Type and Cotype

Posted: 2016-03-14 , Modified: 2016-03-14

Tags: none

  1. Say that \(X\) has type \(p\) if there exists \(C>0\) such that for every \(n, y_1,\ldots, y_n\in X\), [ _{{1}^n} _XC^{p}. ]
    • This is always true for \(p=1\) by the triangle inquality.
    • The RHS decreases as \(p\) increases.
    • Let \(T_p(X)\) be the infimum of valid \(T\).
    • \(X\) has nontrivial type if it has type \(>1\).
  2. Say that \(X\) has cotype \(r\) if there exists \(C>0\) such that for every \(n, x_1,\ldots, x_n\in Y\), [ _{{1}^n} _YC^{p}. ]
    • This is always true for \(p=\iy\) by Jensen.
    • Let \(C_r(X)\) be the infimum of valid \(C\).
    • \(X\) has finite cotype if it has type \(<\iy\).