Bornological space

Short description: Space where bounded operators are continuous

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".

Bornologies and bounded maps

Main page: Bornology

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

1. $\displaystyle{ \mathcal{B} }$ covers $\displaystyle{ X; }$ that is, $\displaystyle{ X = \cup \mathcal{B} }$;
2. $\displaystyle{ \mathcal{B} }$ is stable under inclusions; that is, if $\displaystyle{ B \in \mathcal{B} }$ and $\displaystyle{ A \subseteq B, }$ then $\displaystyle{ A \in \mathcal{B} }$;
3. $\displaystyle{ \mathcal{B} }$ is stable under finite unions; that is, if $\displaystyle{ B_1, \ldots, B_n \in \mathcal{B} }$ then $\displaystyle{ B_1 \cup \cdots \cup B_n \in \mathcal{B} }$;

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

A base or fundamental system of a bornology $\displaystyle{ \mathcal{B} }$ is a subset $\displaystyle{ \mathcal{B}_0 }$ of $\displaystyle{ \mathcal{B} }$ such that each element of $\displaystyle{ \mathcal{B} }$ is a subset of some element of $\displaystyle{ \mathcal{B}_0. }$ Given a collection $\displaystyle{ \mathcal{S} }$ of subsets of $\displaystyle{ X, }$ the smallest bornology containing $\displaystyle{ \mathcal{S} }$ is called the bornology generated by $\displaystyle{ \mathcal{S}. }$[2]

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

Bounded maps

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

Vector bornologies

Main page: Vector bornology

Let $\displaystyle{ X }$ be a vector space over a field $\displaystyle{ \mathbb{K} }$ where $\displaystyle{ \mathbb{K} }$ has a bornology $\displaystyle{ \mathcal{B}_{\mathbb{K}}. }$ A bornology $\displaystyle{ \mathcal{B} }$ on $\displaystyle{ X }$ is called a vector bornology on $\displaystyle{ X }$ 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 $\displaystyle{ X }$ is a topological vector space (TVS) and $\displaystyle{ \mathcal{B} }$ is a bornology on $\displaystyle{ X, }$ then the following are equivalent:

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

A vector bornology $\displaystyle{ \mathcal{B} }$ 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 $\displaystyle{ \mathcal{B}. }$ And a vector bornology $\displaystyle{ \mathcal{B} }$ is called separated if the only bounded vector subspace of $\displaystyle{ X }$ is the 0-dimensional trivial space $\displaystyle{ \{ 0 \}. }$

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

Bornivorous subsets

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

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

Mackey convergence

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

Bornology of a topological vector space

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

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

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

Induced topology

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

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

Theorem[4] — Let $\displaystyle{ X }$ and $\displaystyle{ Y }$ be locally convex TVS and let $\displaystyle{ X_b }$ denote $\displaystyle{ X }$ endowed with the topology induced by von Neumann bornology of $\displaystyle{ X. }$ Define $\displaystyle{ Y_b }$ similarly. Then a linear map $\displaystyle{ L : X \to Y }$ is a bounded linear operator if and only if $\displaystyle{ L : X_b \to Y }$ is continuous.

Moreover, if $\displaystyle{ X }$ is bornological, $\displaystyle{ Y }$ is Hausdorff, and $\displaystyle{ L : X \to Y }$ is continuous linear map then so is $\displaystyle{ L : X \to Y_b. }$ If in addition $\displaystyle{ X }$ is also ultrabornological, then the continuity of $\displaystyle{ L : X \to Y }$ implies the continuity of $\displaystyle{ L : X \to Y_{ub}, }$ where $\displaystyle{ Y_{ub} }$ is the ultrabornological space associated with $\displaystyle{ Y. }$

Quasi-bornological spaces

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

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

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

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) $\displaystyle{ (X, \tau) }$ with a continuous dual $\displaystyle{ X^{\prime} }$ 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 $\displaystyle{ X }$ is a neighborhood of zero.[4]
2. Every bounded linear operator from $\displaystyle{ X }$ 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 $\displaystyle{ 0 }$ 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 $\displaystyle{ X }$ into a seminormed space is continuous.[4]
4. Every bounded linear operator from $\displaystyle{ X }$ into a Banach space is continuous.[4]

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

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

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

Sufficient conditions

Mackey–Ulam theorem[9] — The product of a collection $\displaystyle{ X_\bull = ( X_i)_{i \in I} }$ locally convex bornological spaces is bornological if and only if $\displaystyle{ I }$ does not admit an Ulam measure.

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]
• 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 $\displaystyle{ X }$ that has finite codimension in $\displaystyle{ X }$ 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.

Properties

• 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 $\displaystyle{ X }$ with continuous dual $\displaystyle{ X^{\prime}, }$ the topology of $\displaystyle{ X }$ coincides with the Mackey topology $\displaystyle{ \tau(X, X^{\prime}). }$
• 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 $\displaystyle{ X }$ be a metrizable locally convex space with continuous dual $\displaystyle{ X^{\prime}. }$ Then the following are equivalent:
1. $\displaystyle{ \beta(X^{\prime}, X) }$ is bornological.
2. $\displaystyle{ \beta(X^{\prime}, X) }$ is quasi-barrelled.
3. $\displaystyle{ \beta(X^{\prime}, X) }$ is barrelled.
4. $\displaystyle{ X }$ is a distinguished space.
• If $\displaystyle{ L : X \to Y }$ is a linear map between locally convex spaces and if $\displaystyle{ X }$ is bornological, then the following are equivalent:
1. $\displaystyle{ L : X \to Y }$ is continuous.
2. $\displaystyle{ L : X \to Y }$ is sequentially continuous.[4]
3. For every set $\displaystyle{ B \subseteq X }$ that's bounded in $\displaystyle{ X, }$ $\displaystyle{ L(B) }$ is bounded.
4. If $\displaystyle{ x_{\bull} = (x_i)_{i=1}^\infty }$ is a null sequence in $\displaystyle{ X }$ then $\displaystyle{ L \circ x_\bull = (L(x_i))_{i=1}^\infty }$ is a null sequence in $\displaystyle{ Y. }$
5. If $\displaystyle{ x_\bull = (x_i)_{i=1}^\infty }$ is a Mackey convergent null sequence in $\displaystyle{ X }$ then $\displaystyle{ L \circ x_\bull = (L(x_i))_{i=1}^\infty }$ is a bounded subset of $\displaystyle{ Y. }$
• Suppose that $\displaystyle{ X }$ and $\displaystyle{ Y }$ are locally convex TVSs and that the space of continuous linear maps $\displaystyle{ L_b(X; Y) }$ is endowed with the topology of uniform convergence on bounded subsets of $\displaystyle{ X. }$ If $\displaystyle{ X }$ is a bornological space and if $\displaystyle{ Y }$ is complete then $\displaystyle{ L_b(X; Y) }$ 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 $\displaystyle{ B }$ is a neighborhood of $\displaystyle{ 0 }$ ($\displaystyle{ B }$ 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]

Ultrabornological spaces

Main page: Ultrabornological space

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

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

Properties

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