Cotype

Noun

1. A syntype or paratype.
2. A Banach space ( X , ‖<!-- ‖ --> ⋅<!-- ⋅ --> ‖<!-- ‖ --> ) {\displaystyle (X,\|\cdot \|)} , such that, given a sequence of independent random variables from a Rademacher distribution, ( ε<!-- ε --> i ) {\displaystyle (\varepsilon _{i})} , and a rank q {\displaystyle q} in the range from 2 to infinity, there exists a finite constant C ≥<!-- ≥ --> 1 {\displaystyle C\geq 1} such that for all finite sequences ( x i ) ∈<!-- ∈ --> X {\displaystyle (x_{i})\in X} , E ε<!-- ε --> [ ‖ ∑<!-- ∑ --> i = 1 n ε<!-- ε --> i x i ‖ q ] ≥<!-- ≥ --> 1 C q ( ∑<!-- ∑ --> i = 1 n ‖<!-- ‖ --> x i ‖<!-- ‖ --> q ) , if 2 ≤<!-- ≤ --> q < ∞<!-- ∞ --> {\displaystyle \mathbb {E} _{\varepsilon }\left[\left\|\sum \limits _{i=1}^{n}\varepsilon _{i}x_{i}\right\|^{q}\right]\geq {\frac {1}{C^{q}}}\left(\sum \limits _{i=1}^{n}\|x_{i}\|^{q}\right),\quad {\text{if}}\;2\leq q<\infty } or E ε<!-- ε --> [ ‖ ∑<!-- ∑ --> i = 1 n ε<!-- ε --> i x i ‖ ] ≥<!-- ≥ --> 1 C sup ‖<!-- ‖ --> x i ‖<!-- ‖ --> , if q = ∞<!-- ∞ --> {\displaystyle \mathbb {E} _{\varepsilon }\left[\left\|\sum \limits _{i=1}^{n}\varepsilon _{i}x_{i}\right\|\right]\geq {\frac {1}{C}}\sup \|x_{i}\|,\quad {\text{if}}\;q=\infty } .