Type, cotype and \(K\)-convexity.

*(English)*Zbl 1074.46006
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1299-1332 (2003).

This survey article is devoted to the theory of type and cotype in Banach spaces. This theory was developed in the seventies and early eighties and has found numerous applications later on. The main contributions are due to B. Maurey and G. Pisier.

The paper starts with a lively description of the pre-history of type and cotype. It goes on by explaining the main results and applications of the theory to the factorization of linear operators and to the local theory of Banach spaces. The remaining parts are devoted to concise proofs of Krivine’s theorem, the Maurey-Pisier (+Krivine) theorem and Pisier’s theorem on the equivalence of \(K\)- and \(B\)-convexity. These results are cornerstones of the structure theory of Banach spaces.

Let us recall that a Banach space \(X\) has (Rademacher) type \(p\) for some \(p\in [1,2]\) if there is a constant \(C\) such that \[ {\text{Average}}_{\varepsilon_i=\pm 1} \biggl\| \sum_{i=1}^n \varepsilon_i x_i \biggr\| \leq C \biggl( \sum_{i=1}^n \| x_i\| ^p \biggr)^{1/p} \] holds for every choice of \(n\) and \(x_1,\ldots,x_n\in X\). Similarly, \(X\) has cotype \(q\) for some \(q\in [2,\infty)\) if there is a constant \(C\) such that \[ \biggl( \sum_{i=1}^n \| x_i\| ^q \biggr)^{1/q} \leq C\cdot {\text{Average}}_{\varepsilon_i=\pm 1} \Big\| \sum_{i=1}^n \varepsilon_i x_i \Big\| . \] A Banach space is \(B\)-convex if it does not contain the spaces \(l_1^n\) uniformly. It is \(K\)-convex if the Rademacher projection in \(X\) is bounded.

The theorem of J. L. Krivine [Ann. Math. (2) 104, 1–29 (1976; Zbl 0329.46008)] investigates the numbers \(p\in [1,\infty]\) for which a given infinite dimensional Banach space \(X\) contains \((1+\varepsilon)\)-isomorphic copies of \(l_p^n\) for all \(n\) and \(\varepsilon>0\). Dvoretzky’s theorem says that \(p=2\) is always a possible value. The Maurey-Pisier theorem [B. Maurey and G. Pisier, Stud. Math. 58, 45–90 (1976; Zbl 0344.47014)] gives further quantitative information on those \(p\) in relation to type and cotype. It says that \[ \min \{ p\in [1,2] : X \;\text{contains}\;l_p^n \;\text{uniformly} \} = \sup \{ p\in [1,2] : X \;\text{has type} \;p\}, \] \[ \max \{ q\in [2,\infty] : X \;\text{contains}\;l_q^n \;\text{uniformly} \} = \inf \{ q\in [2,\infty] : X \;\text{has cotype}\;q\}. \] G. Pisier’s theorem [Ann. Math. (2) 115, 375–392 (1982; Zbl 0487.46008)] says that \(X\) is \(K\)-convex if and only if \(X\) does not contain \(l_1^n\)-s uniformly. It illuminates the duality between type and cotype. If \(X\) is \(K\)-convex and \(1/p+1/q=1\), then \(X^*\) has cotype \(q\) if and only if \(X\) has type \(p\).

For the entire collection see [Zbl 1013.46001].

The paper starts with a lively description of the pre-history of type and cotype. It goes on by explaining the main results and applications of the theory to the factorization of linear operators and to the local theory of Banach spaces. The remaining parts are devoted to concise proofs of Krivine’s theorem, the Maurey-Pisier (+Krivine) theorem and Pisier’s theorem on the equivalence of \(K\)- and \(B\)-convexity. These results are cornerstones of the structure theory of Banach spaces.

Let us recall that a Banach space \(X\) has (Rademacher) type \(p\) for some \(p\in [1,2]\) if there is a constant \(C\) such that \[ {\text{Average}}_{\varepsilon_i=\pm 1} \biggl\| \sum_{i=1}^n \varepsilon_i x_i \biggr\| \leq C \biggl( \sum_{i=1}^n \| x_i\| ^p \biggr)^{1/p} \] holds for every choice of \(n\) and \(x_1,\ldots,x_n\in X\). Similarly, \(X\) has cotype \(q\) for some \(q\in [2,\infty)\) if there is a constant \(C\) such that \[ \biggl( \sum_{i=1}^n \| x_i\| ^q \biggr)^{1/q} \leq C\cdot {\text{Average}}_{\varepsilon_i=\pm 1} \Big\| \sum_{i=1}^n \varepsilon_i x_i \Big\| . \] A Banach space is \(B\)-convex if it does not contain the spaces \(l_1^n\) uniformly. It is \(K\)-convex if the Rademacher projection in \(X\) is bounded.

The theorem of J. L. Krivine [Ann. Math. (2) 104, 1–29 (1976; Zbl 0329.46008)] investigates the numbers \(p\in [1,\infty]\) for which a given infinite dimensional Banach space \(X\) contains \((1+\varepsilon)\)-isomorphic copies of \(l_p^n\) for all \(n\) and \(\varepsilon>0\). Dvoretzky’s theorem says that \(p=2\) is always a possible value. The Maurey-Pisier theorem [B. Maurey and G. Pisier, Stud. Math. 58, 45–90 (1976; Zbl 0344.47014)] gives further quantitative information on those \(p\) in relation to type and cotype. It says that \[ \min \{ p\in [1,2] : X \;\text{contains}\;l_p^n \;\text{uniformly} \} = \sup \{ p\in [1,2] : X \;\text{has type} \;p\}, \] \[ \max \{ q\in [2,\infty] : X \;\text{contains}\;l_q^n \;\text{uniformly} \} = \inf \{ q\in [2,\infty] : X \;\text{has cotype}\;q\}. \] G. Pisier’s theorem [Ann. Math. (2) 115, 375–392 (1982; Zbl 0487.46008)] says that \(X\) is \(K\)-convex if and only if \(X\) does not contain \(l_1^n\)-s uniformly. It illuminates the duality between type and cotype. If \(X\) is \(K\)-convex and \(1/p+1/q=1\), then \(X^*\) has cotype \(q\) if and only if \(X\) has type \(p\).

For the entire collection see [Zbl 1013.46001].

Reviewer: Aicke Hinrichs (Jena)

##### MSC:

46B07 | Local theory of Banach spaces |

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B08 | Ultraproduct techniques in Banach space theory |

46B20 | Geometry and structure of normed linear spaces |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |