B-convex space

In functional analysis, the class of B-convex spaces is a class of Banach space. The concept of B-convexity was defined and used to characterize Banach spaces that have the strong law of large numbers 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 every sequence of independent, 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 || ||. X is said to be B-convex if for some ε > 0 and some natural number n, it holds true that whenever x₁, ..., x_n are elements of the closed unit ball of X, there is a choice of signs α₁, ..., α_n ∈ {−1, +1} such that

\left\|\sum _{i=1}^{n}\alpha _{i}x_{i}\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.

References

Beck, Anatole (1962). "A convexity condition in Banach spaces and the strong law of large numbers". Proc. Amer. Math. Soc. 13 (2): 329–334. doi:10.1090/S0002-9939-1962-0133857-9. ISSN 0002-9939. MR 0133857.
Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015. (See chapter 9)

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Functional analysis (topics, glossary)
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Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory