functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, a

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) is said to be countably barrelled if every weakly bounded countable union of

equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable fa ...

subsets of its

continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...

is again equicontinuous. This property is a generalization of

barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a b ...

Definition

A TVS ''X'' with continuous dual space

X^

is said to be countably barrelled if

B^ \subseteq X^

is a weak-* bounded subset of

X^

that is equal to a countable union of

subsets of

X^

, then

B^

is itself equicontinuous. A Hausdorff

locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...

TVS is countably barrelled if and only if each

barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...

in ''X'' that is equal to the countable intersection of closed

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...

neighborhoods of 0 is itself a neighborhood of 0.

σ-barrelled space

A TVS with continuous dual space

X^

is said to be σ-barrelled if every weak-* bounded (countable) sequence in

X^

is equicontinuous.

Sequentially barrelled space

A TVS with continuous dual space

X^

is said to be sequentially barrelled if every weak-* convergent sequence in

X^

is equicontinuous.

Properties

Every countably barrelled space is a

countably quasibarrelled space In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generaliza ...

, a

σ-barrelled space In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of ...

, a

σ-quasi-barrelled space In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generaliza ...

, and a

sequentially barrelled space In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of ...

. An

H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together wit ...

is a TVS whose

strong dual space In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

is countably barrelled. Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled. Every σ-barrelled space is a

. A locally convex

quasi-barrelled space In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied becaus ...

that is also a 𝜎-barrelled space is a

Examples and sufficient conditions

Every

is countably barrelled. However, there exist

semi-reflexive In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual ...

countably barrelled spaces that are not barrelled. The

strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

of a

distinguished space In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are c ...

and of a metrizable locally convex space is countably barrelled.

Counter-examples

There exist σ-barrelled spaces that are not countably barrelled. There exist normed

DF-space In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...

s that are not countably barrelled. There exists a

that is not a 𝜎-barrelled space. There exist σ-barrelled spaces that are not

Mackey space In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still pres ...

s. There exist σ-barrelled spaces that are not

countably quasi-barrelled space In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generaliza ...

s and thus not countably barrelled. There exist sequentially barrelled spaces that are not σ-quasi-barrelled. There exist

quasi-complete In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non- metrizable TVSs. Properties * Eve ...

locally convex TVSs that are not sequentially barrelled.

References

* * * * * {{Topological vector spaces Functional analysis