle D}${\displaystyle D} in ${\displaystyle X}${\displaystyle X} is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).^[4]

If ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is a convex vector bornology on a vector space ${\displaystyle X,}${\displaystyle X,} then the collection ${\displaystyle {\mathcal {N}}_{\mathcal {B}}(0)}${\displaystyle {\mathcal {N}}_{\mathcal {B}}(0)} of all convex balanced subsets of ${\displaystyle X}${\displaystyle X} that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on ${\displaystyle X}${\displaystyle X} called the topology induced by ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}.^[4]

If ${\displaystyle (X,\tau )}${\displaystyle (X,\tau )} is a TVS then the bornological space associated with ${\displaystyle X}${\displaystyle X} is the vector space ${\displaystyle X}${\displaystyle X} endowed with the locally convex topology induced by the von Neumann bornology of ${\displaystyle (X,\tau ).}${\displaystyle (X,\tau ).}^[4]

Quasi-bornological spaces where introduced by S. Iyahen in 1968.^[6]

A topological vector space (TVS) ${\displaystyle (X,\tau )}${\displaystyle (X,\tau )} with a continuous dual ${\displaystyle X^{\prime }}${\displaystyle X^{\prime }} is called a quasi-bornological space^[6] if any of the following equivalent conditions holds:

1. Every bounded linear operator from ${\displaystyle X}${\displaystyle X} into another TVS is continuous.^[6]
2. Every bounded linear operator from ${\displaystyle X}${\displaystyle X} into a complete metrizable TVS is continuous.^{[6] [7]}
3. Every knot in a bornivorous string is a neighborhood of the origin.^[6]

Every pseudometrizable TVS is quasi-bornological. ^[6] A TVS ${\displaystyle (X,\tau )}${\displaystyle (X,\tau )} in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.^[8] If ${\displaystyle X}${\displaystyle X} is a quasi-bornological TVS then the finest locally convex topology on ${\displaystyle X}${\displaystyle X} that is coarser than ${\displaystyle \tau }${\displaystyle \tau } makes ${\displaystyle X}${\displaystyle X} into a locally convex bornological space.

In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.

Every locally convex quasi-bornological space is bornological but there exist bornological spaces that are not quasi-bornological.^[6]

A topological vector space (TVS) ${\displaystyle (X,\tau )}${\displaystyle (X,\tau )} with a continuous dual ${\displaystyle X^{\prime }}${\displaystyle X^{\prime }} is called a bornological space if it is locally convex and any of the following equivalent conditions holds:

1. Every convex, balanced, and bornivorous set in ${\displaystyle X}${\displaystyle X} is a neighborhood of zero.^[4]
2. Every bounded linear operator from ${\displaystyle X}${\displaystyle X} into a locally convex TVS is continuous.^[4]
   • Recall that a linear map is bounded if and only if it maps any sequence converging to ${\displaystyle 0}${\displaystyle 0} in the domain to a bounded subset of the codomain.^[4] In particular, any linear map that is sequentially continuous at the origin is bounded.
3. Every bounded linear operator from ${\displaystyle X}${\displaystyle X} into a seminormed space is continuous.^[4]
4. Every bounded linear operator from ${\displaystyle X}${\displaystyle X} into a Banach space is continuous.^[4]

If ${\displaystyle X}${\displaystyle X} is a Hausdorff locally convex space then we may add to this list:^[7]

5. The locally convex topology induced by the von Neumann bornology on ${\displaystyle X}${\displaystyle X} is the same as ${\displaystyle \tau ,}${\displaystyle \tau ,} ${\displaystyle X}${\displaystyle X}'s given topology.
6. Every bounded seminorm on ${\displaystyle X}${\displaystyle X} is continuous.^[4]
7. Any other Hausdorff locally convex topological vector space topology on ${\displaystyle X}${\displaystyle X} that has the same (von Neumann) bornology as ${\displaystyle (X,\tau )}${\displaystyle (X,\tau )} is necessarily coarser than ${\displaystyle \tau .}${\displaystyle \tau .}
8. ${\displaystyle X}${\displaystyle X} is the inductive limit of normed spaces.^[4]
9. ${\displaystyle X}${\displaystyle X} is the inductive limit of the normed spaces ${\displaystyle X_{D}}${\displaystyle X_{D}} as ${\displaystyle D}${\displaystyle D} varies over the closed and bounded disks of ${\displaystyle X}${\displaystyle X} (or as ${\displaystyle D}${\displaystyle D} varies over the bounded disks of ${\displaystyle X}${\displaystyle X}).^[4]
10. ${\displaystyle X}${\displaystyle X} carries the Mackey topology ${\displaystyle \tau (X,X^{\prime })}${\displaystyle \tau (X,X^{\prime })} and all bounded linear functionals on ${\displaystyle X}${\displaystyle X} are continuous.^[4]
11. ${\displaystyle X}${\displaystyle X} has both of the following properties:
    • ${\displaystyle X}${\displaystyle X} is convex-sequential or C-sequential, which means that every convex sequentially open subset of ${\displaystyle X}${\displaystyle X} is open,
    • ${\displaystyle X}${\displaystyle X} is sequentially bornological or S-bornological, which means that every convex and bornivorous subset of ${\displaystyle X}${\displaystyle X} is sequentially open.
    where a subset ${\displaystyle A}${\displaystyle A} of ${\displaystyle X}${\displaystyle X} is called sequentially open if every sequence converging to ${\displaystyle 0}${\displaystyle 0} eventually belongs to ${\displaystyle A.}${\displaystyle A.}

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,^[4] where recall that a linear operator is sequentially continuous if and only if it is sequentially continuous at the origin. Thus for linear maps from a bornological space into a locally convex space, continuity is equivalent to sequential continuity at the origin. More generally, we even have the following:

• Any linear map ${\displaystyle F:X\to Y}${\displaystyle F:X\to Y} from a locally convex bornological space into a locally convex space ${\displaystyle Y}${\displaystyle Y} that maps null sequences in ${\displaystyle X}${\displaystyle X} to bounded subsets of ${\displaystyle Y}${\displaystyle Y} is necessarily continuous.

As a consequent of the Mackey–Ulam theorem, "for all practical purposes, the product of bornological spaces is bornological."^[9]

The following topological vector spaces are all bornological:

• Any locally convex pseudometrizable TVS is bornological.^{[4] [10]}
  • Thus every normed space and Fréchet space is bornological.
• Any strict inductive limit of bornological spaces, in particular any strict LF-space, is bornological.
  • This shows that there are bornological spaces that are not metrizable.
• A countable product of locally convex bornological spaces is bornological.^{[11] [10]}
• Quotients of Hausdorff locally convex bornological spaces are bornological.^[10]
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.^[10]
• Fréchet Montel spaces have bornological strong duals.
• The strong dual of every reflexive Fréchet space is bornological.^[12]
• If the strong dual of a metrizable locally convex space is separable, then it is bornological.^[12]
• A vector subspace of a Hausdorff locally convex bornological space ${\displaystyle X}${\displaystyle X} that has finite codimension in ${\displaystyle X}${\displaystyle X} is bornological.^{[4] [10]}
• The finest locally convex topology on a vector space is bornological.^[4]

Counterexamples

There exists a bornological LB-space whose strong bidual is not bornological.^[13]

A closed vector subspace of a locally convex bornological space is not necessarily bornological.^{[4] [14]} There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.^[4]

Bornological spaces need not be barrelled and barrelled spaces need not be bornological.^[4] Because every locally convex ultrabornological space is barrelled,^[4] it follows that a bornological space is not necessarily ultrabornological.

• The strong dual space of a locally convex bornological space is complete.^[4]
• Every locally convex bornological space is infrabarrelled.^[4]
• Every Hausdorff sequentially complete bornological TVS is ultrabornological.^[4]
  • Thus every complete Hausdorff bornological space is ultrabornological.
  • In particular, every Fréchet space is ultrabornological.^[4]
• The finite product of locally convex ultrabornological spaces is ultrabornological.^[4]
• Every Hausdorff bornological space is quasi-barrelled.^[15]
• Given a bornological space ${\displaystyle X}${\displaystyle X} with continuous dual ${\displaystyle X^{\prime },}${\displaystyle X^{\prime },} the topology of ${\displaystyle X}${\displaystyle X} coincides with the Mackey topology ${\displaystyle \tau (X,X^{\prime }).}${\displaystyle \tau (X,X^{\prime }).}
  • In particular, bornological spaces are Mackey spaces.
• Every quasi-complete (i.e. all closed and bounded subsets are complete) bornological space is barrelled. There exist, however, bornological spaces that are not barrelled.
• Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete).
• Let ${\displaystyle X}${\displaystyle X} be a metrizable locally convex space with continuous dual ${\displaystyle X^{\prime }.}${\displaystyle X^{\prime }.} Then the following are equivalent:
  1. ${\displaystyle \beta (X^{\prime },X)}${\displaystyle \beta (X^{\prime },X)} is bornological.
  2. ${\displaystyle \beta (X^{\prime },X)}${\displaystyle \beta (X^{\prime },X)} is quasi-barrelled.
  3. ${\displaystyle \beta (X^{\prime },X)}${\displaystyle \beta (X^{\prime },X)} is barrelled.
  4. ${\displaystyle X}${\displaystyle X} is a distinguished space.
• If ${\displaystyle L:X\to Y}${\displaystyle L:X\to Y} is a linear map between locally convex spaces and if ${\displaystyle X}${\displaystyle X} is bornological, then the following are equivalent:
  1. ${\displaystyle L:X\to Y}${\displaystyle L:X\to Y} is continuous.
  2. ${\displaystyle L:X\to Y}${\displaystyle L:X\to Y} is sequentially continuous.^[4]
  3. For every set ${\displaystyle B\subseteq X}${\displaystyle B\subseteq X} that's bounded in ${\displaystyle X,}${\displaystyle X,} ${\displaystyle L(B)}${\displaystyle L(B)} is bounded.
  4. If ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }} is a null sequence in ${\displaystyle X}${\displaystyle X} then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }} is a null sequence in ${\displaystyle Y.}${\displaystyle Y.}
  5. If ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }} is a Mackey convergent null sequence in ${\displaystyle X}${\displaystyle X} then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }} is a bounded subset of ${\displaystyle Y.}${\displaystyle Y.}
• Suppose that ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} are locally convex TVSs and that the space of continuous linear maps ${\displaystyle L_{b}(X;Y)}${\displaystyle L_{b}(X;Y)} is endowed with the topology of uniform convergence on bounded subsets of ${\displaystyle X.}${\displaystyle X.} If ${\displaystyle X}${\displaystyle X} is a bornological space and if ${\displaystyle Y}${\displaystyle Y} is complete then ${\displaystyle L_{b}(X;Y)}${\displaystyle L_{b}(X;Y)} is a complete TVS.^[4]
  • In particular, the strong dual of a locally convex bornological space is complete.^[4] However, it need not be bornological.

Subsets

• In a locally convex bornological space, every convex bornivorous set ${\displaystyle B}${\displaystyle B} is a neighborhood of ${\displaystyle 0}${\displaystyle 0} (${\displaystyle B}${\displaystyle B} is not required to be a disk).^[4]
• Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.^[4]
• Closed vector subspaces of bornological space need not be bornological.^[4]

A disk in a topological vector space ${\displaystyle X}${\displaystyle X} is called infrabornivorous if it absorbs all Banach disks.

If ${\displaystyle X}${\displaystyle X} is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks.

A locally convex space is called ultrabornological if any of the following equivalent conditions hold:

1. Every infrabornivorous disk is a neighborhood of the origin.
2. ${\displaystyle X}${\displaystyle X} is the inductive limit of the spaces ${\displaystyle X_{D}}${\displaystyle X_{D}} as ${\displaystyle D}${\displaystyle D} varies over all compact disks in ${\displaystyle X.}${\displaystyle X.}
3. A seminorm on ${\displaystyle X}${\displaystyle X} that is bounded on each Banach disk is necessarily continuous.
4. For every locally convex space ${\displaystyle Y}${\displaystyle Y} and every linear map ${\displaystyle u:X\to Y,}${\displaystyle u:X\to Y,} if ${\displaystyle u}${\displaystyle u} is bounded on each Banach disk then ${\displaystyle u}${\displaystyle u} is continuous.
5. For every Banach space ${\displaystyle Y}${\displaystyle Y} and every linear map ${\displaystyle u:X\to Y,}${\displaystyle u:X\to Y,} if ${\displaystyle u}${\displaystyle u} is bounded on each Banach disk then ${\displaystyle u}${\displaystyle u} is continuous.

The finite product of ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological.

• Bornology – Mathematical generalization of boundedness
• Bornivorous set – A set that can absorb any bounded subset
• Bounded set (topological vector space) – Generalization of boundedness
• Locally convex topological vector space – 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

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