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.

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]

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]

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

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]

Properties

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