Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant k with the following property. If a_i,j is an n by n (real or complex) matrix with

$\left| \sum_{i,j} a_{ij} s_i t_j \right|\le 1$

for all (real or complex) numbers s_i, t_j of absolute value at most 1, then

$\left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|\le k$,

for all vectors S_i, T_j in the unit ball B(H) of a (real or complex) Hilbert space H. The smallest constant k which satisfies this property for all n by n matrices is called a Grothendieck constant and denoted k(n); in fact there are two Grothendieck constants k_R(n) and k_C(n) for each n depending on whether one works with real or complex numbers, respectively.

The sequences k_R(n) and k_C(n) are easily seen to be increasing, and Alexander Grothendieck's result states that they are bounded, so they have limits.

If we definek_R to be sup_nk_R(n) then Grothendieck proved that: $1.57 \approx \frac{\pi}{2} \leq k_{\R} \leq \mathrm{sinh}(\frac{\pi}{2}) \approx 2.3$.

Later Krivine improved the result by proving: 1.67696... ≤ k_R ≤ 1.7822139781...=$\frac{\pi}{2 \ln(1+\sqrt{2})}$, conjecturing that the upper bound is tight. However, this conjecture was disproved in a preprint by Braverman, Makarychev, Makarychev and Naor.

External links

• Weisstein, Eric W., "Grothendieck's Constant", MathWorld. (NB: the historical part is not exact there.)
• Pisier, Gilles (April 2012). "Grothendieck's Theorem, Past and Present" (PDF). Bulletin of the American Mathematical Society 49 (2): 237–323.