Bornological space

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 article: 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 article: 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 ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$ in a TVS ${\displaystyle X}$ is said to be Mackey convergent to ${\displaystyle 0}$ if there exists a sequence of positive real numbers ${\displaystyle r_{\bullet }=(r_{i})_{i=1}^{\infty }}$ 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 ${\displaystyle r>0}$ 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]

5. The locally convex topology induced by the von Neumann bornology on ${\displaystyle X}$ is the same as ${\displaystyle \tau ,}$${\displaystyle X}$'s given topology.
6. Every bounded seminorm on ${\displaystyle X}$ is continuous. ^[4]
7. 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 ${\displaystyle \tau .}$
8. ${\displaystyle X}$ is the inductive limit of normed spaces. ^[4]
9. ${\displaystyle X}$ 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]
10. ${\displaystyle X}$ carries the Mackey topology ${\displaystyle \tau (X,X^{\prime })}$ and all bounded linear functionals on ${\displaystyle X}$ are continuous. ^[4]
11. ${\displaystyle X}$ 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 ${\displaystyle X_{\bullet }=(X_{i})_{i\in I}}$ 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]}
  • 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 ${\displaystyle X}$ 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 ${\displaystyle \tau (X,X^{\prime }).}$
  • 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 ${\displaystyle X}$ 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 ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$ is a null sequence in ${\displaystyle X}$ then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}$ is a null sequence in ${\displaystyle Y.}$
  5. If ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$ is a Mackey convergent null sequence in ${\displaystyle X}$ then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}$ 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 article: 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.

See also

• 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  – 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

References

1. Narici & Beckenstein 2011, p. 168.
2. Narici & Beckenstein 2011, pp. 156–175.
3. Wilansky 2013, p. 50.
4. Narici & Beckenstein 2011, pp. 441–457.
5. Swartz 1992, pp. 15–16.
6. Narici & Beckenstein 2011, pp. 453–454.
7. Adasch, Ernst & Keim 1978, pp. 60–61.
8. Wilansky 2013, p. 48.
9. Narici & Beckenstein 2011, p. 450.
10. Adasch, Ernst & Keim 1978, pp. 60–65.
11. Narici & Beckenstein 2011, p. 453.
12. Schaefer & Wolff 1999, p. 144.
13. Khaleelulla 1982, pp. 28–63.
14. Schaefer & Wolff 1999, pp. 103–110.
15. Adasch, Ernst & Keim 1978, pp. 70–73.

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