Grothendieck inequality

In mathematics, the Grothendieck inequality relates :$max\_\{-1\; leq\; s\_i\; leq\; 1,\; -1\; leq\; t\_j\; leq\; 1\; \}\; left|\; sum\_\{i,j\}\; a\_\{ij\}\; s\_i\; t\_j\; ight|$ to :$max\_\{S\_i,T\_j\; in\; B(H)\}\; left|\; sum\_\{i,j\}\; a\_\{ij\}\; langle\; S\_i\; ,\; T\_j\; angle\; ight|$,where $B(H)$ is the unit ball of a Hilbert space $H$. The best constant $k(H)$ in:$max\_\{S\_i,T\_j\; in\; B(H)\}\; left|\; sum\_\{i,j\}\; a\_\{ij\}\; langle\; S\_i\; ,\; T\_j\; angle\; ight|\; leq\; k(H)\; max\_\{-1\; leq\; s\_i\; leq\; 1,\; -1\; leq\; t\_j\; leq\; 1\; \}\; left|\; sum\_\{i,j\}\; a\_\{ij\}\; s\_i\; t\_j\; ight|,\; quad\; a\_\{i,j\}\; in\; mathbb\{R\}$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

:$1.67696dotsleq\; k\; leq\; 1.7822139781dots;$

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

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.

External links

*MathWorld|urlname=GrothendiecksConstant|title=Grothendieck's Constant (NB: the historical part is not exact there)