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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, 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 quasi-barrelled if every strongly bounded countable union of equicontinuous 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 const ...

is again equicontinuous. This property is a generalization of quasibarrelled spaces.

Definition

A TVS ''X'' with continuous dual space

X^

is said to be countably quasi-barrelled if

B^ \subseteq X^

is a strongly bounded subset of

X^

that is equal to a countable union of equicontinuous subsets of

X^

, then

B^

is itself equicontinuous. A Hausdorff locally convex TVS is countably quasi-barrelled if and only if each bornivorous

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 stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...

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 polytop ...

balanced neighborhoods of 0 is itself a neighborhood of 0.

σ-quasi-barrelled space

A TVS with continuous dual space

X^

is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in

X^

is equicontinuous.

Sequentially quasi-barrelled space

A TVS with continuous dual space

X^

is said to be sequentially quasi-barrelled if every strongly convergent sequence in

X^

is equicontinuous.

Properties

Every countably quasi-barrelled space is a σ-quasi-barrelled space.

Examples and sufficient conditions

Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space. 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 su ...

of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled. Every σ-barrelled space is a σ-quasi-barrelled space. Every

DF-space In the mathematical 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 ...

is countably quasi-barrelled. A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space. There exist σ-barrelled spaces that are not Mackey spaces. There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces. There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled. There exist sequentially barrelled spaces that are not σ-quasi-barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.

References