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 Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the inequality and the existence of the constants in a paper published in 1953.

Bounds on the constants

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

With k_R defined to be sup_n k_R(n) then Grothendieck proved that: $1.57 \approx \frac{\pi}{2} \leq k_{\R} \leq \mathrm{sinh}(\frac{\pi}{2}) \approx 2.3$.

Krivine (1979) 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 by Braverman et al. (2011).

See also

• Pisier–Ringrose inequality

External links

• Weisstein, Eric W., "Grothendieck's Constant", MathWorld. (NB: the historical part is not exact there.)