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 ...

and related areas of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an ultrabarrelled space is a

topological vector spaces 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) for which every ultrabarrel is a

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...

of the origin.

Definition

A subset

B_0

of a TVS

X

is called an ultrabarrel if it is a closed and

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 ...

subset of

X

and if there exists a sequence

\left(B_i\right)_^

of closed balanced and absorbing subsets of

X

such that

B_ + B_ \subseteq B_i

for all

i = 0, 1, \ldots.

In this case,

\left(B_i\right)_^

is called a defining sequence for

B_0.

A TVS

X

is called ultrabarrelled if every ultrabarrel in

X

is a

of the origin.

Properties

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 ...

ultrabarrelled space is a

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 ...

. Every ultrabarrelled space is a

quasi-ultrabarrelled space In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin. Definition A subset ''B''0 of a TVS ''X'' is ...

Examples and sufficient conditions

Complete and metrizable TVSs are ultrabarrelled. If

X

is a complete

locally bounded In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. Locally bounded f ...

non-locally convex TVS and if

B_0

is a closed

and bounded neighborhood of the origin, then

B_0

is an ultrabarrel that is not

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 ...

and has a defining sequence consisting of non-convex sets.

Counter-examples

There exist

s that are not ultrabarrelled. There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.

Citations

Bibliography