Grothendieck inequality

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In mathematics, the Grothendieck inequality relates

\max_{-1 \leq s_i \leq 1, -1 \leq t_j \leq 1 } \left| \sum_{i,j} a_{ij} s_i t_j \right|

to

\max_{S_i,T_j \in B(H)} \left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|,

where B(H) is the unit ball of a Hilbert space H. The best constant k(H) in

\max_{S_i,T_j \in B(H)} \left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right| \leq k(H) \max_{-1 \leq s_i \leq 1, -1 \leq t_j \leq 1 } \left| \sum_{i,j} a_{ij} s_i t_j \right|, \quad a_{i,j} \in \mathbb{R}

is called the Grothendieck constant of the Hilbert space H.

Alexander Grothendieck showed that k(H) is bounded by a universal constant, independent of H; define

k = \sup_H k(H).

Grothendieck himself proved that

1.57 \leq k \leq 2.3.

Later, Krivine showed that

1.67696\dots\leq k \leq 1.7822139781\dots;

in spite of later efforts, the precise value of k is still unknown.

[edit] References

  • A.Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques (French), Bol. Soc. Mat. São Paulo 8 1953 1--79
  • J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les spheres., Adv. Math. 31, 16-30, 1979.

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