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 ...

, the class of ''B''-convex spaces is a class of

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

. The concept of ''B''-convexity was defined and used to characterize Banach spaces that have the

strong law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

by Anatole Beck in 1962; accordingly, "B-convexity" is understood as an abbreviation of Beck convexity. Beck proved the following theorem: A Banach space is ''B''-convex

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

every sequence of

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

, symmetric, uniformly bounded and Radon random variables in that space satisfies the strong law of large numbers. Let ''X'' be a Banach space with

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

, , , , . ''X'' is said to be ''B''-convex if for some ''ε'' > 0 and some

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

''n'', it holds true that whenever ''x''₁, ..., ''x''_''n'' are elements of the

closed unit ball In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...

of ''X'', there is a choice of signs ''α''₁, ..., ''α''_''n'' ∈ such that :

\left\,  \sum_^ \alpha_ x_ \right\,  \leq (1 - \varepsilon) n.

Later authors have shown that B-convexity is equivalent to a number of other important properties in the theory of Banach spaces. Being B-convex and having Rademacher type

p>1

were shown to be equivalent Banach-space properties by

Gilles Pisier Gilles I. Pisier (born 18 November 1950) is a professor of mathematics at the Pierre and Marie Curie University and a distinguished professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University. He is known for his contribution ...

References

* * (See chapter 9) {{Functional analysis Banach spaces Convex geometry