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Rademacher Cotype


The Rademacher cotype-q constant C_q^r(U) of a bounded operator U:X->Y between Banach spaces, for a real number q>=2, is the least nonnegative constant such that

 (sum_(i)||Ux_i||^q)^(1/q)<=C_q^r(U)E||sum_(i)epsilon_ix_i||,

where E denotes the expectation value, the inequality holds for every finite sequence (x_i) in X, and the epsilon_i are independent and identically distributed random variables taking the values -1 and 1 with equal probability. If no such constant exists, C_q^r(U)=infty (Talagrand 2021, p. 606). The operator U has Rademacher cotype q when C_q^r(U)<infty. The Rademacher cotype of a Banach space X is the Rademacher cotype of its identity operator.


See also

Cotype-Cotype Problem, Gaussian Cotype, K-Convex Space, Operator Cotype

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References

Talagrand, M. Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems, 2nd ed. Cham, Switzerland: Springer, 2021. https://doi.org/10.1007/978-3-030-82595-9.

Cite this as:

Weisstein, Eric W. "Rademacher Cotype." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RademacherCotype.html

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